--- a/thys/BitCoded.thy Tue Jul 23 21:21:49 2019 +0100
+++ b/thys/BitCoded.thy Mon Jul 29 09:37:20 2019 +0100
@@ -1,9 +1,9 @@
-
+
theory BitCoded
imports "Lexer"
begin
-section {* Bit-Encodings *}
+section \<open>Bit-Encodings\<close>
datatype bit = Z | S
@@ -118,6 +118,13 @@
| "erase (ASEQ _ r1 r2) = SEQ (erase r1) (erase r2)"
| "erase (ASTAR _ r) = STAR (erase r)"
+lemma decode_code_erase:
+ assumes "\<Turnstile> v : (erase a)"
+ shows "decode (code v) (erase a) = Some v"
+ using assms
+ by (simp add: decode_code)
+
+
fun nonalt :: "arexp \<Rightarrow> bool"
where
"nonalt (AALTs bs2 rs) = False"
@@ -560,6 +567,13 @@
| "flts ((AALTs bs rs1) # rs) = (map (fuse bs) rs1) @ flts rs"
| "flts (r1 # rs) = r1 # flts rs"
+fun li :: "bit list \<Rightarrow> arexp list \<Rightarrow> arexp"
+ where
+ "li _ [] = AZERO"
+| "li bs [a] = fuse bs a"
+| "li bs as = AALTs bs as"
+
+
fun bsimp_ASEQ :: "bit list \<Rightarrow> arexp \<Rightarrow> arexp \<Rightarrow> arexp"
where
"bsimp_ASEQ _ AZERO _ = AZERO"
@@ -582,6 +596,8 @@
| "bsimp r = r"
+
+
fun
bders_simp :: "arexp \<Rightarrow> string \<Rightarrow> arexp"
where
@@ -1285,26 +1301,34 @@
shows "good (bsimp a) \<or> bsimp a = AZERO"
apply(induct a taking: asize rule: measure_induct)
apply(case_tac x)
- apply(simp)
- apply(simp)
+ apply(simp)
+ apply(simp)
apply(simp)
prefer 3
apply(simp)
prefer 2
+ (* AALTs case *)
apply(simp only:)
apply(case_tac "x52")
apply(simp)
- apply(simp only: good0a)
+ thm good0a
+ (* AALTs list at least one - case *)
+ apply(simp only: )
apply(frule_tac x="a" in spec)
apply(drule mp)
- apply(simp)
+ apply(simp)
+ (* either first element is good, or AZERO *)
apply(erule disjE)
prefer 2
- apply(simp)
+ 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(auto)[1]
+ apply(simp add: asize0)
+ apply(subst (asm) bsimp.simps)
+ apply(subst (asm) bsimp.simps)
+ apply(assumption)
+ (* in the good case *)
apply(frule_tac x="AALTs x51 list" in spec)
apply(drule mp)
apply(simp add: asize0)
@@ -1324,6 +1348,7 @@
apply(auto)[1]
apply auto[1]
apply (metis ex_map_conv nn1b nn1c)
+ (* in the AZERO case *)
apply(simp)
apply(frule_tac x="a" in spec)
apply(drule mp)
@@ -1866,6 +1891,728 @@
apply(auto)
done
+lemma LA:
+ assumes "\<Turnstile> v : ders s (erase r)"
+ shows "retrieve (bders r s) v = retrieve r (flex (erase r) id s v)"
+ using assms
+ apply(induct s arbitrary: r v rule: rev_induct)
+ apply(simp)
+ apply(simp add: bders_append ders_append)
+ apply(subst bder_retrieve)
+ apply(simp)
+ apply(drule Prf_injval)
+ by (simp add: flex_append)
+
+
+lemma LB:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+ using assms
+ apply(induct s arbitrary: r v rule: rev_induct)
+ apply(simp)
+ apply(subgoal_tac "v = mkeps (erase r)")
+ prefer 2
+ apply (simp add: Posix1(1) Posix_determ Posix_mkeps nullable_correctness)
+ apply(simp)
+ apply(simp add: flex_append ders_append)
+ by (metis Posix_determ Posix_flex Posix_injval Posix_mkeps ders_snoc lexer_correctness(2) lexer_flex)
+
+lemma LB_sym:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve r (flex (erase r) id s (mkeps (erase (bders r s))))"
+ using assms
+ by (simp add: LB)
+
+
+lemma LC:
+ assumes "s \<in> (erase r) \<rightarrow> v"
+ shows "retrieve r v = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ apply(simp)
+ by (metis LA LB Posix1(1) assms lexer_correct_None lexer_flex mkeps_nullable)
+
+
+lemma L0:
+ assumes "bnullable a"
+ shows "retrieve (bsimp a) (mkeps (erase (bsimp a))) = retrieve a (mkeps (erase a))"
+ using assms
+ by (metis b3 bmkeps_retrieve bmkeps_simp bnullable_correctness)
+
+thm bmkeps_retrieve
+
+lemma L0a:
+ assumes "s \<in> L(erase a)"
+ shows "retrieve (bsimp (bders a s)) (mkeps (erase (bsimp (bders a s)))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ by (metis L0 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+
+lemma L0aa:
+ assumes "s \<in> L (erase a)"
+ shows "[] \<in> erase (bsimp (bders a s)) \<rightarrow> mkeps (erase (bsimp (bders a s)))"
+ using assms
+ by (metis Posix_mkeps b3 bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+
+lemma L0aaa:
+ assumes "[c] \<in> L (erase a)"
+ shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bder c a)))"
+ using assms
+ by (metis bders.simps(1) bders.simps(2) erase_bders lexer_correct_None lexer_correct_Some lexer_flex option.inject)
+
+lemma L0aaaa:
+ assumes "[c] \<in> L (erase a)"
+ shows "[c] \<in> (erase a) \<rightarrow> flex (erase a) id [c] (mkeps (erase (bders a [c])))"
+ using assms
+ using L0aaa by auto
+
+
+lemma L02:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id [c] (mkeps (erase (bder c (bsimp a))))) =
+ retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a))))"
+ using assms
+ apply(simp)
+ using bder_retrieve L0 bmkeps_simp bmkeps_retrieve L0 LA LB
+ apply(subst bder_retrieve[symmetric])
+ apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder mkeps_nullable nullable_correctness)
+ apply(simp)
