--- a/thys2/ClosedForms.thy Fri Apr 01 23:18:00 2022 +0100
+++ b/thys2/ClosedForms.thy Sun Apr 03 22:12:27 2022 +0100
@@ -2,129 +2,6 @@
"BasicIdentities"
begin
-
-lemma idem_after_simp1:
- shows "rsimp_ALTs (rdistinct (rflts [rsimp aa]) {}) = rsimp aa"
- apply(case_tac "rsimp aa")
- apply simp+
- apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
- by simp
-
-lemma identity_wwo0:
- shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
- by (metis idem_after_simp1 list.exhaust list.simps(8) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
-
-
-lemma distinct_removes_last:
- shows "\<lbrakk>a \<in> set as\<rbrakk>
- \<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
-and "rdistinct (ab # as @ [ab]) rset1 = rdistinct (ab # as) rset1"
- apply(induct as arbitrary: rset ab rset1 a)
- apply simp
- apply simp
- apply(case_tac "aa \<in> rset")
- apply(case_tac "a = aa")
- apply (metis append_Cons)
- apply simp
- apply(case_tac "a \<in> set as")
- apply (metis append_Cons rdistinct.simps(2) set_ConsD)
- apply(case_tac "a = aa")
- prefer 2
- apply simp
- apply (metis append_Cons)
- apply(case_tac "ab \<in> rset1")
- prefer 2
- apply(subgoal_tac "rdistinct (ab # (a # as) @ [ab]) rset1 =
- ab # (rdistinct ((a # as) @ [ab]) (insert ab rset1))")
- prefer 2
- apply force
- apply(simp only:)
- apply(subgoal_tac "rdistinct (ab # a # as) rset1 = ab # (rdistinct (a # as) (insert ab rset1))")
- apply(simp only:)
- apply(subgoal_tac "rdistinct ((a # as) @ [ab]) (insert ab rset1) = rdistinct (a # as) (insert ab rset1)")
- apply blast
- apply(case_tac "a \<in> insert ab rset1")
- apply simp
- apply (metis insertI1)
- apply simp
- apply (meson insertI1)
- apply simp
- apply(subgoal_tac "rdistinct ((a # as) @ [ab]) rset1 = rdistinct (a # as) rset1")
- apply simp
- by (metis append_Cons insert_iff insert_is_Un rdistinct.simps(2))
-
-
-lemma distinct_removes_middle:
- shows "\<lbrakk>a \<in> set as\<rbrakk>
- \<Longrightarrow> rdistinct (as @ as2) rset = rdistinct (as @ [a] @ as2) rset"
-and "rdistinct (ab # as @ [ab] @ as3) rset1 = rdistinct (ab # as @ as3) rset1"
- apply(induct as arbitrary: rset rset1 ab as2 as3 a)
- apply simp
- apply simp
- apply(case_tac "a \<in> rset")
- apply simp
- apply metis
- apply simp
- apply (metis insertI1)
- apply(case_tac "a = ab")
- apply simp
- apply(case_tac "ab \<in> rset")
- apply simp
- apply presburger
- apply (meson insertI1)
- apply(case_tac "a \<in> rset")
- apply (metis (no_types, opaque_lifting) Un_insert_left append_Cons insert_iff rdistinct.simps(2) sup_bot_left)
- apply(case_tac "ab \<in> rset")
- apply simp
- apply (meson insert_iff)
- apply simp
- by (metis insertI1)
-
-
-lemma distinct_removes_middle3:
- shows "\<lbrakk>a \<in> set as\<rbrakk>
- \<Longrightarrow> rdistinct (as @ a #as2) rset = rdistinct (as @ as2) rset"
- using distinct_removes_middle(1) by fastforce
-
-lemma distinct_removes_last2:
- shows "\<lbrakk>a \<in> set as\<rbrakk>
- \<Longrightarrow> rdistinct as rset = rdistinct (as @ [a]) rset"
- by (simp add: distinct_removes_last(1))
-
-lemma distinct_removes_middle2:
- shows "a \<in> set as \<Longrightarrow> rdistinct (as @ [a] @ rs) {} = rdistinct (as @ rs) {}"
- by (metis distinct_removes_middle(1))
-
