| author | Chengsong |
| Tue, 19 Apr 2022 09:08:01 +0100 | |
| changeset 492 | 61eff2abb0b6 |
| parent 491 | 48ce16d61e03 |
| child 493 | 1481f465e6ea |
--- a/thys2/ClosedForms.thy Sat Apr 16 09:52:57 2022 +0100 +++ b/thys2/ClosedForms.thy Tue Apr 19 09:08:01 2022 +0100 @@ -25,62 +25,62 @@ inductive - hrewrite:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ \<leadsto> _" [99, 99] 99) + hrewrite:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto> _" [99, 99] 99) where - "RSEQ RZERO r2 \<leadsto> RZERO" -| "RSEQ r1 RZERO \<leadsto> RZERO" -| "RSEQ RONE r \<leadsto> r" -| "r1 \<leadsto> r2 \<Longrightarrow> RSEQ r1 r3 \<leadsto> RSEQ r2 r3" -| "r3 \<leadsto> r4 \<Longrightarrow> RSEQ r1 r3 \<leadsto> RSEQ r1 r4" -| "r \<leadsto> r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) \<leadsto> (RALTS (rs1 @ [r'] @ rs2))" + "RSEQ RZERO r2 h\<leadsto> RZERO" +| "RSEQ r1 RZERO h\<leadsto> RZERO" +| "RSEQ RONE r h\<leadsto> r" +| "r1 h\<leadsto> r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r2 r3" +| "r3 h\<leadsto> r4 \<Longrightarrow> RSEQ r1 r3 h\<leadsto> RSEQ r1 r4" +| "r h\<leadsto> r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto> (RALTS (rs1 @ [r'] @ rs2))" (*context rule for eliminating 0, alts--corresponds to the recursive call flts r::rs = r::(flts rs)*) -| "RALTS (rsa @ [RZERO] @ rsb) \<leadsto> RALTS (rsa @ rsb)" -| "RALTS (rsa @ [RALTS rs1] @ rsb) \<leadsto> RALTS (rsa @ rs1 @ rsb)" -| "RALTS [] \<leadsto> RZERO" -| "RALTS [r] \<leadsto> r" -| "a1 = a2 \<Longrightarrow> RALTS (rsa@[a1]@rsb@[a2]@rsc) \<leadsto> RALTS (rsa @ [a1] @ rsb @ rsc)" +| "RALTS (rsa @ [RZERO] @ rsb) h\<leadsto> RALTS (rsa @ rsb)" +| "RALTS (rsa @ [RALTS rs1] @ rsb) h\<leadsto> RALTS (rsa @ rs1 @ rsb)" +| "RALTS [] h\<leadsto> RZERO" +| "RALTS [r] h\<leadsto> r" +| "a1 = a2 \<Longrightarrow> RALTS (rsa@[a1]@rsb@[a2]@rsc) h\<leadsto> RALTS (rsa @ [a1] @ rsb @ rsc)" inductive - hrewrites:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ \<leadsto>* _" [100, 100] 100) + hrewrites:: "rrexp \<Rightarrow> rrexp \<Rightarrow> bool" ("_ h\<leadsto>* _" [100, 100] 100) where - rs1[intro, simp]:"r \<leadsto>* r" -| rs2[intro]: "\<lbrakk>r1 \<leadsto>* r2; r2 \<leadsto> r3\<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3" - - -lemma hr_in_rstar : "r1 \<leadsto> r2 \<Longrightarrow> r1 \<leadsto>* r2" + rs1[intro, simp]:"r h\<leadsto>* r" +| rs2[intro]: "\<lbrakk>r1 h\<leadsto>* r2; r2 h\<leadsto> r3\<rbrakk> \<Longrightarrow> r1 h\<leadsto>* r3" + + +lemma hr_in_rstar : "r1 h\<leadsto> r2 \<Longrightarrow> r1 h\<leadsto>* r2" using hrewrites.intros(1) hrewrites.intros(2) by blast lemma hreal_trans[trans]: - assumes a1: "r1 \<leadsto>* r2" and a2: "r2 \<leadsto>* r3" - shows "r1 \<leadsto>* r3" + assumes a1: "r1 h\<leadsto>* r2" and a2: "r2 h\<leadsto>* r3" + shows "r1 h\<leadsto>* r3" using a2 a1 apply(induct r2 r3 arbitrary: r1 rule: hrewrites.induct) apply(auto) done -lemma hmany_steps_later: "\<lbrakk>r1 \<leadsto> r2; r2 \<leadsto>* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>* r3" +lemma hmany_steps_later: "\<lbrakk>r1 h\<leadsto> r2; r2 h\<leadsto>* r3 \<rbrakk> \<Longrightarrow> r1 h\<leadsto>* r3" by (meson hr_in_rstar hreal_trans) lemma hrewrites_seq_context: - shows "r1 \<leadsto>* r2 \<Longrightarrow> RSEQ r1 r3 \<leadsto>* RSEQ r2 r3" + shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r3" apply(induct r1 r2 rule: hrewrites.induct) apply simp using hrewrite.intros(4) by blast lemma hrewrites_seq_context2: - shows "r1 \<leadsto>* r2 \<Longrightarrow> RSEQ r0 r1 \<leadsto>* RSEQ r0 r2" + shows "r1 h\<leadsto>* r2 \<Longrightarrow> RSEQ r0 r1 h\<leadsto>* RSEQ r0 r2" apply(induct r1 r2 rule: hrewrites.induct) apply simp using hrewrite.intros(5) by blast lemma hrewrites_seq_context0: - shows "r1 \<leadsto>* RZERO \<Longrightarrow> RSEQ r1 r3 \<leadsto>* RZERO" - apply(subgoal_tac "RSEQ r1 r3 \<leadsto>* RSEQ RZERO r3") + shows "r1 h\<leadsto>* RZERO \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RZERO" + apply(subgoal_tac "RSEQ r1 r3 h\<leadsto>* RSEQ RZERO r3") using hrewrite.intros(1) apply blast by (simp add: hrewrites_seq_context) lemma hrewrites_seq_contexts: - shows "\<lbrakk>r1 \<leadsto>* r2; r3 \<leadsto>* r4\<rbrakk> \<Longrightarrow> RSEQ r1 r3 \<leadsto>* RSEQ r2 r4" + shows "\<lbrakk>r1 h\<leadsto>* r2; r3 h\<leadsto>* r4\<rbrakk> \<Longrightarrow> RSEQ r1 r3 h\<leadsto>* RSEQ r2 r4" by (meson hreal_trans hrewrites_seq_context hrewrites_seq_context2) @@ -1215,7 +1215,7 @@ -(*a more refined notion of \<leadsto>* is needed, +(*a more refined notion of h\<leadsto>* is needed, this lemma fails when rs1 contains some RALTS rs where elements of rs appear in later parts of rs1, which will be picked up by rs2 and deduplicated*) @@ -1546,11 +1546,11 @@ lemma grewrite_ralts: - shows "rs \<leadsto>g rs' \<Longrightarrow> RALTS rs \<leadsto>* RALTS rs'" + shows "rs \<leadsto>g rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'" by (smt (verit) grewrite_cases_middle hr_in_rstar hrewrite.intros(11) hrewrite.intros(7) hrewrite.intros(8)) lemma grewrites_ralts: - shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs \<leadsto>* RALTS rs'" + shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* RALTS rs'" apply(induct rule: grewrites.induct) apply simp using grewrite_ralts hreal_trans by blast @@ -1558,8 +1558,8 @@ lemma distinct_grewrites_subgoal1: shows " - \<lbrakk>rs1 \<leadsto>g* [a]; RALTS rs1 \<leadsto>* a; [a] \<leadsto>g rs3\<rbrakk> \<Longrightarrow> RALTS rs1 \<leadsto>* rsimp_ALTs rs3" - apply(subgoal_tac "RALTS rs1 \<leadsto>* RALTS rs3") + \<lbrakk>rs1 \<leadsto>g* [a]; RALTS rs1 h\<leadsto>* a; [a] \<leadsto>g rs3\<rbrakk> \<Longrightarrow> RALTS rs1 h\<leadsto>* rsimp_ALTs rs3" + apply(subgoal_tac "RALTS rs1 h\<leadsto>* RALTS rs3") apply (metis hrewrite.intros(10) hrewrite.intros(9) rs2 rsimp_ALTs.cases rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3)) apply(subgoal_tac "rs1 \<leadsto>g* rs3") using grewrites_ralts apply blast @@ -1571,7 +1571,7 @@ lemma grewrites_ralts_rsimpalts: - shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs \<leadsto>* rsimp_ALTs rs' " + shows "rs \<leadsto>g* rs' \<Longrightarrow> RALTS rs h\<leadsto>* rsimp_ALTs rs' " apply(induct rs rs' rule: grewrites.induct) apply(case_tac rs) using hrewrite.intros(9) apply force @@ -1592,7 +1592,7 @@ by (metis (no_types, opaque_lifting) grewrite_ralts hr_in_rstar hreal_trans hrewrite.intros(10) neq_Nil_conv rsimp_ALTs.simps(2) rsimp_ALTs.simps(3)) lemma hrewrites_alts: - shows " r \<leadsto>* r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) \<leadsto>* (RALTS (rs1 @ [r'] @ rs2))" + shows " r h\<leadsto>* r' \<Longrightarrow> (RALTS (rs1 @ [r] @ rs2)) h\<leadsto>* (RALTS (rs1 @ [r'] @ rs2))" apply(induct r r' rule: hrewrites.induct) apply simp using hrewrite.intros(6) by blast @@ -1601,17 +1601,17 @@ srewritescf:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" (" _ scf\<leadsto>* _" [100, 100] 100) where ss1: "[] scf\<leadsto>* []" -| ss2: "\<lbrakk>r \<leadsto>* r'; rs scf\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) scf\<leadsto>* (r'#rs')" +| ss2: "\<lbrakk>r h\<leadsto>* r'; rs scf\<leadsto>* rs'\<rbrakk> \<Longrightarrow> (r#rs) scf\<leadsto>* (r'#rs')" lemma hrewrites_alts_cons: - shows " RALTS rs \<leadsto>* RALTS rs' \<Longrightarrow> RALTS (r # rs) \<leadsto>* RALTS (r # rs')" + shows " RALTS rs h\<leadsto>* RALTS rs' \<Longrightarrow> RALTS (r # rs) h\<leadsto>* RALTS (r # rs')" oops -lemma srewritescf_alt: "rs1 scf\<leadsto>* rs2 \<Longrightarrow> (RALTS (rs@rs1)) \<leadsto>* (RALTS (rs@rs2))" +lemma srewritescf_alt: "rs1 scf\<leadsto>* rs2 \<Longrightarrow> (RALTS (rs@rs1)) h\<leadsto>* (RALTS (rs@rs2))" apply(induct rs1 rs2 arbitrary: rs rule: srewritescf.induct) apply(rule rs1) @@ -1626,7 +1626,7 @@ corollary srewritescf_alt1: assumes "rs1 scf\<leadsto>* rs2" - shows "RALTS rs1 \<leadsto>* RALTS rs2" + shows "RALTS rs1 h\<leadsto>* RALTS rs2" using assms by (metis append_Nil srewritescf_alt) @@ -1634,7 +1634,7 @@ lemma trivialrsimp_srewrites: - "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x \<leadsto>* f x \<rbrakk> \<Longrightarrow> rs scf\<leadsto>* (map f rs)" + "\<lbrakk>\<And>x. x \<in> set rs \<Longrightarrow> x h\<leadsto>* f x \<rbrakk> \<Longrightarrow> rs scf\<leadsto>* (map f rs)" apply(induction rs) apply simp @@ -1643,15 +1643,15 @@ lemma hrewrites_list: shows -" (\<And>xa. xa \<in> set x \<Longrightarrow> xa \<leadsto>* rsimp xa) \<Longrightarrow> RALTS x \<leadsto>* RALTS (map rsimp x)" +" (\<And>xa. xa \<in> set x \<Longrightarrow> xa h\<leadsto>* rsimp xa) \<Longrightarrow> RALTS x h\<leadsto>* RALTS (map rsimp x)" apply(induct x) apply(simp)+ by (simp add: srewritescf_alt1 ss2 trivialrsimp_srewrites) -(* apply(subgoal_tac "RALTS x \<leadsto>* RALTS (map rsimp x)")*) +(* apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)")*) lemma hrewrite_simpeq: - shows "r1 \<leadsto> r2 \<Longrightarrow> rsimp r1 = rsimp r2" + shows "r1 h\<leadsto> r2 \<Longrightarrow> rsimp r1 = rsimp r2" apply(induct rule: hrewrite.induct) apply simp+ apply (simp add: basic_rsimp_SEQ_property3) @@ -1668,7 +1668,7 @@ using grewrite.intros(4) grewrite_equal_rsimp by presburger lemma hrewrites_simpeq: - shows "r1 \<leadsto>* r2 \<Longrightarrow> rsimp r1 = rsimp r2" + shows "r1 h\<leadsto>* r2 \<Longrightarrow> rsimp r1 = rsimp r2" apply(induct rule: hrewrites.induct) apply simp apply(subgoal_tac "rsimp r2 = rsimp r3") @@ -1678,13 +1678,13 @@ lemma simp_hrewrites: - shows "r1 \<leadsto>* rsimp r1" + shows "r1 h\<leadsto>* rsimp r1" apply(induct r1) apply simp+ apply(case_tac "rsimp r11 = RONE") apply simp apply(subst basic_rsimp_SEQ_property1) - apply(subgoal_tac "RSEQ r11 r12 \<leadsto>* RSEQ RONE r12") + apply(subgoal_tac "RSEQ r11 r12 h\<leadsto>* RSEQ RONE r12") using hreal_trans hrewrite.intros(3) apply blast using hrewrites_seq_context apply presburger apply(case_tac "rsimp r11 = RZERO") @@ -1698,8 +1698,8 @@ apply simp+ using hrewrites_seq_contexts apply presburger apply simp - apply(subgoal_tac "RALTS x \<leadsto>* RALTS (map rsimp x)") - apply(subgoal_tac "RALTS (map rsimp x) \<leadsto>* rsimp_ALTs (rdistinct (rflts (map rsimp x)) {}) ") + apply(subgoal_tac "RALTS x h\<leadsto>* RALTS (map rsimp x)") + apply(subgoal_tac "RALTS (map rsimp x) h\<leadsto>* rsimp_ALTs (rdistinct (rflts (map rsimp x)) {}) ") using hreal_trans apply blast apply (meson flts_gstar greal_trans grewrites_ralts_rsimpalts gstar_rdistinct) @@ -1707,9 +1707,9 @@ by simp lemma interleave_aux1: - shows " RALT (RSEQ RZERO r1) r \<leadsto>* r" - apply(subgoal_tac "RSEQ RZERO r1 \<leadsto>* RZERO") - apply(subgoal_tac "RALT (RSEQ RZERO r1) r \<leadsto>* RALT RZERO r") + shows " RALT (RSEQ RZERO r1) r h\<leadsto>* r" + apply(subgoal_tac "RSEQ RZERO r1 h\<leadsto>* RZERO") + apply(subgoal_tac "RALT (RSEQ RZERO r1) r h\<leadsto>* RALT RZERO r") apply (meson grewrite.intros(1) grewrite_ralts hreal_trans hrewrite.intros(10) hrewrites.simps) using rs1 srewritescf_alt1 ss1 ss2 apply presburger by (simp add: hr_in_rstar hrewrite.intros(1)) @@ -1717,7 +1717,7 @@ lemma rnullable_hrewrite: - shows "r1 \<leadsto> r2 \<Longrightarrow> rnullable r1 = rnullable r2" + shows "r1 h\<leadsto> r2 \<Longrightarrow> rnullable r1 = rnullable r2" apply(induct rule: hrewrite.induct) apply simp+ apply blast @@ -1726,7 +1726,7 @@ lemma interleave1: - shows "r \<leadsto> r' \<Longrightarrow> rder c r \<leadsto>* rder c r'" + shows "r h\<leadsto> r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'" apply(induct r r' rule: hrewrite.induct) apply (simp add: hr_in_rstar hrewrite.intros(1)) apply (metis (no_types, lifting) basic_rsimp_SEQ_property3 list.simps(8) list.simps(9) rder.simps(1) rder.simps(5) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) simp_hrewrites) @@ -1758,7 +1758,7 @@ using hrewrite.intros(11) by auto lemma interleave_star1: - shows "r \<leadsto>* r' \<Longrightarrow> rder c r \<leadsto>* rder c r'" + shows "r h\<leadsto>* r' \<Longrightarrow> rder c r h\<leadsto>* rder c r'" apply(induct rule : hrewrites.induct) apply simp by (meson hreal_trans interleave1) @@ -1775,7 +1775,7 @@ using inside_simp_seq_nullable apply blast apply simp apply (smt (verit, del_insts) basic_rsimp_SEQ_property2 basic_rsimp_SEQ_property3 der_simp_nullability rder.simps(1) rder.simps(5) rnullable.simps(2) rsimp.simps(1) rsimp_SEQ.simps(1) rsimp_idem) - apply(subgoal_tac "rder x (RALTS xa) \<leadsto>* rder x (rsimp (RALTS xa))") + apply(subgoal_tac "rder x (RALTS xa) h\<leadsto>* rder x (rsimp (RALTS xa))") using hrewrites_simpeq apply presburger using interleave_star1 simp_hrewrites apply presburger by simp
--- a/thys2/ClosedFormsBounds.thy Sat Apr 16 09:52:57 2022 +0100 +++ b/thys2/ClosedFormsBounds.thy Tue Apr 19 09:08:01 2022 +0100 @@ -752,6 +752,7 @@ lemma rders_simp_bounded: + fixes "r" shows "\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N" apply(induct r) apply(rule_tac x = "Suc 0 " in exI) @@ -767,9 +768,11 @@ apply (metis alts_closed_form_bounded size_list_estimation') using star_closed_form_bounded by blast +corollary rders_simp_finiteness: + shows "\<forall>r. \<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N" + using rders_simp_bounded by auto -unused_thms end
--- a/thys2/SizeBound3.thy Sat Apr 16 09:52:57 2022 +0100 +++ b/thys2/SizeBound3.thy Tue Apr 19 09:08:01 2022 +0100 @@ -1,6 +1,6 @@ theory SizeBound3 - imports "Lexer" + imports "ClosedFormsBounds" begin section \<open>Bit-Encodings\<close> @@ -1160,17 +1160,1025 @@ using blexer_correctness main_blexer_simp by simp +fun + rerase :: "arexp \<Rightarrow> rrexp" +where + "rerase AZERO = RZERO" +| "rerase (AONE _) = RONE" +| "rerase (ACHAR _ c) = RCHAR c" +| "rerase (AALTs bs rs) = RALTS (map rerase rs)" +| "rerase (ASEQ _ r1 r2) = RSEQ (rerase r1) (rerase r2)" +| "rerase (ASTAR _ r) = RSTAR (rerase r)" -export_code blexer_simp blexer lexer bders bders_simp in Scala module_name VerifiedLexers + + +lemma asize_rsize: + shows "rsize (rerase r) = asize r" + apply(induct r) + apply simp+ + + apply (metis (mono_tags, lifting) comp_apply map_eq_conv) + by simp + +lemma rb_nullability: + shows " rnullable (rerase a) = bnullable a" + apply(induct a) + apply simp+ + done + +lemma fuse_anything_doesnt_matter: + shows "rerase a = rerase (fuse bs a)" + by (smt (verit) fuse.elims rerase.simps(2) rerase.simps(3) rerase.simps(4) rerase.simps(5) rerase.simps(6)) + + +lemma rerase_preimage: + shows "rerase r = RZERO \<Longrightarrow> r = AZERO" + apply(case_tac r) + apply simp+ + done + +lemma rerase_preimage2: + shows "rerase r = RONE \<Longrightarrow> \<exists>bs. r = AONE bs" + apply(case_tac r) + apply simp+ + done + +lemma rerase_preimage3: + shows "rerase r= RCHAR c \<Longrightarrow> \<exists>bs. r = ACHAR bs c" + apply(case_tac r) + apply simp+ + done + +lemma rerase_preimage4: + shows "rerase r = RSEQ r1 r2 \<Longrightarrow> \<exists>bs a1 a2. r = ASEQ bs a1 a2 \<and> rerase a1 = r1 \<and> rerase a2 = r2" + apply(case_tac r) + apply(simp)+ + done + +lemma rerase_preimage5: + shows "rerase r = RALTS rs \<Longrightarrow> \<exists>bs as. r = AALTs bs as \<and> map rerase as = rs" + apply(case_tac r) + apply(simp)+ + done + +lemma rerase_preimage6: + shows "rerase r = RSTAR r0 \<Longrightarrow> \<exists>bs a0. r = ASTAR bs a0 \<and> rerase a0 = r0 " + apply(case_tac r) + apply(simp)+ + done + +lemma map_rder_bder: + shows "\<lbrakk> \<And>xa a. \<lbrakk>xa \<in> set x; rerase a = xa\<rbrakk> \<Longrightarrow> rerase (bder c a) = rder c xa; + map rerase as = x\<rbrakk> \<Longrightarrow> + map rerase (map (bder c) as) = map (rder c) x" + apply(induct x arbitrary: as) + apply simp+ + by force -unused_thms +lemma der_rerase: + shows "rerase a = r \<Longrightarrow> rerase (bder c a) = rder c r" + apply(induct r arbitrary: a) + apply simp + using rerase_preimage apply fastforce + using rerase_preimage2 apply force + apply (metis bder.simps(3) rder.simps(3) rerase.simps(1) rerase.simps(2) rerase_preimage3) + apply(insert rerase_preimage4)[1] + apply(subgoal_tac " \<exists>bs a1 a2. a = ASEQ bs a1 a2 \<and> rerase a1 = r1 \<and> rerase a2 = r2") + prefer 2 + apply presburger + apply(erule exE)+ + apply(erule conjE)+ + apply(subgoal_tac " rerase (bder c a1) = rder c r1") + prefer 2 + apply blast + apply(subgoal_tac " rerase (bder c a2) = rder c r2") + prefer 2 + apply blast + apply(case_tac "rnullable r1") + apply(subgoal_tac "bnullable a1") + apply simp + using fuse_anything_doesnt_matter apply presburger + using rb_nullability apply blast + apply (metis bder.simps(5) rb_nullability rder.simps(5) rerase.simps(5)) + apply simp + apply(insert rerase_preimage5)[1] + apply(subgoal_tac "\<exists>bs as. a = AALTs bs as \<and> map rerase as = x") + prefer 2 + + apply blast + apply(erule exE)+ + apply(erule conjE)+ + apply(subgoal_tac "map rerase (map (bder c) as) = map (rder c) x") + using bder.simps(4) rerase.simps(4) apply presburger + using map_rder_bder apply blast + using fuse_anything_doesnt_matter rerase_preimage6 by force + +lemma der_rerase2: + shows "rerase (bder c a) = rder c (rerase a)" + using der_rerase by force + +lemma ders_rerase: + shows "rerase (bders a s) = rders (rerase a) s" + apply(induct s rule: rev_induct) + apply simp + by (simp add: bders_append der_rerase rders_append) + +lemma rerase_bsimp_ASEQ: + shows "rerase (bsimp_ASEQ x1 a1 a2) = rsimp_SEQ (rerase a1) (rerase a2)" + by (smt (verit, ccfv_SIG) SizeBound3.bsimp_ASEQ0 SizeBound3.bsimp_ASEQ2 basic_rsimp_SEQ_property1 basic_rsimp_SEQ_property2 basic_rsimp_SEQ_property3 bsimp_ASEQ.simps(1) bsimp_ASEQ1 fuse_anything_doesnt_matter rerase.simps(1) rerase.simps(2) rerase.simps(5) rerase_preimage rerase_preimage2 rsimp_SEQ.simps(1)) + +lemma rerase_bsimp_AALTs: + shows "rerase (bsimp_AALTs bs rs) = rsimp_ALTs (map rerase rs)" + apply(induct rs arbitrary: bs) + apply simp+ + by (smt (verit, ccfv_threshold) Cons_eq_map_conv bsimp_AALTs.elims fuse_anything_doesnt_matter list.discI list.inject list.simps(9) rerase.simps(4) rsimp_ALTs.elims) + + +fun anonalt :: "arexp \<Rightarrow> bool" + where + "anonalt (AALTs bs2 rs) = False" +| "anonalt r = True" + + +fun agood :: "arexp \<Rightarrow> bool" where + "agood AZERO = False" +| "agood (AONE cs) = True" +| "agood (ACHAR cs