+ done
+
+lemma L02_bders:
+ assumes "bnullable (bders a s)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
+ retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s)))"
+ using assms
+ by (metis LA L_bsimp_erase bnullable_correctness ders_correctness erase_bders mkeps_nullable nullable_correctness)
+
+
+
+
+lemma L03:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ bmkeps (bsimp (bder c (bsimp a)))"
+ using assms
+ by (metis L0 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L04:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
+ using assms
+ by (metis L0 L_bsimp_erase bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L05:
+ assumes "bnullable (bder c a)"
+ shows "retrieve (bder c (bsimp a)) (mkeps (erase (bder c (bsimp a)))) =
+ retrieve (bsimp (bder c (bsimp a))) (mkeps (erase (bsimp (bder c (bsimp a)))))"
+ using assms
+ using L04 by auto
+
+lemma L06:
+ assumes "bnullable (bder c a)"
+ shows "bmkeps (bder c (bsimp a)) = bmkeps (bsimp (bder c (bsimp a)))"
+ using assms
+ by (metis L03 L_bsimp_erase bmkeps_retrieve bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L07:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))
+ = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ using assms
+ using LB LC lexer_correct_Some by auto
+
+lemma LXXX:
+ assumes "s \<in> (erase r) \<rightarrow> v" "s \<in> (erase (bsimp r)) \<rightarrow> v'"
+ shows "retrieve r v = retrieve (bsimp r) v'"
+ using assms
+ apply -
+ thm LC
+ apply(subst LC)
+ apply(assumption)
+ apply(subst L0[symmetric])
+ using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+ apply(subst (2) LC)
+ apply(assumption)
+ apply(subst (2) L0[symmetric])
+ using bnullable_correctness lexer_correctness(2) lexer_flex apply fastforce
+
+ oops
+
+
+lemma L07a:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve (bsimp r) (flex (erase (bsimp r)) id s (mkeps (ders s (erase (bsimp r)))))
+ = retrieve r (flex (erase r) id s (mkeps (ders s (erase r))))"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using L0a apply force
+ apply(drule_tac x="(bder a r)" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(drule sym)
+ apply(simp)
+ apply(subst (asm) bder_retrieve)
+ apply (metis Posix_Prf Posix_flex Posix_mkeps ders.simps(2) lexer_correct_None lexer_flex)
+ apply(simp only: flex_fun_apply)
+ apply(simp)
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+
+
+lemma L08:
+ assumes "s \<in> L (erase r)"
+ shows "retrieve (bders (bsimp r) s) (mkeps (erase (bders (bsimp r) s)))
+ = retrieve (bders r s) (mkeps (erase (bders r s)))"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using L0 bnullable_correctness nullable_correctness apply blast
+ apply(simp add: bders_append)
+ apply(drule_tac x="(bder a (bsimp r))" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(drule sym)
+ apply(simp)
+ apply(subst LA)
+ apply (metis L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
+ apply(subst LA)
+ using lexer_correct_None lexer_flex mkeps_nullable apply force
+
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+
+thm L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars] L07[no_vars]
+ oops
+
+lemma test:
+ assumes "s = [c]"
+ shows "retrieve (bders r s) v = XXX" and "YYY = retrieve r (flex (erase r) id s v)"
+ using assms
+ apply(simp only: bders.simps)
+ defer
+ using assms
+ apply(simp only: flex.simps id_simps)
+ using L0[no_vars] bder_retrieve[no_vars] LA[no_vars] LC[no_vars]
+ find_theorems "retrieve (bders _ _) _"
+ find_theorems "retrieve _ (mkeps _)"
+ oops
+
+lemma L06X:
+ assumes "bnullable (bder c a)"
+ shows "bmkeps (bder c (bsimp a)) = bmkeps (bder c a)"
+ using assms
+ apply(induct a arbitrary: c)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ prefer 2
+ apply(simp)
+
+ defer
+ apply(simp only: bnullable.simps)
+ oops
+
+lemma L06_2:
+ assumes "bnullable (bders a [c,d])"
+ shows "bmkeps (bders (bsimp a) [c,d]) = bmkeps (bsimp (bders (bsimp a) [c,d]))"
+ using assms
+ apply(simp)
+ by (metis L_bsimp_erase bmkeps_simp bnullable_correctness der_correctness erase_bder nullable_correctness)
+
+lemma L06_bders:
+ assumes "bnullable (bders a s)"
+ shows "bmkeps (bders (bsimp a) s) = bmkeps (bsimp (bders (bsimp a) s))"
+ using assms
+ by (metis L_bsimp_erase bmkeps_simp bnullable_correctness ders_correctness erase_bders nullable_correctness)
+
+lemma LLLL:
+ shows "L (erase a) = L (erase (bsimp a))"
+ and "L (erase a) = {flat v | v. \<Turnstile> v: (erase a)}"
+ and "L (erase a) = {flat v | v. \<Turnstile> v: (erase (bsimp a))}"
+ using L_bsimp_erase apply(blast)
+ apply (simp add: L_flat_Prf)
+ using L_bsimp_erase L_flat_Prf apply(auto)[1]
+ done
+
+
+
+lemma LXX_bders:
+ assumes ""
+ shows "bmkeps (bders (bsimp a) s) = bmkeps (bders a s)"
+ using assms
+ apply(induct s arbitrary: a rule: rev_induct)
+ apply(simp add: bmkeps_simp)
+ apply(simp add: bders_append)
+
+
+
+lemma L07:
+ assumes "s \<in> L (erase a)"