-lemma distinct_removes_list:
- shows "\<lbrakk> \<forall>r \<in> set rs. r \<in> set as\<rbrakk> \<Longrightarrow> rdistinct (as @ rs) {} = rdistinct as {}"
- apply(induct rs)
- apply simp+
- apply(subgoal_tac "rdistinct (as @ a # rs) {} = rdistinct (as @ rs) {}")
- prefer 2
- apply (metis append_Cons append_Nil distinct_removes_middle(1))
- by presburger
-
-
-lemma spawn_simp_rsimpalts:
- shows "rsimp (rsimp_ALTs rs) = rsimp (rsimp_ALTs (map rsimp rs))"
- apply(cases rs)
- apply simp
- apply(case_tac list)
- apply simp
- apply(subst rsimp_idem[symmetric])
- apply simp
- apply(subgoal_tac "rsimp_ALTs rs = RALTS rs")
- apply(simp only:)
- apply(subgoal_tac "rsimp_ALTs (map rsimp rs) = RALTS (map rsimp rs)")
- apply(simp only:)
- prefer 2
- apply simp
- prefer 2
- using rsimp_ALTs.simps(3) apply presburger
- apply auto
- apply(subst rsimp_idem)+
- by (metis comp_apply rsimp_idem)
-
lemma map_concat_cons:
shows "map f rsa @ f a # rs = map f (rsa @ [a]) @ rs"
by simp
@@ -133,98 +10,14 @@
shows " \<not> a \<notin> aset \<Longrightarrow> a \<in> aset"
by simp
-lemma flts_removes0:
- shows " rflts (rs @ [RZERO]) =
- rflts rs"
- apply(induct rs)
- apply simp
- by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-
-lemma flts_keeps1:
- shows " rflts (rs @ [RONE]) =
- rflts rs @ [RONE] "
- apply (induct rs)
- apply simp
- by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-
-lemma flts_keeps_others:
- shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow>rflts (rs @ [a]) = rflts rs @ [a]"
- apply(induct rs)
- apply simp
- apply (simp add: rflts_def_idiot)
- apply(case_tac a)
- apply simp
- using flts_keeps1 apply blast
- apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
- apply (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
- apply blast
- by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-lemma rflts_def_idiot2:
- shows "\<lbrakk>a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1; a \<in> set rs\<rbrakk> \<Longrightarrow> a \<in> set (rflts rs)"
- apply(induct rs)
- apply simp
- by (metis append.assoc in_set_conv_decomp insert_iff list.simps(15) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-
-lemma rflts_spills_last:
- shows "a = RALTS rs \<Longrightarrow> rflts (rs1 @ [a]) = rflts rs1 @ rs"
- apply (induct rs1)
- apply simp
- by (metis append.assoc append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
-lemma spilled_alts_contained:
- shows "\<lbrakk>a = RALTS rs ; a \<in> set rs1\<rbrakk> \<Longrightarrow> \<forall>r \<in> set rs. r \<in> set (rflts rs1)"
- apply(induct rs1)
- apply simp
- apply(case_tac "a = aa")
- apply simp
- apply(subgoal_tac " a \<in> set rs1")
- prefer 2
- apply (meson set_ConsD)
- apply(case_tac aa)
- using rflts.simps(2) apply presburger
- apply fastforce
- apply fastforce
- apply fastforce
- apply fastforce
- by fastforce
+
-lemma distinct_removes_duplicate_flts:
- shows " a \<in> set rsa
- \<Longrightarrow> rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
- rdistinct (rflts (map rsimp rsa)) {}"
- apply(subgoal_tac "rsimp a \<in> set (map rsimp rsa)")
- prefer 2
- apply simp
- apply(induct "rsimp a")
- apply simp
- using flts_removes0 apply presburger
- apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