c) = True" +| "agood (AALTs cs []) = False" +| "agood (AALTs cs [r]) = False" +| "agood (AALTs cs (r1#r2#rs)) = (distinct (map erase (r1 # r2 # rs)) \<and>(\<forall>r' \<in> set (r1#r2#rs). agood r' \<and> anonalt r'))" +| "agood (ASEQ _ AZERO _) = False" +| "agood (ASEQ _ (AONE _) _) = False" +| "agood (ASEQ _ _ AZERO) = False" +| "agood (ASEQ cs r1 r2) = (agood r1 \<and> agood r2)" +| "agood (ASTAR cs r) = True" + + +fun anonnested :: "arexp \<Rightarrow> bool" + where + "anonnested (AALTs bs2 []) = True" +| "anonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False" +| "anonnested (AALTs bs2 (r # rs2)) = anonnested (AALTs bs2 rs2)" +| "anonnested r = True" -inductive aggressive:: "arexp \<Rightarrow> arexp \<Rightarrow> bool" ("_ \<leadsto>? _" [99, 99] 99) - where - "ASEQ bs (AALTs bs1 rs) r \<leadsto>? AALTs (bs@bs1) (map (\<lambda>r'. ASEQ [] r' r) rs) " +lemma k0: + shows "flts (r # rs1) = flts [r] @ flts rs1" + apply(induct r arbitrary: rs1) + apply(auto) + done + +lemma k00: + shows "flts (rs1 @ rs2) = flts rs1 @ flts rs2" + apply(induct rs1 arbitrary: rs2) + apply(auto) + by (metis append.assoc k0) + + +lemma bbbbs: + assumes "agood r" "r = AALTs bs1 rs" + shows "bsimp_AALTs bs (flts [r]) = AALTs bs (map (fuse bs1) rs)" + using assms + by (smt (verit, ccfv_SIG) Cons_eq_map_conv agood.simps(4) agood.simps(5) append.right_neutral bsimp_AALTs.elims flts.simps(1) flts.simps(3) list.map_disc_iff) + +lemma bbbbs1: + shows "anonalt r \<or> (\<exists>bs rs. r = AALTs bs rs)" + by (meson anonalt.elims(3)) + + + +lemma good_fuse: + shows "agood (fuse bs r) = agood r" + apply(induct r arbitrary: bs) + apply(auto) + apply(case_tac r1) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r1) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac x2a) + apply(simp_all) + apply(case_tac list) + apply(simp_all) + apply(case_tac x2a) + apply(simp_all) + apply(case_tac list) + apply(simp_all) + done + +lemma good0: + assumes "rs \<noteq> Nil" "\<forall>r \<in> set rs. anonalt r" "distinct (map erase rs)" + shows "agood (bsimp_AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. agood r)" + using assms + apply(induct bs rs rule: bsimp_AALTs.induct) + apply simp+ + apply (simp add: good_fuse) + apply(subgoal_tac "bsimp_AALTs bs1 (v # vb # vc) = AALTs bs1 (v # vb # vc)") + prefer 2 + + + using bsimp_AALTs.simps(3) apply presburger + apply(simp only:) + apply(subgoal_tac "agood (AALTs bs1 (v # vb # vc)) = ((\<forall>r \<in> (set (v # vb # vc)). agood r \<and> anonalt r) \<and> distinct (map erase (v # vb # vc)))") + prefer 2 + using agood.simps(6) apply blast + apply(simp only:) + apply(subgoal_tac "((\<forall>r\<in>set (v # vb # vc). agood r \<and> anonalt r) \<and> distinct (map erase (v # vb # vc))) = ((\<forall>r\<in>set (v # vb # vc). agood r) \<and> distinct (map erase (v # vb # vc)))") + prefer 2 + apply blast + apply(simp only:) + apply(subgoal_tac "((\<forall>r \<in> set (v # vb # vc). agood r) \<and> distinct (map erase (v # vb # vc))) = (\<forall> r \<in> set (v # vb # vc). agood r) ") + prefer 2 + apply(subgoal_tac "distinct (map erase (v # vb # vc)) = True") + prefer 2 + apply blast + prefer 2 + apply force + apply simp + done + + +lemma nn11a: + assumes "anonalt r" + shows "anonalt (fuse bs r)" + using assms + apply(induct r) + apply(auto) + done + +lemma flts0: + assumes "r \<noteq> AZERO" "anonalt r" + shows "flts [r] \<noteq> []" + using assms + apply(induct r) + apply(simp_all) + done + +lemma flts1: + assumes "agood r" + shows "flts [r] \<noteq> []" + using assms + apply(induct r) + apply(simp_all) + apply(case_tac x2a) + apply(simp) + apply(simp) + done + +lemma flts2: + assumes "agood r" + shows "\<forall>r' \<in> set (flts [r]). agood r' \<and> anonalt r'" + using assms + apply(induct r) + apply(simp) + apply(simp) + apply(simp) + prefer 2 + apply(simp) + apply(auto)[1] + apply (metis bsimp_AALTs.elims agood.simps(4) agood.simps(5) agood.simps(6) good_fuse) + apply (metis bsimp_AALTs.elims agood.simps(4) agood.simps(5) agood.simps(6) nn11a) + apply fastforce + apply(simp) + done + + +lemma flts3: + assumes "\<forall>r \<in> set rs. agood r \<or> r = AZERO" + shows "\<forall>r \<in> set (flts rs). agood r" + using assms + apply(induct rs arbitrary: rule: flts.induct) + apply(simp_all) + by (metis SizeBound3.flts2 UnE append.right_neutral flts.simps(1) flts.simps(3) list.set_map) + + +lemma flts3b: + assumes "\<exists>r\<in>set rs. agood r" + shows "flts rs \<noteq> []" + using assms + apply(induct rs arbitrary: rule: flts.induct) + apply(simp) + apply(simp) + apply(simp) + apply(auto) + done + +lemma k0a: + shows "flts [AALTs bs rs] = map (fuse bs) rs" + apply(simp) + done + + +lemma k0b: + assumes "anonalt r" "r \<noteq> AZERO" + shows "flts [r] = [r]" + using assms + apply(case_tac r) + apply(simp_all) + done + +lemma flts4_nogood: + shows "bsimp_AALTs bs0 rs = AZERO \<Longrightarrow> \<forall>r \<in> set rs. \<not> agood r" + by (metis SizeBound3.flts3b arexp.distinct(7) bsimp_AALTs.elims flts.simps(1) flts.simps(2) fuse_anything_doesnt_matter rerase.simps(1) rerase_preimage) + +lemma flts4_append: + shows "bsimp_AALTs bs0 (rs @ flts rsa) = AZERO \<Longrightarrow> bsimp_AALTs bs0 rs = AZERO" + by (metis append_is_Nil_conv arexp.simps(13) bsimp_AALTs.elims bsimp_AALTs.simps(1) butlast_append butlast_snoc) + +lemma flts4: + assumes "bsimp_AALTs bs (flts rs) = AZERO" + shows "\<forall>r \<in> set rs. \<not> agood r" + using assms + apply(induct rs arbitrary: bs rule: flts.induct) + apply(auto) + defer + apply (metis (no_types, lifting) Nil_is_append_conv append_self_conv2 bsimp_AALTs.elims butlast.simps(2) butlast_append flts3b anonalt.simps(1) anonalt.simps(2)) + apply (metis agood.simps(1) agood.simps(2) arexp.distinct(7) bsimp_AALTs.elims good_fuse list.distinct(1) list.inject) + apply (metis agood.simps(1) agood.simps(3) arexp.distinct(7) bsimp_AALTs.elims good_fuse list.distinct(1) list.inject) + apply (metis agood.simps(1) arexp.distinct(7) bsimp_AALTs.elims good_fuse list.discI list.inject) + apply (metis arexp.distinct(7) bsimp_AALTs.elims list.distinct(1) list.inject) + apply (metis agood.simps(1) agood.simps(33) arexp.distinct(7) bsimp_AALTs.elims good_fuse list.discI list.inject) + by (metis SizeBound3.bbbbs SizeBound3.k0a arexp.simps(13) flts4_append) + + +lemma flts_nil: + assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> + agood (bsimp y) \<or> bsimp y = AZERO" + and "\<forall>r\<in>set rs. \<not> agood (bsimp r)" + shows "flts (map bsimp rs) = []" + using assms + apply(induct rs) + apply(simp) + apply(simp) + apply(subst k0) + apply(simp) + by force + +lemma flts_nil2: + assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> + agood (bsimp y) \<or> bsimp y = AZERO" + and "bsimp_AALTs bs (flts (map bsimp rs)) = AZERO" + shows "flts (map bsimp rs) = []" + using assms + apply(induct rs arbitrary: bs) + apply(simp) + apply(simp) + apply(subst k0) + apply(simp) + apply(subst (asm) k0) + apply(auto) + apply (metis flts.simps(1) flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1)) + by (metis flts.simps(2) flts4 k0 less_add_Suc1 list.set_intros(1)) + + + +lemma good_SEQ: + assumes "r1 \<noteq> AZERO" "r2 \<noteq> AZERO" "\<forall>bs. r1 \<noteq> AONE bs" + shows "agood (ASEQ bs r1 r2) = (agood r1 \<and> agood r2)" + using assms + apply(case_tac r1) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + apply(case_tac r2) + apply(simp_all) + done + +lemma asize0: + shows "0 < asize r" + apply(induct r) + apply(auto) + done + +lemma nn1qq: + assumes "anonnested (AALTs bs rs)" + shows "\<nexists>bs1 rs1. AALTs bs1 rs1 \<in> set rs" + using assms + apply(induct rs rule: flts.induct) + apply(auto) + done + +lemma nn1c: + assumes "\<forall>r \<in> set rs. anonnested r" + shows "\<forall>r \<in> set (flts rs). anonalt r" + using assms + apply(induct rs rule: flts.induct) + apply(auto) + apply(rule nn11a) + by (metis nn1qq anonalt.elims(3)) + +lemma n0: + shows "anonnested (AALTs bs rs) \<longleftrightarrow> (\<forall>r \<in> set rs. anonalt r)" + apply(induct rs arbitrary: bs) + apply(auto) + apply (metis SizeBound3.nn1qq anonalt.elims(3) list.set_intros(1)) + apply (metis SizeBound3.bbbbs1 SizeBound3.nn1qq list.set_intros(2)) + by (metis anonalt.elims(2) anonnested.simps(3) anonnested.simps(4) anonnested.simps(5) anonnested.simps(6) anonnested.simps(7)) + + + +lemma nn1bb: + assumes "\<forall>r \<in> set rs. anonalt r" + shows "anonnested (bsimp_AALTs bs rs)" + using assms + apply(induct bs rs rule: bsimp_AALTs.induct) + apply(auto) + apply (metis nn11a anonalt.simps(1) anonnested.elims(3)) + using n0 by auto + +lemma nn10: + assumes "anonnested (AALTs cs rs)" + shows "anonnested (AALTs (bs @ cs) rs)" + using assms + apply(induct rs arbitrary: cs bs) + apply(simp_all) + apply(case_tac a) + apply(simp_all) + done + +lemma nn1a: + assumes "anonnested r" + shows "anonnested (fuse bs r)" + using assms + apply(induct bs r arbitrary: rule: fuse.induct) + apply(simp_all add: nn10) + done + +lemma dB_keeps_property: + shows "(\<forall>r \<in> set rs. P r) \<longrightarrow> (\<forall>r \<in> set (distinctBy rs erase rset). P r)" + apply(induct