+ shows "s \<in> erase a \<rightarrow> flex (erase a) id s (mkeps (ders s (erase a)))"
+ using assms
+ by (meson lexer_correct_None lexer_correctness(1) lexer_flex)
+
+lemma LX0:
+ assumes "s \<in> L r"
+ shows "decode (bmkeps (bders (intern r) s)) r = Some(flex r id s (mkeps (ders s r)))"
+ by (metis assms blexer_correctness blexer_def lexer_correct_None lexer_flex)
+
+
+lemma L02_bders2:
+ assumes "bnullable (bders a s)" "s = [c]"
+ shows "retrieve (bders (bsimp a) s) (mkeps (erase (bders (bsimp a) s))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ apply(simp)
+
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using L0 apply auto[1]
+ apply(simp)
+
+thm bmkeps_retrieve bmkeps_simp Posix_mkeps
+
+lemma WQ1:
+ assumes "s \<in> L (der c r)"
+ shows "s \<in> der c r \<rightarrow> mkeps (ders s (der c r))"
+ using assms
+ sorry
+
+lemma L02_bsimp:
+ assumes "bnullable (bders a s)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (erase (bders (bsimp a) s)))) =
+ retrieve a (flex (erase a) id s (mkeps (erase (bders a s))))"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ apply (simp add: L0)
+ apply(simp)
+ apply(drule_tac x="bder a aa" in meta_spec)
+ apply(simp)
+ apply(subst (asm) bder_retrieve)
+ using Posix_Prf Posix_flex Posix_mkeps bnullable_correctness apply fastforce
+ apply(simp add: flex_fun_apply)
+ apply(drule sym)
+ apply(simp)
+ apply(subst flex_injval)
+ apply(subst bder_retrieve[symmetric])
+ apply (metis L_bsimp_erase Posix_Prf Posix_flex Posix_mkeps bders.simps(2) bnullable_correctness ders.simps(2) erase_bders lexer_correct_None lexer_flex option.distinct(1))
+ apply(simp only: erase_bder[symmetric] erase_bders[symmetric])
+ apply(subst LB_sym[symmetric])
+ apply(simp)
+ apply(rule WQ1)
+ apply(simp only: erase_bder[symmetric])
+ apply(rule L07)
+ apply (metis L_bsimp_erase bnullable_correctness der_correctness erase_bder erase_bders lexer_correct_None lexer_flex option.simps(3))
+
+ apply(simp)
+ (*sledgehammer [timeout = 6000]*)
+
+
+ using bder_retrieve
+
+
+
+lemma LX0MAIN:
+ assumes "s \<in> r \<rightarrow> v"
+ shows "decode (bmkeps (bders_simp (intern r) s)) r = Some(flex r id s v)"
+ using assms
+ apply(induct s arbitrary: r v)
+ apply(simp)
+ apply (metis LX0 Posix1(1) bders.simps(1) lexer_correctness(1) lexer_flex option.simps(3))
+ apply(simp add: bders_simp_append ders_append flex_append)
+ apply(subst bmkeps_simp[symmetric])
+ apply (met is L_bders_simp Posix1(1) bnullable_correctness der_correctness ders_snoc erase_bder erase_bders erase_intern lexer_correct_None lexer_flex nullable_correctness)
+ (*apply(subgoal_tac "v = flex r id xs (injval (ders xs r) x (mkeps (ders (xs @ [x]) r)))")
+ prefer 2
+ apply (metis Posix1(1) flex.simps(1) flex.simps(2) flex_append lexer_correct_None lexer_correctness(1) lexer_flex option.inject)
+*) apply(drule_tac x="r" in meta_spec)
+ apply(drule_tac x="(mkeps (ders xs r))" in meta_spec)
+ apply(drule meta_mp)
+
+ defer
+ apply(simp)
+ apply(drule sym)
+ apply(simp)
+ using bder_retrieve MAIN_decode
+ oops
+
+lemma LXaa:
+ assumes "s \<in> L (erase a)"
+ shows "decode (bmkeps (bsimp (bders (bsimp a) s))) (erase a) = decode (bmkeps (bders a s)) (erase a)"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using bmkeps_simp bnullable_correctness bsimp_idem nullable_correctness apply auto[1]
+ apply(simp)
+ apply(drule_tac x="bsimp (bder a (bsimp aa))" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(subst (asm) bsimp_idem)
+ apply(subst bmkeps_simp[symmetric])
+ apply (metis L0aa L_bsimp_erase Posix1(1) bders.simps(2) bnullable_correctness nullable_correctness)
+ apply(subst (asm) bmkeps_simp[symmetric])
+ apply (metis L0aa L_bsimp_erase Posix1(1) bnullable_correctness ders.simps(2) ders_correctness erase_bder erase_bders nullable_correctness)
+ apply(subst bmkeps_retrieve)
+ apply (metis L0aa L_bsimp_erase Posix1(1) bders.simps(2) nullable_correctness)
+ apply(subst LA)
+ apply(simp)
+ apply (metis L_bsimp_erase ders.simps(2) lexer_correct_None lexer_flex mkeps_nullable)
+
+
+
+lemma LXaa:
+ assumes "s \<in> L (erase a)"
+ shows "bmkeps (bsimp (bders (bsimp a) s)) = bmkeps (bders a s)"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using bmkeps_simp bnullable_correctness bsimp_idem nullable_correctness apply auto[1]
+ apply(simp)
+ apply(drule_tac x="bsimp (bder a (bsimp aa))" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(subst (asm) bsimp_idem)
+ apply(subst bmkeps_simp[symmetric])
+ apply (metis L0aa L_bsimp_erase Posix1(1) bders.simps(2) bnullable_correctness nullable_correctness)
+ apply(subst (asm) bmkeps_simp[symmetric])
+ apply (metis L0aa L_bsimp_erase Posix1(1) bnullable_correctness ders.simps(2) ders_correctness erase_bder erase_bders nullable_correctness)
+
+
+lemma LXaa:
+ assumes "s \<in> L (erase a)"
+ shows "bmkeps (bders (bsimp a) s) = bmkeps (bders a s)"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using bmkeps_simp bnullable_correctness nullable_correctness apply auto[1]