- rdistinct (rflts (map rsimp rsa @ [RONE])) {}")
- apply (simp only:)
- apply(subst flts_keeps1)
- apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.simps(20) rrexp.simps(6))
- apply presburger
- apply(subgoal_tac " rdistinct (rflts (map rsimp rsa @ [rsimp a])) {} =
- rdistinct ((rflts (map rsimp rsa)) @ [RCHAR x]) {}")
- apply (simp only:)
- prefer 2
- apply (metis flts_keeps_others rrexp.distinct(21) rrexp.distinct(3))
- apply (metis distinct_removes_last2 rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
- apply (metis distinct_removes_last2 flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
- prefer 2
- apply (metis distinct_removes_last2 flts_keeps_others flts_removes0 rflts_def_idiot2 rrexp.distinct(29))
- apply(subgoal_tac "rflts (map rsimp rsa @ [rsimp a]) = rflts (map rsimp rsa) @ x")
- prefer 2
- apply (simp add: rflts_spills_last)
- apply(simp only:)
- apply(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
- prefer 2
- using spilled_alts_contained apply presburger
- by (metis append_self_conv distinct_removes_list in_set_conv_decomp rev_exhaust)
+
lemma flts_middle0:
shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
@@ -246,13 +39,7 @@
apply simp+
done
-lemma flts_append:
- shows "rflts (rs1 @ rs2) = rflts rs1 @ rflts rs2"
- apply(induct rs1)
- apply simp
- apply(case_tac a)
- apply simp+
- done
+
lemma simp_removes_duplicate1:
shows " a \<in> set rsa \<Longrightarrow> rsimp (RALTS (rsa @ [a])) = rsimp (RALTS (rsa))"
@@ -351,18 +138,6 @@
-lemma distinct_flts_no0:
- shows " rflts (map rsimp (rdistinct rs (insert RZERO rset))) =
- rflts (map rsimp (rdistinct rs rset)) "
-
- apply(induct rs arbitrary: rset)
- apply simp
- apply(case_tac a)
- apply simp+
- apply (smt (verit, ccfv_SIG) rflts.simps(2) rflts.simps(3) rflts_def_idiot)
- prefer 2
- apply simp
- by (smt (verit, ccfv_threshold) Un_insert_right insert_iff list.simps(9) rdistinct.simps(2) rflts.simps(2) rflts.simps(3) rflts_def_idiot rrexp.distinct(7))
@@ -387,12 +162,20 @@
| "rs1 \<leadsto>g rs2 \<Longrightarrow> (r # rs1) \<leadsto>g (r # rs2)"
| "rsa @ [a] @ rsb @ [a] @ rsc \<leadsto>g rsa @ [a] @ rsb @ rsc"
+lemma grewrite_variant1:
+ shows "a \<in> set rs1 \<Longrightarrow> rs1 @ a # rs \<leadsto>g rs1 @ rs"
+ apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
+ done
+
inductive
grewrites:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ \<leadsto>g* _" [10, 10] 10)
where
[intro, simp]:"rs \<leadsto>g* rs"
| [intro]: "\<lbrakk>rs1 \<leadsto>g* rs2; rs2 \<leadsto>g rs3\<rbrakk> \<Longrightarrow> rs1 \<leadsto>g* rs3"
+
+
+
(*
inductive
frewrites2:: "rrexp list \<Rightarrow> rrexp list \<Rightarrow> bool" ("_ <\<leadsto>f* _" [10, 10] 10)
@@ -431,6 +214,26 @@
lemma gmany_steps_later: "\<lbrakk>r1 \<leadsto>g r2; r2 \<leadsto>g* r3 \<rbrakk> \<Longrightarrow> r1 \<leadsto>g* r3"
by (meson gr_in_rstar greal_trans)
+lemma gstar_rdistinct_general:
+ shows "rs1 @ rs \<leadsto>g* rs1 @ (rdistinct rs (set rs1))"
+ apply(induct rs arbitrary: rs1)
+ apply simp
+ apply(case_tac " a \<in> set rs1")
+ apply simp
+ apply(subgoal_tac "rs1 @ a # rs \<leadsto>g rs1 @ rs")
+ using gmany_steps_later apply auto[1]