rs arbitrary: rset) + apply simp+ + done + +lemma dB_filters_out: + shows "erase a \<in> rset \<Longrightarrow> a \<notin> set (distinctBy rs erase (rset))" + apply(induct rs arbitrary: rset) + apply simp + apply(case_tac "a = aa") + apply simp+ + done + +lemma dB_will_be_distinct: + shows "distinct (distinctBy rs erase (insert (erase a) rset)) \<Longrightarrow> distinct (a # (distinctBy rs erase (insert (erase a) rset)))" + using dB_filters_out by force + + + +lemma dB_does_the_job2: + shows "distinct (distinctBy rs erase rset)" + apply(induct rs arbitrary: rset) + apply simp + apply(case_tac "erase a \<in> rset") + apply simp + apply(drule_tac x = "insert (erase a) rset" in meta_spec) + apply(subgoal_tac "distinctBy (a # rs) erase rset = a # distinctBy rs erase (insert (erase a ) rset)") + apply(simp only:) + using dB_will_be_distinct apply presburger + by auto + +lemma dB_does_more_job: + shows "distinct (map erase (distinctBy rs erase rset))" + apply(induct rs arbitrary:rset) + apply simp + apply(case_tac "erase a \<in> rset") + apply simp+ + using dB_filters_out by force + +lemma dB_mono2: + shows "rset \<subseteq> rset' \<Longrightarrow> distinctBy rs erase rset = [] \<Longrightarrow> distinctBy rs erase rset' = []" + apply(induct rs arbitrary: rset rset') + apply simp+ + by (meson in_mono list.discI) + + +lemma nn1b: + shows "anonnested (bsimp r)" + apply(induct r) + apply(simp_all) + apply(case_tac "bsimp r1 = AZERO") + apply(simp) + apply(case_tac "bsimp r2 = AZERO") + apply(simp) + apply(case_tac "\<exists>bs. bsimp r1 = AONE bs") + apply(auto)[1] + apply (simp add: nn1a) + apply(subst bsimp_ASEQ1) + apply(auto) + apply(rule nn1bb) + apply(auto) + apply(subgoal_tac "\<forall>r \<in> set (flts (map bsimp x2a)). anonalt r") + prefer 2 + apply (metis (mono_tags, lifting) SizeBound3.nn1c image_iff list.set_map) + apply(subgoal_tac "\<forall>r \<in> set (distinctBy (flts (map bsimp x2a)) erase {}). anonalt r") + prefer 2 + using dB_keeps_property apply presburger + by blast + + +lemma induct_smaller_elem_list: + shows "\<forall>r \<in> set list. asize r < Suc (sum_list (map asize list))" + apply(induct list) + apply simp+ + by fastforce + +lemma no0s_fltsbsimp: + shows "\<forall>r \<in> set (flts (map bsimp rs)). r \<noteq> AZERO" + oops + +lemma flts_all0s: + shows "\<forall>r \<in> set rs. r = AZERO \<Longrightarrow> flts rs = []" + apply(induct rs) + apply simp+ + done + + + + + +lemma good1: + shows "agood (bsimp a) \<or> bsimp a = AZERO" + apply(induct a taking: asize rule: measure_induct) + apply(case_tac x) + apply(simp) + apply(simp) + apply(simp) + prefer 3 + apply(simp) + prefer 2 + (* AALTs case *) + apply(simp only:) + apply(case_tac "x52") + apply(simp) + (* AALTs list at least one - case *) + apply(simp only: ) + apply(frule_tac x="a" in spec) + apply(drule mp) + apply(simp) + (* either first element is agood, or AZERO *) + apply(erule disjE) + prefer 2 + apply(simp) + (* in the AZERO case, the size is smaller *) + apply(drule_tac x="AALTs x51 list" in spec) + apply(drule mp) + apply(simp add: asize0) + apply(subst (asm) bsimp.simps) + apply(subst (asm) bsimp.simps) + apply(assumption) + (* in the agood case *) + apply(frule_tac x="AALTs x51 list" in spec) + apply(drule mp) + apply(simp add: asize0) + apply(erule disjE) + apply(rule disjI1) + apply(simp add: good0) + apply(subst good0) + using SizeBound3.flts3b distinctBy.elims apply force + apply(subgoal_tac " Ball (set ( (flts (bsimp a # map bsimp list)))) anonalt") + prefer 2 + apply (metis Cons_eq_map_conv SizeBound3.nn1b SizeBound3.nn1c ex_map_conv) + using dB_keeps_property apply presburger + using dB_does_more_job apply presburger + apply(subgoal_tac " Ball (set ( (flts (bsimp a # map bsimp list)) )) agood") + using dB_keeps_property apply presburger + apply(subgoal_tac "\<forall>r \<in> (set (bsimp a # map bsimp list)). (agood r) \<or> (r = AZERO)") + using SizeBound3.flts3 apply blast + + apply(subgoal_tac "\<forall>r \<in> set list. asize r < Suc (asize a + sum_list (map asize list))") + + apply simp + + apply(subgoal_tac "\<forall>r \<in> set list. asize r < Suc (sum_list (map asize list))") + + apply fastforce + using induct_smaller_elem_list apply blast + + + + + apply simp + apply(subgoal_tac "agood (bsimp a)") + prefer 2 + apply blast + apply(subgoal_tac "\<forall>r \<in> (set (map bsimp list)). r = AZERO \<or> agood r") + prefer 2 + apply (metis add_Suc_right image_iff induct_smaller_elem_list list.set_map trans_less_add2) + apply(subgoal_tac "\<not>(\<exists> r\<in>set (map bsimp list). agood r)") + prefer 2 + apply (metis SizeBound3.flts3 SizeBound3.flts3b distinctBy.elims empty_iff flts4_nogood list.set_intros(1)) + apply(subgoal_tac "\<forall>r \<in> set (map bsimp list). r = AZERO") + prefer 2 + apply blast + apply(subgoal_tac "flts (map bsimp list) = []") + prefer 2 + using flts_all0s apply presburger + apply (smt (verit, del_insts) SizeBound3.flts2 SizeBound3.good0 SizeBound3.k0 dB_does_more_job dB_keeps_property distinctBy.simps(1) flts.simps(1)) + apply(subgoal_tac "agood (bsimp x42) \<or> bsimp x42 = AZERO") + apply(subgoal_tac "agood (bsimp x43) \<or> bsimp x43 = AZERO") + apply(case_tac "bsimp x42 = AZERO") + apply simp + apply(case_tac "bsimp x43 = AZERO") + apply simp + apply(case_tac "\<exists>bs'. bsimp x42 = AONE bs'") + apply(erule exE) + apply simp + using good_fuse apply presburger + apply simp + apply(subgoal_tac "bsimp_ASEQ x41 (bsimp x42) (bsimp x43) = ASEQ x41 (bsimp x42) (bsimp x43)") + prefer 2 + using bsimp_ASEQ1 apply presburger + using SizeBound3.good_SEQ apply presburger + using asize.simps(5) less_add_Suc2 apply presburger + by simp + + + + + + +lemma good1a: + assumes "L (erase a) \<noteq> {}" + shows "agood (bsimp a)" + using good1 assms + using L_bsimp_erase by force + + + +lemma flts_append: + "flts (xs1 @ xs2) = flts xs1 @ flts xs2" + apply(induct xs1 arbitrary: xs2 rule: rev_induct) + apply(auto) + apply(case_tac xs) + apply(auto) + apply(case_tac x) + apply(auto) + apply(case_tac x) + apply(auto) + done + +lemma g1: + assumes "agood (bsimp_AALTs bs rs)" + shows "bsimp_AALTs bs rs = AALTs bs rs \<or> (\<exists>r. rs = [r] \<and> bsimp_AALTs bs [r] = fuse bs r)" +using assms + apply(induct rs arbitrary: bs) + apply(simp) + apply(case_tac rs) + apply(simp only:) + apply(simp) + apply(case_tac list) + apply(simp) + by simp + +lemma flts_0: + assumes "anonnested (AALTs bs rs)" + shows "\<forall>r \<in> set (flts rs). r \<noteq> AZERO" + using assms + apply(induct rs arbitrary: bs rule: flts.induct) + apply(simp) + apply(simp) + defer + apply(simp) + apply(simp) + apply(simp) +apply(simp) + apply(rule ballI) + apply(simp) + done + +lemma flts_0a: + assumes "anonnested (AALTs bs rs)" + shows "AZERO \<notin> set (flts rs)" + using assms + using flts_0 by blast + +lemma qqq1: + shows "AZERO \<notin> set (flts (map bsimp rs))" + by (metis ex_map_conv flts3 agood.simps(1) good1) + + +fun nonazero :: "arexp \<Rightarrow> bool" + where + "nonazero AZERO = False" +| "nonazero r = True" + +lemma flts_concat: + shows "flts rs = concat (map (\<lambda>r. flts [r]) rs)" + apply(induct rs) + apply(auto) + apply(subst k0) + apply(simp) + done + +lemma flts_single1: + assumes "anonalt r" "nonazero r" + shows "flts [r] = [r]" + using assms + apply(induct r) + apply(auto) + done + +lemma flts_qq: + assumes "\<forall>y. asize y < Suc (sum_list (map asize rs)) \<longrightarrow> agood y \<longrightarrow> bsimp y = y" + "\<forall>r'\<in>set rs. agood r' \<and> anonalt r'" + shows "flts (map bsimp rs) = rs" + using assms + apply(induct rs) + apply(simp) + apply(simp) + apply(subst k0) + apply(subgoal_tac "flts [bsimp a] = [a]") + prefer 2 + apply(drule_tac x="a" in spec) + apply(drule mp) + apply(simp) + apply(auto)[1] + using agood.simps(1) k0b apply blast + apply(auto)[1] + done + +lemma test: + assumes "agood r" + shows "bsimp r = r" + using assms + apply(induct r taking: "asize" rule: measure_induct) + apply(erule agood.elims) + apply(simp_all) + apply(subst k0) + apply(subst (2) k0) + apply(subst flts_qq) + apply(auto)[1] + apply(auto)[1] + oops + + + +lemma bsimp_idem: + shows "bsimp (bsimp r) = bsimp r" + using test good1 + oops + + +lemma erase_preimage1: + assumes "anonalt r" + shows "erase r = ONE \<Longrightarrow> \<exists>bs. r = AONE bs" + apply(case_tac r) + apply simp+ + using anonalt.simps(1) assms apply presburger + by fastforce + +lemma erase_preimage_char: + assumes "anonalt r" + shows "erase r = CH c \<Longrightarrow> \<exists>bs. r = ACHAR bs c" + apply(case_tac r) + apply simp+ + using assms apply fastforce + by simp + +lemma erase_preimage_seq: + assumes "anonalt r" + shows "erase r = SEQ r1 r2 \<Longrightarrow> \<exists>bs a1 a2. r = ASEQ bs a1 a2 \<and> erase a1 = r1 \<and> erase a2 = r2" + apply(case_tac r) + apply simp+ + using assms apply fastforce + by simp + +lemma rerase_preimage_seq: + assumes "anonalt r" + shows "rerase r = RSEQ r1 r2 \<Longrightarrow> \<exists>bs a1 a2. r = ASEQ bs a1 a2 \<and> rerase a1 = r1 \<and> rerase a2 = r2 " + using rerase_preimage4 by presburger + +lemma seq_recursive_equality: + shows "\<lbrakk>r1 = a1; r2 = a2\<rbrakk> \<Longrightarrow> SEQ r1 r2 = SEQ a1 a2" + by meson + +lemma almost_identical_image: + assumes "agood r" "\<forall>r \<in> rset. agood r" + shows "erase r \<in> erase ` rset \<Longrightarrow> \<exists>r' \<in> rset. erase r' = erase r" + by auto + +lemma goodalts_never_change: + assumes "r = AALTs bs rs" "agood r" + shows "\<exists>r1 r2. erase r = ALT r1 r2" + by (metis agood.simps(4) agood.simps(5) assms(1) assms(2) bmkepss.cases erase.simps(6)) + + +fun shape_preserving :: "arexp \<Rightarrow> bool" where + "shape_preserving AZERO = True" +| "shape_preserving (AONE bs) = True" +| "shape_preserving (ACHAR bs c) = True" +| "shape_preserving (AALTs bs []) = False" +| "shape_preserving (AALTs bs [a]) = False" +| "shape_preserving (AALTs bs (a1 # a2 # rs)) = (\<forall>r \<in> set (a1 # a2 # rs). shape_preserving r)" +| "shape_preserving (ASEQ bs r1 r2) = (shape_preserving r1 \<and> shape_preserving r2)" +| "shape_preserving (ASTAR bs r) = shape_preserving r" + + +lemma good_shape_preserving: + assumes "\<nexists>bs r0. r = ASTAR bs r0" + shows "agood r \<Longrightarrow> shape_preserving r" + using assms + + apply(induct r) + prefer 6 + + apply blast + apply simp + + oops + + + + + +lemma shadow_arexp_rerase_erase: + shows "\<lbrakk>agood r; agood r'; erase r = erase r'\<rbrakk> \<Longrightarrow> rerase r = rerase r'" + apply(induct r ) + apply simp + apply(induct r') + apply simp+ + apply (metis goodalts_never_change rexp.distinct(15)) + apply simp+ + apply (metis anonalt.elims(3) erase_preimage_char goodalts_never_change rerase.simps(3) rexp.distinct(21)) + apply(induct r') + apply simp + apply simp + apply simp + apply(subgoal_tac "agood r1") + prefer 2 + apply (metis SizeBound3.good_SEQ agood.simps(10) agood.simps(11) agood.simps(2) agood.simps(3) agood.simps(33) agood.simps(7) bsimp.cases) + apply(subgoal_tac "agood r2") + apply(subgoal_tac "agood r'1") + apply(subgoal_tac "agood r'2") + apply(subgoal_tac "rerase r'1 = rerase r1") + apply(subgoal_tac "rerase r'2 = rerase r2") + + using rerase.simps(5) apply presburger + sledgehammer + + +lemma rerase_erase_good: + assumes "agood r" "\<forall>r \<in> rset. agood r" + shows "erase r \<in> erase ` rset \<Longrightarrow> rerase r \<in> rerase ` rset" + using assms + apply(case_tac r) + apply simp+ + oops + +lemma rerase_erase_both: + assumes "\<forall>r \<in> rset. agood r" "agood r" + shows "(rerase r \<in> (rerase ` rset)) \<longleftrightarrow> (erase r \<in> (erase ` rset))" + using assms + apply(induct r ) + apply force + apply simp + apply(case_tac "RONE \<in> rerase ` rset") + apply(subgoal_tac "\<exists>bs. (AONE bs) \<in> rset") + apply (metis erase.simps(2) rev_image_eqI) + apply (metis image_iff rerase_preimage2) + apply(subgoal_tac "\<nexists>bs. (AONE bs) \<in> rset") + apply(subgoal_tac "ONE \<notin> erase ` rset") + + apply blast + sledgehammer + apply (metis erase_preimage1 image_iff) + apply (metis rerase.simps(2) rev_image_eqI) + apply (smt (verit, best) erase.simps(3) erase_preimage_char image_iff rerase.simps(3) rerase_preimage3) +(* apply simp + apply(subgoal_tac "(RSEQ (rerase r1) (rerase r2) \<in> rerase ` rset) \<Longrightarrow> (SEQ (erase r1) (erase r2) \<in> erase ` rset)") + prefer 2 + apply(subgoal_tac "\<exists>bs a1 a2. (ASEQ bs a1 a2) \<in> rset \<and> rerase a1 = rerase r1 \<and> rerase a2 = rerase r2") + prefer 2 + apply (metis (full_types) image_iff rerase_preimage4) + apply(erule exE)+ + apply(subgoal_tac "erase (ASEQ bs a1 a2) \<in> (erase ` rset) ") + apply(subgoal_tac "SEQ (erase a1) (erase a2) \<in> (erase ` rset)") + apply(subgoal_tac "SEQ (erase a1) (erase a2) = SEQ (erase r1) (erase r2)") + apply metis + apply(erule conjE)+*) + apply(drule_tac x = "rset" in meta_spec)+ + + + + + oops + + +lemma rerase_pushin1_general: + assumes "\<forall>r \<in> set rs. agood r" + shows "map rerase (distinctBy rs erase (erase ` rset)) = rdistinct (map rerase rs) (rerase ` rset)" + apply(induct rs arbitrary: rset) + apply simp+ + apply(case_tac "erase a \<in> erase ` rset") + apply simp + + + + oops + +lemma rerase_erase: + assumes "\<forall>r \<in> set as. agood r \<and> anonalt r" + shows "rdistinct (map rerase as) (rerase ` rset) = map rerase (distinctBy as erase (erase ` rset))" + using assms + apply(induct as) + apply simp+ + + sorry + + +lemma rerase_pushin1: + assumes "\<forall>r \<in> set rs. anonalt r \<and> agood r" + shows "map rerase (distinctBy rs erase {}) = rdistinct (map rerase rs) {}" + apply(insert rerase_erase) + by (metis assms image_empty) + + + + + + + +lemma nonalt_flts : shows + "\<forall>r \<in> set (flts (map bsimp rs)). r \<noteq> AZERO" + using SizeBound3.qqq1 by force + + + + +lemma rerase_list_ders: + shows "\<And>x1 x2a. + (\<And>x2aa. x2aa \<in> set x2a \<Longrightarrow> rerase (bsimp x2aa) = rsimp (rerase x2aa)) \<Longrightarrow> + (map rerase (distinctBy (flts (map bsimp x2a)) erase {})) = (rdistinct (rflts (map (rsimp \<circ> rerase) x2a)) {})" + apply(subgoal_tac "\<forall>r \<in> set (flts (map bsimp x2a)). anonalt r ") + prefer 2 + apply (metis SizeBound3.nn1b SizeBound3.nn1c ex_map_conv) + sledgehammer + + sorry + + +lemma simp_rerase: + shows "rerase (bsimp a) = rsimp (rerase a)" + apply(induct a) + apply simp+ + using rerase_bsimp_ASEQ apply presburger + apply simp + apply(subst rerase_bsimp_AALTs) + apply(subgoal_tac "map rerase (distinctBy (flts (map bsimp x2a)) erase {}) = rdistinct (rflts (map (rsimp \<circ> rerase) x2a)) {}") + apply simp + using rerase_list_ders apply blast + by simp + +lemma rders_simp_size: + shows " rders_simp (rerase r) s = rerase (bders_simp r s)" + apply(induct s rule: rev_induct) + apply simp + apply(subst rders_simp_append) + apply(subst bders_simp_append) + apply(subgoal_tac "rders_simp (rerase r ) xs = rerase (bders_simp r xs)") + apply(simp only:) + apply simp + apply (simp add: der_rerase2 simp_rerase) + by simp + + + + +corollary aders_simp_finiteness: + assumes "\<exists>N. \<forall>s. rsize (rders_simp (rerase r) s) \<le> N" + shows " \<exists>N. \<forall>s. asize (bders_simp r s) \<le> N" + using assms + apply(subgoal_tac "\<forall>s. asize (bders_simp r s) = rsize (rerase (bders_simp r s))") + apply(erule exE) + apply(rule_tac x = "N" in exI) + using rders_simp_size apply auto[1] + using asize_rsize by auto + +theorem annotated_size_bound: + shows "\<exists>N. \<forall>s. asize (bders_simp r s) \<le> N" + apply(insert aders_simp_finiteness) + by (simp add: rders_simp_bounded) + end
--- a/thys2/SizeBoundStrong.thy Sat Apr 16 09:52:57 2022 +0100 +++ b/thys2/SizeBoundStrong.thy Tue Apr 19 09:08:01 2022 +0100 @@ -502,6 +502,31 @@ decode (retrieve (bders (intern r) (s @ [c])) v) r" by (simp add: bders_append) qed +corollary main_decode1_horribly_wrong: + assumes "\<Turnstile> v : ders s r" + shows "Some(v) = decode (retrieve (bders (intern r) s) v) r" + using assms +proof (induct s arbitrary: v rule: rev_induct) + case Nil + have "\<Turnstile> v : ders [] r" by fact + then have "\<Turnstile> v: r" by simp + then have "Some v = decode (retrieve (intern r) v) r" + using decode_code retrieve_code by force + then show "Some (v) = decode (retrieve (bders (intern r) []) v) r" + by simp +next + case (snoc c s v) + have IH: "\<And>v. \<Turnstile> v : ders s r \<Longrightarrow> + Some ( v) = decode (retrieve (bders (intern r) s) v ) r" by fact + have asm: "\<Turnstile> v: ders (s @ [c]) r" by fact + then have asm2: "\<Turnstile> injval (ders s r) c v : ders s r" + using Prf_injval ders_snoc by force + then have injv: "Some (injval (ders s r) c v) = +decode (retrieve (bders (intern r) s) (injval (ders s r) c v)) r" + by (simp add: IH asm2) + oops + + definition blex where "blex a s \<equiv> if bnullable (bders a s) then Some (bmkeps (bders a s)) else None" @@ -592,17 +617,17 @@ | \<open>collect erasedR2 (r # rs) = collect erasedR2 rs\<close> -fun pruneRexp where +fun pruneRexp :: "arexp \<Rightarrow> rexp list \<Rightarrow> arexp" + where "pruneRexp (ASEQ bs r1 r2) allowableTerms = (let termsTruncated = (collect (erase r2) allowableTerms) in (let pruned = pruneRexp r1 termsTruncated in - (bsimp_ASEQ1 bs pruned r2)))" + (bsimp_ASEQ bs pruned r2)))" | \<open>pruneRexp (AALTs bs rs) allowableTerms = (let rsp = (filter (\<lambda>r. r \<noteq> AZERO) (map (\<lambda>r. pruneRexp r allowableTerms) rs) ) in bsimp_AALTs bs rsp ) \<close> | \<open>pruneRexp r allowableTerms = (if (erase r) \<in> (set allowableTerms) then r else AZERO)\<close> - fun oneSimp :: \<open>rexp \<Rightarrow> rexp\<close> where \<open> oneSimp (SEQ ONE r) = r \<close> | \<open> oneSimp (SEQ r1 r2) = SEQ (oneSimp r1) r2 \<close> @@ -617,14 +642,16 @@ where \<open>addToAcc r acc = filter (\<lambda>r1. oneSimp r1 \<notin> set acc) (breakIntoTerms (erase r)) \<close> + +(*newSubRexps: terms that are new--have not existed in acc before*) fun dBStrong :: "arexp list \<Rightarrow> rexp list \<Rightarrow> arexp list" where -"dBStrong [] acc = []" + "dBStrong [] acc = []" | "dBStrong (r # rs) acc = (if (erase r) \<in> (set acc) then dBStrong rs acc else let newSubRexps = (addToAcc r acc) in (case (pruneRexp r newSubRexps) of - AZERO \<Rightarrow> dBStrong rs (newSubRexps @ acc) | - r1 \<Rightarrow> r1 # (dBStrong rs (newSubRexps @ acc)) + AZERO \<Rightarrow> dBStrong rs (map oneSimp newSubRexps @ acc) | + r1 \<Rightarrow> r1 # (dBStrong rs (map oneSimp newSubRexps @ acc)) ) ) "