+ apply(simp)
+ apply(drule_tac x="bder a aa" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(drule sym)
+ apply(simp)
+ (* transformation to retrieve *)
+ apply(subst bmkeps_retrieve)
+ apply (metis L0aa L_bsimp_erase Posix1(1) bders.simps(2) ders_correctness erase_bders nullable_correctness)
+ apply(subst bmkeps_retrieve)
+ apply (metis b4 bders_simp.simps(2) bnullable_correctness erase_bders lexer_correct_None lexer_flex)
+ apply(subst (2) LC[symmetric])
+ prefer 2
+ apply(subst L0)
+ apply(subst LA)
+ apply(simp)
+ apply (m etis L_bsimp_erase bders.simps(2) erase_bder erase_bders lexer_correct_None lexer_flex mkeps_nullable)
+ apply(subst LA)
+ apply(simp)
+ apply (m etis L0aa b3 b4 bders_simp.simps(2) bnullable_correctness erase_bders lexer.simps(1) lexer_correctness(2) mkeps_nullable)
+ apply(subst (2) LB_sym[symmetric])
+ prefer 2
+ apply(subst L0)
+ apply(simp)
+ using lexer_correct_None lexer_flex apply fastforce
+ apply(subst (asm) bmkeps_retrieve)
+ apply (metis L_bsimp_erase bders.simps(2) erase_bders lexer_correct_None lexer_flex)
+ apply(subst (asm) bmkeps_retrieve)
+ apply (metis L0aa L_bsimp_erase b3 b4 bders_simp.simps(2) bnullable_correctness lexer.simps(1) lexer_correctness(2))
+ apply(subst LA)
+ apply (met is L0aa L_bsimp_erase Posix1(1) ders.simps(2) ders_correctness erase_bder erase_bders mkeps_nullable nullable_correctness)
+ apply(subst LA)
+ using lexer_correct_None lexer_flex mkeps_nullable apply force
+
+ using L0aaaa LB[OF L0aaaa]
+ apply(subst LB[OF L0aaaa])
+ using L0aaaa
+ apply(rule L0aaaa)
+
+
+
+
+
+
+
+
+lemma L00:
+ assumes "s \<in> L (erase a)"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (ders s (erase (bsimp a))))) =
+ retrieve a (flex (erase a) id s (mkeps (ders s (erase a))))"
+ using assms
+ apply(induct s arbitrary: a taking: length rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ using L0 b3 bnullable_correctness nullable_correctness apply blast
+ apply(simp)
+ apply(drule_tac x="[]" in spec)
+ apply(simp)
+ apply(drule_tac x="bders (bsimp a) (aa#list)" in spec)
+ apply(simp)
+ apply(drule mp)
+ apply (metis L_bsimp_erase ders.simps(2) erase_bder erase_bders lexer_correct_None lexer_flex nullable_correctness)
+ using bder_retrieve bmkeps_simp bmkeps_retrieve L0 LA LB LC L02 L03 L04 L05
+
+ oops
+
+lemma L00:
+ assumes "s \<in> L (erase (bsimp a))"
+ shows "retrieve (bsimp a) (flex (erase (bsimp a)) id s (mkeps (ders s (erase (bsimp a))))) =
+ retrieve a (flex (erase a) id s (mkeps (ders s (erase a))))"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ using L0 b3 bnullable_correctness nullable_correctness apply blast
+ apply(simp add: flex_append)
+ apply(drule_tac x="bder a aa" in meta_spec)
+ apply(drule meta_mp)
+ apply (metis L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(simp)
+
+ (*using bmkeps_retrieve bder_retrieve*)
+ apply(subst (asm) bder_retrieve)
+ apply (metis L_bsimp_erase Posix_Prf Posix_flex Posix_mkeps ders.simps(2) lexer_correct_None lexer_flex)
+ apply(simp add: flex_fun_apply)
+ apply(drule sym)
+ apply(simp)
+ using bmkeps_retrieve bder_retrieve
+ oops
+
+lemma L0a:
+ assumes "s \<in> L (erase a)"
+ shows "retrieve (bders_simp a s) (mkeps (erase (bders_simp a s))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ apply(induct s arbitrary: a)
+ apply(simp)
+ apply(simp)
+ apply(drule_tac x="bsimp (bder a aa)" in meta_spec)
+ apply(drule meta_mp)
+ using L_bsimp_erase lexer_correct_None apply fastforce
+ apply(simp)
+ apply(subst LA)
+ apply (metis L_bsimp_erase ders.simps(2) ders_correctness erase_bder lexer_correct_None lexer_flex mkeps_nullable nullable_correctness)
+ apply(subst LA)
+ apply(simp)
+ apply (metis ders.simps(2) lexer_correct_None lexer_flex mkeps_nullable)
+
+ apply(simp)
+ (* MAIN *)
+
+ apply(subst L00)
+ apply (met is L_bsimp_erase erase_bder lexer.simps(2) lexer_correct_None option.case(1))
+ apply(simp)
+ done
+
+lemma L0a:
+ assumes "s \<in> L (erase a)"
+ shows "retrieve (bders_simp a s) (mkeps (erase (bders_simp a s))) =
+ retrieve (bders a s) (mkeps (erase (bders a s)))"
+ using assms
+ apply(induct s arbitrary: a rule: rev_induct)
+ apply(simp)
+ apply(simp add: bders_simp_append)
+ apply(subst L0)
+ apply (metis L_bders_simp bnullable_correctness der_correctness ders_snoc erase_bder erase_bders lexer_correct_None lexer_flex nullable_correctness)
+ apply(subst bder_retrieve)
+ apply (metis L_bders_simp der_correctness ders_snoc erase_bder erase_bders lexer_correct_None lexer_flex mkeps_nullable nullable_correctness)
+ apply(simp add: bders_append)
+ apply(subst bder_retrieve)
+ apply (metis ders_snoc erase_bders lexer_correct_None lexer_flex mkeps_nullable)
+ apply(simp add: ders_append)
+ using bder_retrieve
+
+ find_theorems "retrieve _ (injval _ _ _)"
+find_theorems "mkeps (erase _)"
+
+
+lemma L1:
+ assumes "s \<in> r \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) s)) r = Some v"
+ using assms
+ by (metis blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))
+
+lemma L2:
+ assumes "s \<in> (der c r) \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) (c # s))) r = Some (injval r c v)"
+ using assms
+ apply(subst bmkeps_retrieve)
+ using Posix1(1) lexer_correct_None lexer_flex apply fastforce
+ using MAIN_decode
+ apply(subst MAIN_decode[symmetric])
+ apply(simp)
+ apply (meson Posix1(1) lexer_correct_None lexer_flex mkeps_nullable)
+ apply(simp)
+ apply(subgoal_tac "v = flex (der c r) id s (mkeps (ders s (der c r)))")
+ prefer 2
+ apply (metis Posix_determ lexer_correctness(1) lexer_flex option.distinct(1))
+ apply(simp)
+ apply(subgoal_tac "injval r c (flex (der c r) id s (mkeps (ders s (der c r)))) =
+ (flex (der c r) ((\<lambda>v. injval r c v) o id) s (mkeps (ders s (der c r))))")
+ apply(simp)
+ using flex_fun_apply by blast
+
+lemma L3:
+ assumes "s2 \<in> (ders s1 r) \<rightarrow> v"
+ shows "decode (bmkeps (bders (intern r) (s1 @ s2))) r = Some (flex r id s1 v)"
+ using assms
+ apply(induct s1 arbitrary: r s2 v rule: rev_induct)
+ apply(simp)
+ using L1 apply blast
+ apply(simp add: ders_append)
+ apply(drule_tac x="r" in meta_spec)
+ apply(drule_tac x="x # s2" in meta_spec)
+ apply(drule_tac x="injval (ders xs r) x v" in meta_spec)
+ apply(drule meta_mp)
+ defer
+ apply(simp)
+ apply(simp add: flex_append)
+ by (simp add: Posix_injval)
+
+lemma L4:
+ assumes "[c] \<in> r \<rightarrow> v"
+ shows "bmkeps (bder c (bsimp (intern r))) = code (flex r id [c] v)"
+ using assms
+ apply(subst bmkeps_retrieve)
+ apply (metis L_bsimp_erase Posix1(1) bders.simps(1) bders.simps(2) erase_bders erase_intern lexer_correct_None lexer_flex)
+ apply(subst bder_retrieve)
+ apply (metis L_bsimp_erase Posix1(1) bders.simps(1) bders.simps(2) erase_bder erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable)
+ apply(simp)
+
+
+
+
+
+lemma
+ assumes "s \<in> L r"
+ shows "bmkeps (bders_simp (intern r) s) = bmkeps (bders (intern r) s)"
+ using assms
+ apply(induct s arbitrary: r rule: rev_induct)
+ apply(simp)
+ apply(simp add: bders_simp_append bders_append)
+ apply(subst bmkeps_simp[symmetric])
+ apply (metis b4 bders.simps(1) bders.simps(2) bders_simp_append blexer_correctness blexer_def lexer_correct_None)
+ apply(subst bmkeps_retrieve)
+ apply(simp)
+ apply (metis L_bders_simp der_correctness ders_snoc erase_bders erase_intern lexer_correct_None lexer_flex nullable_correctness)
+ apply(subst bmkeps_retrieve)
+ apply(simp)
+ apply (metis ders_snoc lexer_correct_None lexer_flex)
+ apply(subst bder_retrieve)
+ apply (metis L_bders_simp der_correctness ders_snoc erase_bder erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable nullable_correctness)
+ apply(subst bder_retrieve)
+ apply (metis ders_snoc erase_bder erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable)
+ using bmkeps_retrieve
+ find_theorems "retrieve (bders _ _) = _"
+ find_theorems "retrieve _ _ = _"
+ oops
+
+lemma
+ assumes "s \<in> r \<rightarrow> v"
+ shows "decode (bmkeps (bders_simp (intern r) s)) = Some v"
+ using assms
+ apply(induct s arbitrary: r v taking: length rule: measure_induct)
+ apply(subgoal_tac "x = [] \<or> (\<exists>a xs. x = xs @ [a])")
+ prefer 2
+ apply (meson rev_exhaust)
+ apply(erule disjE)
+ apply(simp add: blexer_simp_def)
+ apply (metis Posix1(1) Posix_Prf Posix_determ Posix_mkeps bmkeps_retrieve erase_intern lexer.simps(1) lexer_correct_None retrieve_code)
+ apply(clarify)
+ apply(simp add: bders_simp_append)
+ apply(subst bmkeps_simp[symmetric])
+ apply (metis b3 b4 bders_simp.simps(1) bders_simp.simps(2) bders_simp_append blexer_correctness blexer_def lexer_correctness(1) option.distinct(1))
+ apply(subst bmkeps_retrieve)
+ apply (metis L_bders_simp Posix1(1) der_correctness ders_snoc erase_bder erase_bders erase_intern lexer_correct_None lexer_flex nullable_correctness)
+ apply(simp)
+ apply(subst bder_retrieve)
+ apply (metis L_bders_simp Posix1(1) der_correctness ders_snoc erase_bders erase_intern lexer_correct_None lexer_flex mkeps_nullable nullable_correctness)
+
+
+ find_theorems "bmkeps _ = _"
+
+ apply(subgoal_tac "lexer r (xs @ [a]) = Some v")
+ prefer 2
+ using lexer_correctness(1) apply blast
+ apply(simp (no_asm_simp) add: bders_simp_append)
+ apply(subst bmkeps_simp[symmetric])
+ apply (m etis b4 bders.simps(1) bders.simps(2) bders_simp_append bnullable_correctness erase_bders erase_intern lexer_flex option.distinct(1))
+ apply(subgoal_tac "nullable (ders (xs @ [a]) r)")
+ prefer 2
+ apply (metis lexer_flex option.distinct(1))
+ apply(simp add: lexer_flex)
+ apply(simp add: flex_append ders_append)
+ apply(drule_tac sym)
+ apply(simp)
+ using Posix_flex flex.simps Posix_flex2
+ apply(drule_tac Posix_flex2)
+ apply (simp add: Prf_injval mkeps_nullable)
+ apply(drule_tac x="[a]" in spec)
+ apply(drule mp)
+ defer