+ apply (metis append.assoc append_Cons append_Nil grewrite.intros(4) split_list_first)
+ apply simp
+ apply(drule_tac x = "rs1 @ [a]" in meta_spec)
+ by simp
+
+
+lemma gstar_rdistinct:
+ shows "rs \<leadsto>g* rdistinct rs {}"
+ apply(induct rs)
+ apply simp
+ by (metis append.left_neutral empty_set gstar_rdistinct_general)
+
lemma frewrite_append:
@@ -540,29 +343,27 @@
apply simp
oops
+lemma grewrite_cases_middle:
+ shows "rs1 \<leadsto>g rs2 \<Longrightarrow>
+(\<exists>rsa rsb rsc. rs1 = (rsa @ [RALTS rsb] @ rsc) \<and> rs2 = (rsa @ rsb @ rsc)) \<or>
+(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc) \<or>
+(\<exists>rsa rsb rsc a. rs1 = rsa @ [a] @ rsb @ [a] @ rsc \<and> rs2 = rsa @ [a] @ rsb @ rsc)"
+ apply( induct rs1 rs2 rule: grewrite.induct)
+ apply simp
+ apply blast
+ apply (metis append_Cons append_Nil)
+ apply (metis append_Cons)
+ by blast
lemma grewrite_equal_rsimp:
shows "rs1 \<leadsto>g rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
- apply(induct rs1 rs2 rule: grewrite.induct)
- apply simp
- using simp_flatten apply blast
- prefer 2
- apply (smt (verit) append.assoc append_Cons in_set_conv_decomp simp_removes_duplicate2)
- apply simp
- apply(case_tac "rdistinct (rflts (map rsimp rs1)) {}")
- apply(case_tac "rsimp r = RZERO")
- apply simp
- apply(case_tac "\<exists>rs. rsimp r = RALTS rs")
- prefer 2
-
- apply(subgoal_tac "rdistinct (rflts (rsimp r # map rsimp rs1)) {} =
- rsimp r # rdistinct (rflts (map rsimp rs1)) {rsimp r}")
- prefer 2
- apply (simp add: list.inject rflts_def_idiot)
- apply(simp only:)
-
- sorry
+ apply(frule grewrite_cases_middle)
+ apply(case_tac "(\<exists>rsa rsb rsc. rs1 = rsa @ [RALTS rsb] @ rsc \<and> rs2 = rsa @ rsb @ rsc)")
+ using simp_flatten3 apply auto[1]
+ apply(case_tac "(\<exists>rsa rsc. rs1 = rsa @ [RZERO] @ rsc \<and> rs2 = rsa @ rsc)")
+ apply (metis (mono_tags, opaque_lifting) append_Cons append_Nil list.set_intros(1) list.simps(9) rflts.simps(2) rsimp.simps(2) rsimp.simps(3) simp_removes_duplicate3)
+ by (smt (verit) append.assoc append_Cons append_Nil in_set_conv_decomp simp_removes_duplicate3)
lemma grewrites_equal_rsimp:
@@ -576,7 +377,10 @@
lemma grewrites_equal_simp_2:
shows "rsimp (RALTS rs1) = rsimp (RALTS rs2) \<Longrightarrow> rs1 \<leadsto>g* rs2"
- sorry
+ oops
+
+
+
lemma grewrites_last:
shows "r # [RALTS rs] \<leadsto>g* r # rs"
@@ -615,7 +419,6 @@
-
lemma with_wo0_distinct:
shows "rdistinct rs rset \<leadsto>f* rdistinct rs (insert RZERO rset)"
apply(induct rs arbitrary: rset)
@@ -661,93 +464,391 @@
r = +rs
[] \<leadsto>g* rs which is wrong
*)
-lemma frewrite_with_distinct:
- shows " \<lbrakk>rs2 \<leadsto>f rs3\<rbrakk>
- \<Longrightarrow> rdistinct rs2
- (insert RZERO (rset \<union> \<Union> (alt_set ` rset))) \<leadsto>g*
- rdistinct rs3
- (insert RZERO (rset \<union> \<Union> (alt_set ` rset)))"
- apply(induct rs2 rs3 arbitrary: rset rule: frewrite.induct)
- apply(case_tac "RZERO \<in> (rset \<union> \<Union> (alt_set ` rset))")
- apply simp
- apply simp
-
-
- oops
-lemma frewrites_with_distinct:
- shows "\<lbrakk>rsa \<leadsto>f rsb \<rbrakk> \<Longrightarrow>
-( (rs1 @ (rdistinct rsa (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))) \<leadsto>g*
- rs1 @ (rdistinct rsb (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))))