--- a/thys2/blexer2.sc Sat Apr 16 09:52:57 2022 +0100 +++ b/thys2/blexer2.sc Tue Apr 19 09:08:01 2022 +0100 @@ -10,6 +10,7 @@ // // amm lexer.sc all +import scala.util.Try // regular expressions including records abstract class Rexp @@ -28,6 +29,7 @@ // values abstract class Val +case object Failure extends Val case object Empty extends Val case class Chr(c: Char) extends Val case class Sequ(v1: Val, v2: Val) extends Val @@ -58,6 +60,166 @@ case class ANOT(bs: Bits, r: ARexp) extends ARexp case class AANYCHAR(bs: Bits) extends ARexp +trait Generator[+T] { + self => // an alias for "this" + def generate(): T + + def gen(n: Int) : List[T] = + if (n == 0) Nil else self.generate() :: gen(n - 1) + + def map[S](f: T => S): Generator[S] = new Generator[S] { + def generate = f(self.generate()) + } + def flatMap[S](f: T => Generator[S]): Generator[S] = new Generator[S] { + def generate = f(self.generate()).generate() + } + + +} + + // tests a property according to a given random generator + def test[T](r: Generator[T], amount: Int = 100)(pred: T => Boolean) { + for (_ <- 0 until amount) { + val value = r.generate() + assert(pred(value), s"Test failed for: $value") + } + println(s"Test passed $amount times") + } + def test2[T, S](r: Generator[T], s: Generator[S], amount: Int = 100)(pred: (T, S) => Boolean) { + for (_ <- 0 until amount) { + val valueR = r.generate() + val valueS = s.generate() + assert(pred(valueR, valueS), s"Test failed for: $valueR, $valueS") + } + println(s"Test passed $amount times") + } + +// random integers +val integers = new Generator[Int] { + val rand = new java.util.Random + def generate() = rand.nextInt() +} + +// random booleans +val booleans = integers.map(_ > 0) + +// random integers in the range lo and high +def range(lo: Int, hi: Int): Generator[Int] = + for (x <- integers) yield (lo + x.abs % (hi - lo)).abs + +// random characters +def chars_range(lo: Char, hi: Char) = range(lo, hi).map(_.toChar) +val chars = chars_range('a', 'z') + + +def oneOf[T](xs: T*): Generator[T] = + for (idx <- range(0, xs.length)) yield xs(idx) + +def single[T](x: T) = new Generator[T] { + def generate() = x +} + +def pairs[T, U](t: Generator[T], u: Generator[U]): Generator[(T, U)] = + for (x <- t; y <- u) yield (x, y) + +// lists +def emptyLists = single(Nil) + +def nonEmptyLists : Generator[List[Int]] = + for (head <- integers; tail <- lists) yield head :: tail + +def lists: Generator[List[Int]] = for { + kind <- booleans + list <- if (kind) emptyLists else nonEmptyLists +} yield list + +def char_list(len: Int): Generator[List[Char]] = { + if(len <= 0) single(Nil) + else{ + for { + c <- chars + tail <- char_list(len - 1) + } yield c :: tail + } +} + +def strings(len: Int): Generator[String] = for(cs <- char_list(len)) yield cs.toString + + +// (simple) binary trees +trait Tree[T] +case class Inner[T](left: Tree[T], right: Tree[T]) extends Tree[T] +case class Leaf[T](x: T) extends Tree[T] + +def leafs[T](t: Generator[T]): Generator[Leaf[T]] = + for (x <- t) yield Leaf(x) + +def inners[T](t: Generator[T]): Generator[Inner[T]] = + for (l <- trees(t); r <- trees(t)) yield Inner(l, r) + +def trees[T](t: Generator[T]): Generator[Tree[T]] = + for (kind <- range(0, 2); + tree <- if (kind == 0) leafs(t) else inners(t)) yield tree + +// regular expressions + +// generates random leaf-regexes; prefers CHAR-regexes +def leaf_rexp() : Generator[Rexp] = + for (kind <- range(0, 5); + c <- chars_range('a', 'd')) yield + kind match { + case 0 => ZERO + case 1 => ONE + case _ => CHAR(c) + } + +// generates random inner regexes with maximum depth d +def inner_rexp(d: Int) : Generator[Rexp] = + for (kind <- range(0, 3); + l <- rexp(d); + r <- rexp(d)) + yield kind match { + case 0 => ALTS(l, r) + case 1 => SEQ(l, r) + case 2 => STAR(r) + } + +// generates random regexes with maximum depth d; +// prefers inner regexes in 2/3 of the cases +def rexp(d: Int = 100): Generator[Rexp] = + for (kind <- range(0, 3); + r <- if (d <= 0 || kind == 0) leaf_rexp() else inner_rexp(d - 1)) yield r + + +// some test functions for rexps +def height(r: Rexp) : Int = r match { + case ZERO => 1 + case ONE => 1 + case CHAR(_) => 1 + case ALTS(r1, r2) => 1 + List(height(r1), height(r2)).max + case SEQ(r1, r2) => 1 + List(height(r1), height(r2)).max + case STAR(r) => 1 + height(r) +} + +// def size(r: Rexp) : Int = r match { +// case ZERO => 1 +// case ONE => 1 +// case CHAR(_) => 1 +// case ALTS(r1, r2) => 1 + size(r1) + size(r2) +// case SEQ(r1, r2) => 1 + size(r1) + size(r2) +// case STAR(r) => 1 + size(r) +// } + +// randomly subtracts 1 or 2 from the STAR case +def size_faulty(r: Rexp) : Int = r match { + case ZERO => 1 + case ONE => 1 + case CHAR(_) => 1 + case ALTS(r1, r2) => 1 + size_faulty(r1) + size_faulty(r2) + case SEQ(r1, r2) => 1 + size_faulty(r1) + size_faulty(r2) + case STAR(r) => 1 + size_faulty(r) - range(0, 2).generate +} + // some convenience for typing in regular expressions @@ -184,104 +346,12 @@ } // some "rectification" functions for simplification -def F_ID(v: Val): Val = v -def F_RIGHT(f: Val => Val) = (v:Val) => Right(f(v)) -def F_LEFT(f: Val => Val) = (v:Val) => Left(f(v)) -def F_ALT(f1: Val => Val, f2: Val => Val) = (v:Val) => v match { - case Right(v) => Right(f2(v)) - case Left(v) => Left(f1(v)) -} -def F_SEQ(f1: Val => Val, f2: Val => Val) = (v:Val) => v match { - case Sequ(v1, v2) => Sequ(f1(v1), f2(v2)) -} -def F_SEQ_Empty1(f1: Val => Val, f2: Val => Val) = - (v:Val) => Sequ(f1(Empty), f2(v)) -def F_SEQ_Empty2(f1: Val => Val, f2: Val => Val) = - (v:Val) => Sequ(f1(v), f2(Empty)) -def F_ERROR(v: Val): Val = throw new Exception("error") -// simplification -def simp(r: Rexp): (Rexp, Val => Val) = r match { - case ALTS(r1, r2) => { - val (r1s, f1s) = simp(r1) - val (r2s, f2s) = simp(r2) - (r1s, r2s) match { - case (ZERO, _) => (r2s, F_RIGHT(f2s)) - case (_, ZERO) => (r1s, F_LEFT(f1s)) - case _ => if (r1s == r2s) (r1s, F_LEFT(f1s)) - else (ALTS (r1s, r2s), F_ALT(f1s, f2s)) - } - } - case SEQ(r1, r2) => { - val (r1s, f1s) = simp(r1) - val (r2s, f2s) = simp(r2) - (r1s, r2s) match { - case (ZERO, _) => (ZERO, F_ERROR) - case (_, ZERO) => (ZERO, F_ERROR) - case (ONE, _) => (r2s, F_SEQ_Empty1(f1s, f2s)) - case (_, ONE) => (r1s, F_SEQ_Empty2(f1s, f2s)) - case _ => (SEQ(r1s,r2s), F_SEQ(f1s, f2s)) - } - } - case r => (r, F_ID) -} -// lexing functions including simplification -def lex_simp(r: Rexp, s: List[Char]) : Val = s match { - case Nil => if (nullable(r)) mkeps(r) else - { throw new Exception(s"lexing error $r not nullable") } - case c::cs => { - val (r_simp, f_simp) = simp(der(c, r)) - inj(r, c, f_simp(lex_simp(r_simp, cs))) - } -} - -def lexing_simp(r: Rexp, s: String) = - env(lex_simp(r, s.toList)) // The Lexing Rules for the WHILE Language -def PLUS(r: Rexp) = r ~ r.% - -def Range(s : List[Char]) : Rexp = s match { - case Nil => ZERO - case c::Nil => CHAR(c) - case c::s => ALTS(CHAR(c), Range(s)) -} -def RANGE(s: String) = Range(s.toList) - -val SYM = RANGE("ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_") -val DIGIT = RANGE("0123456789") -val ID = SYM ~ (SYM | DIGIT).% -val NUM = PLUS(DIGIT) -val KEYWORD : Rexp = "skip" | "while" | "do" | "if" | "then" | "else" | "read" | "write" -val SEMI: Rexp = ";" -val OP: Rexp = ":=" | "=" | "-" | "+" | "*" | "!=" | "<" | ">" -val WHITESPACE = PLUS(" " | "\n" | "\t" | "\r") -val RPAREN: Rexp = "{" -val LPAREN: Rexp = "}" -val STRING: Rexp = "\"" ~ SYM.% ~ "\"" - - -//ab \ a --> 1b -// -val WHILE_REGS = (("k" $ KEYWORD) | - ("i" $ ID) | - ("o" $ OP) | - ("n" $ NUM) | - ("s" $ SEMI) | - ("str" $ STRING) | - ("p" $ (LPAREN | RPAREN)) | - ("w" $ WHITESPACE)).% - -val NREGS = NTIMES(5, OPTIONAL(SYM)) -val NREGS1 = ("test" $ NREGS) -// Two Simple While Tests -//======================== -val NOTREG = "hehe" ~ NOT((ANYCHAR.%) ~ "haha" ~ (ANYCHAR.