+ apply(drule_tac x="ders xs r" in spec)
+ apply(drule_tac x="injval (ders xs r) a (mkeps (der a (ders xs r)))" in spec)
+ apply(simp)
+ apply(subst (asm) bmkeps_simp[symmetric])
+ using bnullable_correctness apply fastforce
+
+
+ find_theorems "good (bsimp _)"
+ oops
+
+lemma
+ assumes "s \<in> L (erase a)"
+ shows "bmkeps (bders (bsimp a) s) = bmkeps (bders a s)"
+ using assms
+ apply(induct s arbitrary: a taking : length rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ using bmkeps_simp bnullable_correctness nullable_correctness apply auto[1]
+ using bsimp_idem apply auto[1]
+ apply(simp add: bders_append bders_simp_append)
+ apply(drule_tac x="bder a (bsimp aa)" in meta_spec)
+ apply(drule meta_mp)
+ defer
+ apply(simp)
+ oops
+
+
+lemma
+ assumes "s \<in> L (erase r)"
+ shows "bmkeps (bders_simp r s) = bmkeps (bders r s)"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ apply(simp add: bders_simp_append bders_append)
+ apply(drule_tac x="bsimp (bder a r)" in meta_spec)
+ apply(drule meta_mp)
+ defer
+ apply(simp)
+
+ find_theorems "good (bsimp _)"
+ oops
+
+lemma
+ assumes "s \<in> L r"
+ shows "decode (bmkeps (bders_simp (intern r) s)) r = Some (flex r id s (mkeps (ders s r)))"
+ using assms
+ apply(induct s arbitrary: r)
+ apply(simp)
+ using bmkeps_retrieve decode_code mkeps_nullable nullable_correctness retrieve_code apply auto[1]
+ apply(simp add: bders_simp_append)
+ apply(subst bmkeps_simp[symmetric])
+ apply(subgoal_tac "[] \<in> L (ders (xs @ [x]) r)")
+ prefer 2
+ apply (meson Posix1(1) lexer_correct_None lexer_flex nullable_correctness)
+ apply(simp add: ders_append)
+ using L_bders_simp bnullable_correctness der_correctness nullable_correctness apply f orce
+ apply(simp add: flex_append ders_append)
+ apply(drule_tac x="der x r" in meta_spec)
+ apply(simp add: ders_append)
+ find_theorems "intern _"
+ find_theorems "bmkeps (bsimp _)"
+
+ find_theorems "(_ # _) \<in> _ \<rightarrow> _"
+ oops
+
lemma ZZZ:
assumes "\<forall>y. asize y < Suc (sum_list (map asize x52)) \<longrightarrow> bsimp (bder c (bsimp y)) = bsimp (bder c y)"
@@ -1899,39 +2646,6 @@
apply(simp add: bders_append)
done
-lemma
- assumes "bnullable (bders a s)" "good a"
- shows "bmkeps (bders_simp a s) = bmkeps (bders a s)"
- using assms
- apply(induct s arbitrary: a)
- apply(simp)
- apply(simp add: bders_append bders_simp_append)
- apply(drule_tac x="bsimp (bsimp (bder a aa))" in meta_spec)
- apply(drule meta_mp)
- apply (metis b4 bders.simps(2) bders_simp.simps(2) bsimp_idem)
- apply(subgoal_tac "good (bsimp (bder a aa)) \<or> bsimp (bder a aa) = AZERO")
- apply(auto simp add: bders_AZERO)
- prefer 2
- apply (metis b4 bders.simps(2) bders_AZERO(2) bders_simp.simps(2) bnullable.simps(1))
- prefer 2
- using good1 apply auto[1]
- apply(drule meta_mp)
- apply (simp add: bsimp_idem)
- apply (subst (asm) (1) bsimp_idem)
- apply(simp)
- apply(subst (asm) (2) bmkeps_simp)
- apply (metis b4 bders.simps(2) bders_simp.simps(2) bsimp_idem)
- apply (subst (asm) (1) bsimp_idem)
- oops
-
-
-lemma
- shows "bmkeps (bders (bders_simp a s2) s1) = bmkeps (bders_simp a (s2 @ s1))"
- apply(induct s2 arbitrary: a s1)
- apply(simp)
- defer
- apply(simp add: bders_append bders_simp_append)
- oops
lemma QQ1:
shows "bsimp (bders (bsimp a) []) = bders_simp (bsimp a) []"
@@ -2039,19 +2753,6 @@
apply (metis n0 nn1b test2)
by (metis flts_fuse flts_nothing)
-lemma "good (bsimp a) \<or> bsimp a = AZERO"
- by (simp add: good1)
-
-
-lemma
- assumes "bnullable (bders r s)" "good r"
- shows "bmkeps (bders (bsimp r) s) = bmkeps (bders r s)"
- using assms
- apply(induct s arbitrary: r)
- apply(simp)
- using bmkeps_simp apply auto[1]
- apply(simp)
- by (simp add: test2)
lemma PP:
assumes "bnullable (bders r s)"
@@ -2101,7 +2802,7 @@
using assms
apply(induct x52 arbitrary: x51)
apply(simp)
-
+ oops
lemma
@@ -2160,15 +2861,7 @@
apply (metis bsimp_AALTs.simps(3) neq_Nil_conv)
apply(auto)
apply (metis add.commute bsimp_size leD le_neq_implies_less less_add_Suc1 less_add_eq_less rt)
-
- apply(drule_tac x="AALTs x51 list" in spec)
- apply(drule mp)
- apply(auto simp add: asize0)[1]
-
-
-
-(* HERE*)
-
+ oops
@@ -2211,94 +2904,133 @@
prefer 2
apply (metis bbbbs1 bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) good.simps(5) good1 length_0_conv length_Suc_conv less_one list.simps(8) nat_neq_iff not_less_eq)
apply(auto)
- sledgehammer [timeout=6000]
-
-
-
-
- defer
- apply(case_tac list)
- apply(simp)
- prefer 2
- apply(simp)
- apply (simp add: bsimp_fuse bsimp_idem)
- apply(case_tac a)
- apply(simp_all)
+ oops
+
+
+lemma
+ assumes "rs = [AALTs bsa [AONE bsb, AONE bsb]]"
+ shows "bsimp (bsimp_AALTs bs rs) = bsimp_AALTs bs (flts (map bsimp rs))"
+ using assms
+ apply(simp)
+ oops
+
+lemma
+ assumes "rs = [AALTs bs0 [AALTs bsa [AONE bsb, AONE bsb]]]"
+ shows "flts [bsimp_AALTs bs (flts rs)] = flts (map (fuse bs) rs)"
+ using assms
+ apply(simp only:)
+ apply(simp only: flts.simps append_Nil2 List.list.map fuse.simps)