-\<or> ( rs1 @ (rdistinct rsb (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))) \<leadsto>g*
- rs1 @ (rdistinct rsa (insert RZERO (set rs1 \<union> \<Union>(alt_set ` (set rs1) )))))
- )
-"
- apply(induct rsa rsb arbitrary: rs1 rule: frewrite.induct)
- apply simp
- apply(case_tac "RALTS rs \<in> set rs1")
- apply(subgoal_tac "set rs \<subseteq> \<Union> (alt_set `set rs1)")
- apply (metis (full_types) Un_iff Un_insert_left
-Un_insert_right grewrites.intros(1) le_supI2 rdistinct.simps(2) rdistinct_concat)
- apply (metis Un_subset_iff Union_upper alt_set.simps(1) imageI)
-
- apply(case_tac "RALTS rs \<in> \<Union> (alt_set ` set rs1)")
- apply simp
- apply (smt (z3) UN_insert Un_iff alt_set.simps(1) alt_set_has_all dual_order.trans grewrites.intros(1) insert_absorb rdistinct_concat subset_insertI)
-
-
- oops
-lemma rd_flts_set:
- shows "rs1 \<leadsto>f* rs2 \<Longrightarrow> rdistinct rs1 ({RZERO} \<union> (rset \<union> \<Union>(alt_set ` rset))) \<leadsto>g*
- rdistinct rs2 ({RZERO} \<union> (rset \<union> \<Union>(alt_set ` rset)))"
- apply(induct rs1 rs2 rule: frewrites.induct)
- apply simp
- oops
lemma frewrite_simpeq:
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
apply(induct rs1 rs2 rule: frewrite.induct)
apply simp
using simp_flatten apply presburger
- by (meson grewrites_cons grewrites_equal_rsimp grewrites_equal_simp_2)
+ by (metis (no_types, opaque_lifting) grewrites_equal_rsimp grewrites_last list.simps(9) rsimp.simps(2))
+
+lemma gstar0:
+ shows "rsa @ (rdistinct rs (set rsa)) \<leadsto>g* rsa @ (rdistinct rs (insert RZERO (set rsa)))"
+ apply(induct rs arbitrary: rsa)
+ apply simp
+ apply(case_tac "a = RZERO")
+ apply simp
+
+ using gr_in_rstar grewrite.intros(1) grewrites_append apply presburger
+ apply(case_tac "a \<in> set rsa")
+ apply simp+
+ apply(drule_tac x = "rsa @ [a]" in meta_spec)
+ by simp
+
+lemma gstar01:
+ shows "rdistinct rs {} \<leadsto>g* rdistinct rs {RZERO}"
+ by (metis empty_set gstar0 self_append_conv2)
+
+
+lemma grewrite_rdistinct_aux:
+ shows "rs @ rdistinct rsa rset \<leadsto>g* rs @ rdistinct rsa (rset \<union> set rs)"
+ sorry
+
+lemma grewrite_rdistinct_worth1:
+ shows "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)"
+ by (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
+
+lemma grewrite_rdisitinct:
+ shows "rs @ rdistinct rsa {RALTS rs} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))"
+ apply(induct rsa arbitrary: rs)
+ apply simp
+ apply(case_tac "a = RALTS rs")
+ apply simp
+ apply(case_tac "a \<in> set rs")
+ apply simp
+ apply(subgoal_tac "rs @
+ a # rdistinct rsa {RALTS rs, a} \<leadsto>g rs @ rdistinct rsa {RALTS rs, a}")
+ apply(subgoal_tac
+"rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa (insert (RALTS rs) (set rs))")
+ using gmany_steps_later apply blast
+ apply(subgoal_tac
+" rs @ rdistinct rsa {RALTS rs, a} \<leadsto>g* rs @ rdistinct rsa ({RALTS rs, a} \<union> set rs)")
+ apply (simp add: insert_absorb)
+ using grewrite_rdistinct_aux apply blast
+ using grewrite_variant1 apply blast
+ by (metis grewrite_rdistinct_aux insert_is_Un)
+
+
+lemma frewrite_rd_grewrites_general:
+ shows "\<lbrakk>rs1 \<leadsto>f rs2; \<And>rs. \<exists>rs3.