%)) - - // bnullable function: tests whether the aregular // expression can recognise the empty string def bnullable (r: ARexp) : Boolean = r match { @@ -371,252 +441,186 @@ } case r => r } - } - def strongBsimp(r: ARexp): ARexp = - { - r match { - case ASEQ(bs1, r1, r2) => (strongBsimp(r1), strongBsimp(r2)) match { - case (AZERO, _) => AZERO - case (_, AZERO) => AZERO - case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s) - case (r1s, r2s) => ASEQ(bs1, r1s, r2s) - } - case AALTS(bs1, rs) => { - val rs_simp = rs.map(strongBsimp(_)) - val flat_res = flats(rs_simp) - val dist_res = distinctBy4(flat_res)//distinctBy(flat_res, erase) - dist_res match { - case Nil => AZERO - case s :: Nil => fuse(bs1, s) - case rs => AALTS(bs1, rs) - } - - } - case r => r +} +def strongBsimp(r: ARexp): ARexp = +{ + r match { + case ASEQ(bs1, r1, r2) => (strongBsimp(r1), strongBsimp(r2)) match { + case (AZERO, _) => AZERO + case (_, AZERO) => AZERO + case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s) + case (r1s, r2s) => ASEQ(bs1, r1s, r2s) } + case AALTS(bs1, rs) => { + val rs_simp = rs.map(strongBsimp(_)) + val flat_res = flats(rs_simp) + val dist_res = distinctBy4(flat_res)//distinctBy(flat_res, erase) + dist_res match { + case Nil => AZERO + case s :: Nil => fuse(bs1, s) + case rs => AALTS(bs1, rs) + } + + } + case r => r } - - def bders (s: List[Char], r: ARexp) : ARexp = s match { - case Nil => r - case c::s => bders(s, bder(c, r)) - } - - def flats(rs: List[ARexp]): List[ARexp] = rs match { - case Nil => Nil - case AZERO :: rs1 => flats(rs1) - case AALTS(bs, rs1) :: rs2 => rs1.map(fuse(bs, _)) ::: flats(rs2) - case r1 :: rs2 => r1 :: flats(rs2) - } - - def distinctBy[B, C](xs: List[B], f: B => C, acc: List[C] = Nil): List[B] = xs match { - case Nil => Nil - case (x::xs) => { - val res = f(x) - if (acc.contains(res)) distinctBy(xs, f, acc) - else x::distinctBy(xs, f, res::acc) - } - } - - def projectFirstChild(rs: List[ARexp]) : List[ARexp] = rs match { - case (ASEQ(bs, r1, r2p)::rs1) => r1::projectFirstChild(rs1) - case Nil => Nil - } +} - - def distinctBy2(xs: List[ARexp], acc: List[Rexp] = Nil): List[ARexp] = xs match { - case Nil => Nil - case (x::xs) => { - val res = erase(x) - if(acc.contains(res)) - distinctBy2(xs, acc) - else - x match { - case ASEQ(bs0, AALTS(bs1, rs), r2) => - val newTerms = distinctBy2(rs.map(r1 => ASEQ(Nil, r1, r2)), acc) - val rsPrime = projectFirstChild(newTerms) - newTerms match { - case Nil => distinctBy2(xs, acc) - case t::Nil => ASEQ(bs0, fuse(bs1, rsPrime.head), r2)::distinctBy2(xs, erase(t)::acc) - case ts => ASEQ(bs0, AALTS(bs1, rsPrime), r2)::distinctBy2(xs, newTerms.map(erase(_)):::acc) - } - case x => x::distinctBy2(xs, res::acc) - } - } - } - - def deepFlts(r: ARexp): List[ARexp] = r match{ +def bders (s: List[Char], r: ARexp) : ARexp = s match { + case Nil => r + case c::s => bders(s, bder(c, r)) +} - case ASEQ(bs, r1, r2) => deepFlts(r1).map(r1p => ASEQ(bs, r1p, r2)) - case ASTAR(bs, r0) => List(r) - case ACHAR(_, _) => List(r) - case AALTS(bs, rs) => rs.flatMap(deepFlts(_))//throw new Error("doubly nested alts in bsimp r") - } - - - def distinctBy3(xs: List[ARexp], acc: List[Rexp] = Nil): List[ARexp] = xs match { +def flats(rs: List[ARexp]): List[ARexp] = rs match { case Nil => Nil - case (x::xs) => { - val res = erase(x) - if(acc.contains(res)) - distinctBy3(xs, acc) - else - x match { - case ASEQ(bs0, AALTS(bs1, rs), r2) => - val newTerms = distinctBy3(rs.flatMap(deepFlts(_)).map(r1 => ASEQ(Nil, r1, r2)), acc)//deepFlts(rs.flatMap(r1 => ASEQ(Nil, r1, r2)), acc) - val rsPrime = projectFirstChild(newTerms) - newTerms match { - case Nil => distinctBy3(xs, acc) - case t::Nil => ASEQ(bs0, fuse(bs1, rsPrime.head), r2)::distinctBy3(xs, erase(t)::acc) - case ts => ASEQ(bs0, AALTS(bs1, rsPrime), r2)::distinctBy3(xs, newTerms.map(erase(_)):::acc) - } - case x => x::distinctBy3(xs, res::acc) - } - } + case AZERO :: rs1 => flats(rs1) + case AALTS(bs, rs1) :: rs2 => rs1.map(fuse(bs, _)) ::: flats(rs2) + case r1 :: rs2 => r1 :: flats(rs2) } - def pruneRexp(r: ARexp, allowableTerms: List[Rexp]) : ARexp = { - r match { - case ASEQ(bs, r1, r2) => - val termsTruncated = allowableTerms.collect(rt => rt match { - case SEQ(r1p, r2p) if(r2p == erase(r2)) => r1p//if(r2p == erase(r2)) - }) - val pruned : ARexp = pruneRexp(r1, termsTruncated) - pruned match { - case AZERO => AZERO - case AONE(bs1) => fuse(bs ++ bs1, r2) - case pruned1 => ASEQ(bs, pruned1, r2) - } - - case AALTS(bs, rs) => - //allowableTerms.foreach(a => println(shortRexpOutput(a))) - val rsp = rs.map(r => - pruneRexp(r, allowableTerms) - ) - .filter(r => r != AZERO) - rsp match { - case Nil => AZERO - case r1::Nil => fuse(bs, r1) - case rs1 => AALTS(bs, rs1) - } - case r => - if(allowableTerms.contains(erase(r))) r else AZERO //assert(r != AZERO) - } +def distinctBy[B, C](xs: List[B], f: B => C, acc: List[C] = Nil): List[B] = xs match { + case Nil => Nil + case (x::xs) => { + val res = f(x) + if (acc.contains(res)) distinctBy(xs, f, acc) + else x::distinctBy(xs, f, res::acc) } - - def oneSimp(r: Rexp) : Rexp = r match { - case SEQ(ONE, r) => r - case SEQ(r1, r2) => SEQ(oneSimp(r1), r2) - case r => r//assert r != 0 - - } +} - def distinctBy4(xs: List[ARexp], acc: List[Rexp] = Nil) : List[ARexp] = xs match { +def pruneRexp(r: ARexp, allowableTerms: List[Rexp]) : ARexp = { + r match { + case ASEQ(bs, r1, r2) => + val termsTruncated = allowableTerms.collect(rt => rt match { + case SEQ(r1p, r2p) if(r2p == erase(r2)) => r1p//if(r2p == erase(r2)) + }) + val pruned : ARexp = pruneRexp(r1, termsTruncated) + pruned match { + case AZERO => AZERO + case AONE(bs1) => fuse(bs ++ bs1, r2) + case pruned1 => ASEQ(bs, pruned1, r2) + } + + case AALTS(bs, rs) => + //allowableTerms.foreach(a => println(shortRexpOutput(a))) + val rsp = rs.map(r => + pruneRexp(r, allowableTerms) + ) + .filter(r => r != AZERO) + rsp match { + case Nil => AZERO + case r1::Nil => fuse(bs, r1) + case rs1 => AALTS(bs, rs1) + } + case r => + if(allowableTerms.contains(erase(r))) r else AZERO //assert(r != AZERO) + } +} + +def oneSimp(r: Rexp) : Rexp = r match { + case SEQ(ONE, r) => r + case SEQ(r1, r2) => SEQ(oneSimp(r1), r2) + case r => r//assert r != 0 - case Nil => Nil - case x :: xs => - //assert(acc.distinct == acc) - val erased = erase(x) - if(acc.contains(erased)) - distinctBy4(xs, acc) - else{ - val addToAcc = breakIntoTerms(erased).filter(r => !acc.contains(oneSimp(r))) - //val xp = pruneRexp(x, addToAcc) - pruneRexp(x, addToAcc) match { - case AZERO => distinctBy4(xs, addToAcc.map(oneSimp(_)) ::: acc) - case xPrime => xPrime :: distinctBy4(xs, addToAcc.map(oneSimp(_)) ::: acc) - } - - } - } +} - def breakIntoTerms(r: Rexp) : List[Rexp] = r match { - case SEQ(r1, r2) => breakIntoTerms(r1).map(r11 => SEQ(r11, r2)) - // breakIntoTerms(r1).map(r11 => r11 match { - // case ONE => r2 - // case r => SEQ(r11, r2) - // } - //) - case ALTS(r1, r2) => breakIntoTerms(r1) ::: breakIntoTerms(r2) - case _ => r::Nil - } - - def prettyRexp(r: Rexp) : String = r match { - case STAR(r0) => s"${prettyRexp(r0)}*" - case SEQ(CHAR(c), r2) => c.toString ++ prettyRexp(r2) - case SEQ(r1, r2) => s"${prettyRexp(r1)}~${prettyRexp(r2)}" - case CHAR(c) => c.toString - case ANYCHAR => "." - // case NOT(r0) => s - } - - def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match { - case (ONE, bs) => (Empty, bs) - case (CHAR(f), bs) => (Chr(f), bs) - case (ALTS(r1, r2), Z::bs1) => { - val (v, bs2) = decode_aux(r1, bs1) - (Left(v), bs2) +def distinctBy4(xs: List[ARexp], acc: List[Rexp] = Nil) : List[ARexp] = xs match { + case Nil => Nil + case x :: xs => + //assert(acc.distinct == acc) + val erased = erase(x) + if(acc.contains(erased)) + distinctBy4(xs, acc) + else{ + val addToAcc = breakIntoTerms(erased).filter(r => !acc.contains(oneSimp(r))) + //val xp = pruneRexp(x, addToAcc) + pruneRexp(x, addToAcc) match { + case AZERO => distinctBy4(xs, addToAcc.map(oneSimp(_)) ::: acc) + case xPrime => xPrime :: distinctBy4(xs, addToAcc.map(oneSimp(_)) ::: acc) + } + } - case (ALTS(r1, r2), S::bs1) => { - val (v, bs2) = decode_aux(r2, bs1) - (Right(v), bs2) - } - case (SEQ(r1, r2), bs) => { - val (v1, bs1) = decode_aux(r1, bs) - val (v2, bs2) = decode_aux(r2, bs1) - (Sequ(v1, v2), bs2) - } - case (STAR(r1), S::bs) => { - val (v, bs1) = decode_aux(r1, bs) - //println(v) - val (Stars(vs), bs2) = decode_aux(STAR(r1), bs1) - (Stars(v::vs), bs2) - } - case (STAR(_), Z::bs) => (Stars(Nil), bs) - case (RECD(x, r1), bs) => { - val (v, bs1) = decode_aux(r1, bs) - (Rec(x, v), bs1) - } - case (NOT(r), bs) => (Nots(prettyRexp(r)), bs) - } +} + - def decode(r: Rexp, bs: Bits) = decode_aux(r, bs) match { - case (v, Nil) => v - case _ => throw new Exception("Not decodable") - } +def breakIntoTerms(r: Rexp) : List[Rexp] = r match { + case SEQ(r1, r2) => breakIntoTerms(r1).map(r11 => SEQ(r11, r2)) + case ALTS(r1, r2) => breakIntoTerms(r1) ::: breakIntoTerms(r2) + case _ => r::Nil +} -def blexSimp(r: Rexp, s: String) : List[Bit] = { - blex_simp(internalise(r), s.toList) +def decode_aux(r: Rexp, bs: Bits) : (Val, Bits) = (r, bs) match { + case (ONE, bs) => (Empty, bs) + case (CHAR(f), bs) => (Chr(f), bs) + case (ALTS(r1, r2), Z::bs1) => { + val (v, bs2) = decode_aux(r1, bs1) + (Left(v), bs2) + } + case (ALTS(r1, r2), S::bs1) => { + val (v, bs2) = decode_aux(r2, bs1) + (Right(v), bs2) + } + case (SEQ(r1, r2), bs) => { + val (v1, bs1) = decode_aux(r1, bs) + val (v2, bs2) = decode_aux(r2, bs1) + (Sequ(v1, v2), bs2) + } + case (STAR(r1), S::bs) => { + val (v, bs1) = decode_aux(r1, bs) + //println(v) + val (Stars(vs), bs2) = decode_aux(STAR(r1), bs1) + (Stars(v::vs), bs2) + } + case (STAR(_), Z::bs) => (Stars(Nil), bs) + case (RECD(x, r1), bs) => { + val (v, bs1) = decode_aux(r1, bs) + (Rec(x, v), bs1) + } + case (NOT(r), bs) => (Nots(r.toString), bs) } +def decode(r: Rexp, bs: Bits) = decode_aux(r, bs) match { + case (v, Nil) => v + case _ => throw new Exception("Not decodable") +} + + + def blexing_simp(r: Rexp, s: String) : Val = { val bit_code = blex_simp(internalise(r), s.toList) decode(r, bit_code) - } +} +def simpBlexer(r: Rexp, s: String) : Val = { + Try(blexing_simp(r, s)).getOrElse(Failure) +} - def strong_blexing_simp(r: Rexp, s: String) : Val = { - decode(r, strong_blex_simp(internalise(r), s.toList)) - } +def strong_blexing_simp(r: Rexp, s: String) : Val = { + decode(r, strong_blex_simp(internalise(r), s.toList)) +} + +def strongBlexer(r: Rexp, s: String) : Val = { + Try(strong_blexing_simp(r, s)).getOrElse(Failure) +} - def strong_blex_simp(r: ARexp, s: List[Char]) :Bits = s match { - case Nil => { - if (bnullable(r)) { - //println(asize(r)) - val bits = mkepsBC(r) - bits - } - else - throw new Exception("Not matched") +def strong_blex_simp(r: ARexp, s: List[Char]) : Bits = s match { + case Nil => { + if (bnullable(r)) { + //println(asize(r)) + val bits = mkepsBC(r) + bits } - case c::cs => { - val der_res = bder(c,r) - val simp_res = strongBsimp(der_res) - strong_blex_simp(simp_res, cs) - } + else + throw new Exception("Not matched") } + case c::cs => { + val der_res = bder(c,r) + val simp_res = strongBsimp(der_res) + strong_blex_simp(simp_res, cs) + } +} @@ -629,12 +633,12 @@ def bdersSimp(s: String, r: Rexp) : ARexp = bders_simp(s.toList, internalise(r)) - def bders_simpS(s: List[Char], r: ARexp) : ARexp = s match { + def bdersStrong(s: List[Char], r: ARexp) : ARexp = s match { case Nil => r - case c::s => bders_simpS(s, strongBsimp(bder(c, r))) + case c::s => bdersStrong(s, strongBsimp(bder(c, r))) } - def bdersSimpS(s: String, r: Rexp) : ARexp = bders_simpS(s.toList, internalise(r)) + def bdersStrongRexp(s: String, r: Rexp) : ARexp = bdersStrong(s.toList, internalise(r)) def erase(r:ARexp): Rexp = r match{ case AZERO => ZERO @@ -650,59 +654,63 @@ } -def allCharSeq(r: Rexp) : Boolean = r match { - case CHAR(c) => true - case SEQ(r1, r2) => allCharSeq(r1) && allCharSeq(r2) - case _ => false -} + def allCharSeq(r: Rexp) : Boolean = r match { + case CHAR(c) => true + case SEQ(r1, r2) => allCharSeq(r1) && allCharSeq(r2) + case _ => false + } + + def flattenSeq(r: Rexp) : String = r match { + case CHAR(c) => c.toString + case SEQ(r1, r2) => flattenSeq(r1) ++ flattenSeq(r2) + case _ => throw new Error("flatten unflattenable rexp") + } -def flattenSeq(r: Rexp) : String = r match { - case CHAR(c) => c.toString - case SEQ(r1, r2) => flattenSeq(r1) ++ flattenSeq(r2) - case _ => throw new Error("flatten unflattenable rexp") -} + def shortRexpOutput(r: Rexp) : String = r match { + case CHAR(c) => c.toString + case ONE => "1" + case ZERO => "0" + case SEQ(r1, r2) if(allCharSeq(r)) => flattenSeq(r)//"[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]" + case SEQ(r1, r2) => "[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]" + case ALTS(r1, r2) => "(" ++ shortRexpOutput(r1) ++ "+" ++ shortRexpOutput(r2) ++ ")" + case STAR(STAR(r)) => "(..)*"// ++ shortRexpOutput(r) ++ "]*" + case STAR(r) => "STAR(" ++ shortRexpOutput(r) ++ ")" + //case RTOP => "RTOP" + } + -def shortRexpOutput(r: Rexp) : String = r match { - case CHAR(c) => c.toString - case ONE => "1" - case ZERO => "0" - case SEQ(r1, r2) if(allCharSeq(r)) => flattenSeq(r)//"[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]" - case SEQ(r1, r2) => "[" ++ shortRexpOutput(r1) ++ "~" ++ shortRexpOutput(r2) ++ "]" - case ALTS(r1, r2) => "(" ++ shortRexpOutput(r1) ++ "+" ++ shortRexpOutput(r2) ++ ")" - case STAR(STAR(r)) => "(..)*"// ++ shortRexpOutput(r) ++ "]*" - case STAR(r) => "STAR(" ++ shortRexpOutput(r) ++ ")" - //case RTOP => "RTOP" + def blex_simp(r: ARexp, s: List[Char]) : Bits = s match { + case Nil => { + if (bnullable(r)) { + //println(asize(r)) + val bits = mkepsBC(r) + bits + } + else + throw new Exception("Not matched") + } + case c::cs => { + val der_res = bder(c,r) + val simp_res = bsimp(der_res) + blex_simp(simp_res, cs) + } } -def blex_simp(r: ARexp, s: List[Char]) : Bits = s match { - case Nil => { - if (bnullable(r)) { - //println(asize(r)) - val bits = mkepsBC(r) - bits - } - else - throw new Exception("Not matched") + + + def size(r: Rexp) : Int = r match { + case ZERO => 1 + case ONE => 1 + case CHAR(_) => 1 + case ANYCHAR => 1 + case NOT(r0) => 1 + size(r0) + case SEQ(r1, r2) => 1 + size(r1) + size(r2) + case ALTS(r1, r2) => 1 + List(r1, r2).map(size).sum + case STAR(r) => 1 + size(r) } - case c::cs => { - val der_res = bder(c,r) - val simp_res = bsimp(der_res) - blex_simp(simp_res, cs) - } - } - def size(r: Rexp) : Int = r match { - case ZERO => 1 - case ONE => 1 - case CHAR(_) => 1 - case ANYCHAR => 1 - case NOT(r0) => 1 + size(r0) - case SEQ(r1, r2) => 1 + size(r1) + size(r2) - case ALTS(r1, r2) => 1 + List(r1, r2).map(size).sum - case STAR(r) => 1 + size(r) - } - def asize(a: ARexp) = size(erase(a)) + def asize(a: ARexp) = size(erase(a)) //pder related type Mon = (Char, Rexp) @@ -808,6 +816,9 @@ (((STAR("a") | ( STAR("aaa")) | STAR("aaaaa"| ( STAR("aaaaaaa")) | STAR("aaaaaaaaaaa"))).%).%).% // @main + + + def small() = { @@ -834,56 +845,89 @@ // //println(refSize) // println(pderSTAR.size) - val A : Rexp= ("c" | (ONE | "b") ~ "d") ~((ONE).%) - val B : Rexp = ((ONE).%) - val C : Rexp = ("d") ~ ((ONE).%) - val PRUNE_REG : Rexp = (C | B | A) - val APRUNE_REG = internalise(PRUNE_REG) - // // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG)) - // // println("program executes and gives: as disired!") - // // println(shortRexpOutput(erase(program_solution))) + // val A : Rexp= ("c" | (ONE | "b") ~ "d") ~((ONE).%) + // val B : Rexp = ((ONE).%) + // val C : Rexp = ("d") ~ ((ONE).%) + // val PRUNE_REG : Rexp = (C | B | A) + // val APRUNE_REG = internalise(PRUNE_REG) + // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG)) + // println("program executes and gives: as disired!") + // println(shortRexpOutput(erase(program_solution))) // val simpedPruneReg = strongBsimp(APRUNE_REG) // println(shortRexpOutput(erase(simpedPruneReg))) - // for(i <- List(1,2 ) ){// 100, 400, 800, 840, 841, 900 - // val prog0 = "a" * i - // //println(s"test: $prog0") - // println(s"testing with $i a's" ) - // //val bd = bdersSimp(prog0, STARREG)//DB - // val sbd = bdersSimpS(prog0, STARREG)//strongDB - // starPrint(erase(sbd)) - // val subTerms = breakIntoTerms(erase(sbd)) - // //val subTermsLarge = breakIntoTerms(erase(bd)) + + + for(i <- List(1,2,3,4,10, 100, 400, 841, 900 ) ){// 100, 400, 800, 840, 841, 900 + val prog0 = "a" * i + //println(s"test: $prog0") + println(s"testing with $i a's" ) + //val bd = bdersSimp(prog0, STARREG)//DB + val sbd = bdersStrongRexp(prog0, STARREG)//strongDB + starPrint(erase(sbd)) + val subTerms = breakIntoTerms(erase(sbd)) + //val subTermsLarge = breakIntoTerms(erase(bd)) - // println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}") + println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}") - // println("the number of distinct subterms for bsimp with strongDB") - // println(subTerms.distinct.size) - // //println(subTermsLarge.distinct.size) - // println("which coincides with the number of PDER terms") + println("the number of distinct subterms for bsimp with strongDB") + println(subTerms.distinct.size) + //println(subTermsLarge.distinct.size) + if(pderSTAR.size > subTerms.length) + println(s"which is less than or equal to the number of PDER terms: ${pderSTAR.size}") - // // println(shortRexpOutput(erase(sbd))) - // // println(shortRexpOutput(erase(bd))) + // println(shortRexpOutput(erase(sbd))) + // println(shortRexpOutput(erase(bd))) - // println("pdersize, original, strongSimp") - // println(refSize, size(STARREG), asize(sbd)) + println("pdersize, original, strongSimp") + println(refSize, size(STARREG), asize(sbd)) - // val vres = strong_blexing_simp( STARREG, prog0) - // println(vres) - // } -// println(vs.length) -// println(vs) + //val vres = strong_blexing_simp( STARREG, prog0) + //println(vres) + } + // println(vs.length) + // println(vs) // val prog1 = """read n; write n""" // println(s"test: $prog1") // println(lexing_simp(WHILE_REGS, prog1)) - val display = ("a"| "ab").% - val adisplay = internalise(display) - bders_simp( "aaaaa".toList, adisplay) + // val display = ("a"| "ab").% + // val adisplay = internalise(display) + // bders_simp( "aaaaa".toList, adisplay) } -small() +def generator_test() { + // println("XXX generates 10 random integers in the range 0..2") + // println(range(0, 3).gen(10).mkString("\n")) + + // println("XXX gnerates random lists and trees") + // println(lists.generate()) + // println(trees(chars).generate()) + + // println("XXX generates 10 random characters") + // println(chars.gen(10).mkString("\n")) + + // println("XXX generates 10 random leaf-regexes") + // println(leaf_rexp().gen(10).mkString("\n")) + + // println("XXX generates 1000 regexes of maximal 10 height") + // println(rexp(10).gen(1000).mkString("\n")) + + // println("XXX generates 1000 regexes and tests an always true-test") + // test(rexp(10), 1000) { _ => true } + // println("XXX generates regexes and tests a valid predicate") + // test(rexp(10), 1000) { r => height(r) <= size(r) } + // println("XXX faulty test") + // test(rexp(10), 100) { r => height(r) <= size_faulty(r) } + println("testing strongbsimp against bsimp") + test2(rexp(10), strings(2), 100) { (r : Rexp, s: String) => + (simpBlexer(r, s) == strongBlexer(r, s)) + } + +} +// small() +generator_test()