+ apply(simp only: bsimp_AALTs.simps)
+ apply(simp only: fuse.simps)
+ apply(simp only: flts.simps append_Nil2)
+ find_theorems "nonnested (bsimp_AALTs _ _)"
+ find_theorems "map _ (_ # _) = _"
+ apply(induct rs arbitrary: bs rule: flts.induct)
+ apply(auto)
+ apply(case_tac rs)
+ apply(simp)
-
- apply(subst k0b)
- apply(simp)
- apply(subst (asm) k0)
- apply(subst (asm) (2) k0)
-
-
- find_theorems "asize _ < asize _"
- using bsimp_AALTs_size3
- apply -
- apply(drule_tac x="a # list" in meta_spec)
- apply(drule_tac x="bs" in meta_spec)
- apply(drule meta_mp)
+ oops
+
+ find_theorems "asize (bsimp _)"
+
+lemma CT1:
+ shows "bsimp (AALTs bs as) = bsimp(AALTs bs (map bsimp as))"
+ apply(induct as arbitrary: bs)
apply(simp)
apply(simp)
- apply(drule_tac x="(bsimp a) # list" in spec)
- apply(drule mp)
+ by (simp add: bsimp_idem comp_def)
+
+lemma CT1a:
+ shows "bsimp (AALT bs a1 a2) = bsimp(AALT bs (bsimp a1) (bsimp a2))"
+ by (metis CT1 list.simps(8) list.simps(9))
+
+(* CT *)
+
+lemma CTU:
+ shows "bsimp_AALTs bs as = li bs as"
+ apply(induct bs as rule: li.induct)
+ apply(auto)
+ done
+
+thm flts.simps
+
+
+lemma CTa:
+ assumes "\<forall>r \<in> set as. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts as = as"
+ using assms
+ apply(induct as)
apply(simp)
- apply(case_tac list)
- apply(simp)
- prefer 2
- apply(simp)
- apply(subst (asm) k0)
- apply(subst (asm) (2) k0)
- thm k0
- apply(subst (asm) k0b)
- apply(simp)
- apply(simp)
-
- defer
+ apply(case_tac as)
apply(simp)
- apply(case_tac list)
- apply(simp)
- defer
+ apply (simp add: k0b)
+ using flts_nothing by auto
+
+lemma CT0:
+ assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts [bsimp_AALTs bs1 as1] = flts (map (fuse bs1) as1)"
+ using assms CTa
+ apply(induct as1 arbitrary: bs1)
apply(simp)
- find_theorems "asize _ < asize _"
- find_theorems "asize _ < asize _"
-apply(subst k0b)
- apply(simp)
- apply(simp)
-
-
- apply(rule nn11a)
- apply(simp)
- apply (metis good.simps(1) good1 good_fuse)
- apply(simp)
- using fuse_append apply blast
- apply(subgoal_tac "\<exists>bs rs. bsimp x43 = AALTs bs rs")
- prefer 2
- using nonalt.elims(3) apply blast
- apply(clarify)
+ apply(simp)
+ apply(case_tac as1)
apply(simp)
- apply(case_tac rs)
- apply(simp)
- apply (metis arexp.distinct(7) good.simps(4) good1)
- apply(simp)
- apply(case_tac list)
- apply(simp)
- apply (metis arexp.distinct(7) good.simps(5) good1)
apply(simp)
+proof -
+fix a :: arexp and as1a :: "arexp list" and bs1a :: "bit list" and aa :: arexp and list :: "arexp list"
+ assume a1: "nonalt a \<and> a \<noteq> AZERO \<and> nonalt aa \<and> aa \<noteq> AZERO \<and> (\<forall>r\<in>set list. nonalt r \<and> r \<noteq> AZERO)"
+ assume a2: "\<And>as. \<forall>r\<in>set as. nonalt r \<and> r \<noteq> AZERO \<Longrightarrow> flts as = as"
+ assume a3: "as1a = aa # list"
+ have "flts [a] = [a]"
+using a1 k0b by blast
+then show "fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list = flts (fuse bs1a a # fuse bs1a aa # map (fuse bs1a) list)"
+ using a3 a2 a1 by (metis (no_types) append.left_neutral append_Cons flts_fuse k00 k0b list.simps(9))
+qed
-
-
-
+lemma CT01:
+ assumes "\<forall>r \<in> set as1. nonalt r \<and> r \<noteq> AZERO" "\<forall>r \<in> set as2. nonalt r \<and> r \<noteq> AZERO"
+ shows "flts [bsimp_AALTs bs1 as1, bsimp_AALTs bs2 as2] = flts ((map (fuse bs1) as1) @ (map (fuse bs2) as2))"
+ using assms CT0
+ by (metis k0 k00)
- find_theorems "flts [_] = [_]"
- apply(case_tac x52)
+
+lemma CTT:
+ assumes "\<forall>r\<in>set (flts (map (bsimp \<circ> bder c) as1)). nonalt r \<and> r \<noteq> AZERO"
+ "\<forall>r\<in>set (flts (map (bsimp \<circ> bder c) as2)). nonalt r \<and> r \<noteq> AZERO"
+ shows "bsimp (AALT bs (AALTs bs1 (map (bder c) as1)) (AALTs bs2 (map (bder c) as2)))
+ = XXX"
+ apply(subst bsimp_idem[symmetric])
apply(simp)
- apply(simp)
- apply(case_tac list)
- apply(simp)
- apply(case_tac a)
- apply(simp_all)
+ apply(subst CT01)
+ apply(rule assms)
+ apply(rule assms)
+ (* CT *)
+
+lemma
+ shows "bsimp (AALT bs (AALTs bs1 (map (bder c) as1)) (AALTs bs2 (map (bder c) as2)))
+ = bsimp (AALTs bs ((map (fuse bs1) (map (bder c) as1)) @
+ (map (fuse bs2) (map (bder c) as2))))"
+ apply(subst bsimp_idem[symmetric])
+ apply(simp)
+ oops
+
+lemma CT_exp:
+ assumes "\<forall>a \<in> set as. bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ shows "map bsimp (map (bder c) as) = map bsimp (map (bder c) (map bsimp as))"
+ using assms
+ apply(induct as)
+ apply(auto)
+ done
+
+lemma asize_set:
+ assumes "a \<in> set as"
+ shows "asize a < Suc (sum_list (map asize as))"
+ using assms
+ apply(induct as arbitrary: a)
+ apply(auto)
+ using le_add2 le_less_trans not_less_eq by blast
lemma XXX2a_long_without_good:
@@ -2311,15 +3043,191 @@
prefer 3
apply(simp)
(* AALT case *)
+ prefer 2
+ apply(simp del: bsimp.simps)
+ apply(subst (2) CT1)
+ apply(subst CT_exp)
+ apply(auto)[1]