+(rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3)\<rbrakk>
+ \<Longrightarrow>
+\<exists>rs3. (rs @ (r # rdistinct rs1 (set rs \<union> {r})) \<leadsto>g* rs3) \<and> (rs @ (r # rdistinct rs2 (set rs \<union> {r})) \<leadsto>g* rs3)"
+ apply(drule_tac x = "rs @ [r]" in meta_spec )
+ by simp
+
+
+lemma grewrites_middle_distinct:
+ shows "RALTS rs \<in> set rsb \<Longrightarrow>
+ rsb @
+ rdistinct ( rs @ rsa)
+ (set rsb) \<leadsto>g* rsb @ rdistinct rsa (set rsb)"
+ sorry
+
+
+
+lemma frewrite_rd_grewrites_aux:
+ shows " rsb @
+ rdistinct (RALTS rs # rsa)
+ (set rsb) \<leadsto>g* rsb @
+ rdistinct rs (set rsb) @ rdistinct rsa (insert (RALTS rs) (set rs) \<union> set rsb)"
+
+
+ sorry
+
+lemma flts_gstar:
+ shows "rs \<leadsto>g* rflts rs"
+ sorry
+
+lemma list_dlist_union:
+ shows "set rs \<subseteq> set rsb \<union> set (rdistinct rs (set rsb))"
+ by (metis rdistinct_concat_general rdistinct_set_equality set_append sup_ge2)
+
+lemma subset_distinct_rewrite1:
+ shows "set1 \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs set1)"
+ apply(induct rs arbitrary: rsb)
+ apply simp
+ apply(case_tac "a \<in> set1")
+ apply simp
+
+ using gmany_steps_later grewrite_variant1 apply blast
+ apply simp
+ apply(drule_tac x = "rsb @ [a]" in meta_spec)
+ apply(subgoal_tac "set1 \<subseteq> set (rsb @ [a])")
+ apply (simp only:)
+ apply(subgoal_tac "(rsb @ [a]) @ rdistinct rs set1 \<leadsto>g* (rsb @ [a]) @ rdistinct rs (insert a set1)")
+ apply (metis (no_types, opaque_lifting) append.assoc append_Cons append_Nil greal_trans)
+ apply (metis append.assoc empty_set grewrite_rdistinct_aux grewrites_append inf_sup_aci(5) insert_is_Un list.simps(15))
+ by auto
+
+
+lemma subset_distinct_rewrite:
+ shows "set rsb' \<subseteq> set rsb \<Longrightarrow> rsb @ rs \<leadsto>g* rsb @ (rdistinct rs (set rsb'))"
+ by (simp add: subset_distinct_rewrite1)
+
+
+
+lemma distinct_3list:
+ shows "rsb @ (rdistinct rs (set rsb)) @ rsa \<leadsto>g*
+ rsb @ (rdistinct rs (set rsb)) @ (rdistinct rsa (set rs))"
+ by (metis append.assoc list_dlist_union set_append subset_distinct_rewrite)
+
+
+
+
+lemma grewrites_shape1:
+ shows " RALTS rs \<notin> set rsb \<Longrightarrow>
+ rsb @
+ RALTS rs #
+ rdistinct rsa
+ (
+ (set rsb)) \<leadsto>g* rsb @
+ rdistinct rs (set rsb) @
+ rdistinct (rflts (rdistinct rsa ( (set rsb \<union> set rs)))) (set rs)"
+
+
+ apply (subgoal_tac " rsb @
+ RALTS rs #
+ rdistinct rsa
+ (
+ (set rsb)) \<leadsto>g* rsb @
+ rs @
+ rdistinct rsa
+ (
+ (set rsb)) ")
+ prefer 2