+ using asize_set apply blast
+ apply(subst CT1[symmetric])
+ apply(simp)
+ oops
+
+lemma big:
+ shows "bsimp (AALT x51 (AALTs bs1 as1) (AALTs bs2 as2)) =
+ bsimp (AALTs x51 ((map (fuse bs1) as1) @ (map (fuse bs2) as2)))"
+ apply(simp add: comp_def bder_fuse)
+ apply(simp add: flts_append)
+
+ sorry
+
+lemma XXX2a_long_without_good:
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ (* AALT case *)
prefer 2
+ apply(case_tac "\<exists>a1 a2. x52 = [a1, a2]")
+ apply(clarify)
+ apply(simp del: bsimp.simps)
+ apply(subst (2) CT1)
+ apply(simp del: bsimp.simps)
+ apply(rule_tac t="bsimp (bder c a1)" and s="bsimp (bder c (bsimp a1))" in subst)
+ apply(simp del: bsimp.simps)
+ apply(rule_tac t="bsimp (bder c a2)" and s="bsimp (bder c (bsimp a2))" in subst)
+ apply(simp del: bsimp.simps)
+ apply(subst CT1a[symmetric])
+ apply(subst bsimp.simps)
+ apply(simp del: bsimp.simps)
+ apply(case_tac "\<exists>bs1 as1. bsimp a1 = AALTs bs1 as1")
+ apply(case_tac "\<exists>bs2 as2. bsimp a2 = AALTs bs2 as2")
+ apply(clarify)
+ apply(simp only:)
+ apply(simp del: bsimp.simps bder.simps)
+ apply(subst bsimp_AALTs_qq)
+ prefer 2
+ apply(simp del: bsimp.simps)
+ apply(subst big)
+ prefer 2
+ apply (metis (no_types, lifting) Nil_is_append_conv Nil_is_map_conv One_nat_def Suc_lessI arexp.distinct(7) bsimp.simps(2) bsimp_AALTs.simps(1) bsimp_idem flts.simps(1) length_0_conv length_append length_greater_0_conv less_Suc0 less_add_same_cancel1)
+ apply(simp add: comp_def bder_fuse)
+
+ thm bder_fuse
+
+ find_theorems "1 < length _"
+
+(* CT *)
+(*
+ under assumption that bsimp a1 = AALT; bsimp a = AALT
+
+ bsimp (AALT x51 (AALTs bs1 (map (bder c) as1)) (AALTs bs2 (map (bder c) as2))) =
+ bsimp (AALTs x51 (map (fuse bs1) (map (bder c) as1)) @ ( map (fuse bs2) (map (bder c) as2)))
+
+ bsimp_AALTs x51 (flts (map bsimp [a1, a2])))
+ = bsimp_AALTs x51 (flts [bsimp a1, bsimp a2])
+
+*)
+
+ apply(case_tac "bsimp a1")
+ prefer 5
+ apply(simp only:)
+ apply(case_tac "bsimp a2")
+ apply(simp)
+
+ prefer 5
+ apply(simp only:)
+ apply(simp only: bder.simps)
+ apply(simp)
+
+
+
+
+ prefer 2
+ using nn1b apply blast
apply(simp only:)
- apply(simp)
+ apply(case_tac x52)
+ apply(simp)
+ apply(simp only:)
+ apply(case_tac "\<exists>bsa rsa. a = AALTs bsa rsa")
+ apply(clarify)
+ apply(simp)
+ apply(frule_tac x="AALTs x51 ((map (fuse bsa) rsa) @ list)" in spec)
+ apply(drule mp)
+ apply(simp)
+ apply (simp add: qq)
+ apply(drule_tac x="c" in spec)
+ apply(simp only: bder.simps)
+ apply(simp)
+ apply(subst k0)
+ apply(subst (2) k0)
+ apply(subst (asm) (2) flts_append)
+ apply(rotate_tac 2)
+
+
+lemma XXX2a_long_without_good:
+ shows "bsimp (bder c (bsimp a)) = bsimp (bder c a)"
+ apply(induct a arbitrary: c taking: "\<lambda>a. asize a" rule: measure_induct)
+ apply(case_tac x)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ prefer 3
+ apply(simp)
+ (* AALT case *)
+ prefer 2
+ apply(subgoal_tac "nonnested (bsimp x)")
+ prefer 2
+ using nn1b apply blast
+ apply(simp only:)
+ apply(drule_tac x="AALTs x51 (flts x52)" in spec)
+ apply(drule mp)
+ defer
+ apply(drule_tac x="c" in spec)
+ apply(simp)
+ apply(rotate_tac 2)
+
+ apply(drule sym)
+ apply(simp)
+
+ apply(simp only: bder.simps)
+ apply(simp only: bsimp.simps)
apply(subst bder_bsimp_AALTs)
apply(case_tac x52)
apply(simp)
apply(simp)
+ apply(case_tac list)
+ apply(simp)
+ apply(case_tac a)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+ defer
+ apply(simp)
+
+
+ (* case AALTs list is not empty *)
+ apply(simp)
+ apply(subst k0)
+ apply(subst (2) k0)
+ apply(simp)
+ apply(case_tac "bsimp a = AZERO")
+ apply(subgoal_tac "bsimp (bder c a) = AZERO")
+ prefer 2
+ using less_iff_Suc_add apply auto[1]
+ apply(simp)
+ apply(drule_tac x="AALTs x51 list" in spec)
+ apply(drule mp)
+ apply(simp add: asize0)
+ apply(drule_tac x="c" in spec)
+ apply(simp add: bder_bsimp_AALTs)
+ apply(case_tac "nonalt (bsimp a)")
+ prefer 2
+ apply(drule_tac x="bsimp (AALTs x51 (a#list))" in spec)
+ apply(drule mp)
+ apply(rule order_class.order.strict_trans2)
+ apply(rule bsimp_AALTs_size3)
+ apply(auto)[1]
+ apply(simp)
+ apply(subst (asm) bsimp_idem)
+ apply(drule_tac x="c" in spec)
+ apply(simp)
+ find_theorems "_ < _ \<Longrightarrow> _ \<le> _ \<Longrightarrow>_ < _"
+ apply(rule le_trans)
+ apply(subgoal_tac "flts [bsimp a] = [bsimp a]")
+ prefer 2
+ using k0b apply blast
+ apply(simp)
+ find_theorems "asize _ < asize _"
+
+ using bder_bsimp_AALTs
apply(case_tac list)
- apply(simp)
+ apply(simp)
+ sledgeha mmer [timeout=6000]
apply(case_tac "\<exists>r \<in> set (map bsimp x52). \<not>nonalt r")
apply(drule_tac x="bsimp (AALTs x51 x52)" in spec)
@@ -2581,6 +3489,19 @@
apply(simp only: bders.simps)
apply(subst bders_simp.simps)
apply(simp)
-
+ oops
+
+
+lemma
+ fixes n :: nat
+ shows "(\<Sum>i \<in> {0..n}. i) = n * (n + 1) div 2"
+ apply(induct n)
+ apply(simp)
+ apply(simp)
+ done
+
+
+
+
end
\ No newline at end of file