+ using gr_in_rstar grewrite.intros(2) grewrites_append apply presburger
+ apply(subgoal_tac "rsb @ rs @ rdistinct rsa ( (set rsb)) \<leadsto>g* rsb @
+(rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb)))")
+ prefer 2
+ apply (metis append_assoc grewrites.intros(1) grewritess_concat gstar_rdistinct_general)
+ apply(subgoal_tac " rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb))
+\<leadsto>g* rsb @ rdistinct rs (set rsb) @ rdistinct rsa ( (set rsb) \<union> (set rs))")
+ prefer 2
+ apply (smt (verit, best) append.assoc append_assoc boolean_algebra_cancel.sup2 grewrite_rdistinct_aux inf_sup_aci(5) insert_is_Un rdistinct_concat_general rdistinct_set_equality set_append sup.commute sup.right_idem sup_commute)
+ apply(subgoal_tac "rdistinct rsa ( (set rsb) \<union> set rs) \<leadsto>g*
+rflts (rdistinct rsa ( (set rsb) \<union> set rs))")
+ apply(subgoal_tac "rsb @ (rdistinct rs (set rsb)) @ rflts (rdistinct rsa ( (set rsb) \<union> set rs)) \<leadsto>g*
+rsb @ (rdistinct rs (set rsb)) @ (rdistinct (rflts (rdistinct rsa ( (set rsb) \<union> set rs))) (set rs))")
+ apply (smt (verit, ccfv_SIG) Un_insert_left greal_trans grewrites_append)
+ using distinct_3list apply presburger
+ using flts_gstar apply blast
+ done
+
+lemma r_finite1:
+ shows "r = RALTS (r # rs) = False"
+ apply(induct r)
+ apply simp+
+ apply (metis list.set_intros(1))
+ by blast
+
+
+
+lemma grewrite_singleton:
+ shows "[r] \<leadsto>g r # rs \<Longrightarrow> rs = []"
+ apply (induct "[r]" "r # rs" rule: grewrite.induct)
+ apply simp
+ apply (metis r_finite1)
+ using grewrite.simps apply blast
+ by simp
+
+lemma impossible_grewrite1:
+ shows "\<not>( [RONE] \<leadsto>g [])"
+ using grewrite.cases by fastforce
+
+
+lemma impossible_grewrite2:
+ shows "\<not> ([RALTS rs] \<leadsto>g (RALTS rs) # a # rs)"
+
+ using grewrite_singleton by blast
+thm grewrite.cases
+lemma impossible_grewrite3:
+ shows "\<not> (RALTS rs # rs1 \<leadsto>g (RALTS rs) # a # rs1)"
+ oops
+
+
+lemma grewrites_singleton:
+ shows "[r] \<leadsto>g* r # rs \<Longrightarrow> rs = []"
+ apply(induct "[r]" "r # rs" rule: grewrites.induct)
+ apply simp
+
+ oops
+
+lemma grewrite_nonequal_elem:
+ shows "r # rs2 \<leadsto>g r # rs3 \<Longrightarrow> rs2 \<leadsto>g rs3"
+ oops
+
+lemma grewrites_nonequal_elem:
+ shows "r # rs2 \<leadsto>g* r # rs3 \<Longrightarrow> rs2 \<leadsto>g* rs3"
+ apply(induct r)
+
+ oops
+
+
+
+
+lemma :
+ shows "rs1 @ rs2 \<leadsto>g* rs1 @ rs3 \<Longrightarrow> rs2 \<leadsto>g* rs3"
+ apply(induct rs1 arbitrary: rs2 rs3 rule: rev_induct)
+ apply simp
+ apply(drule_tac x = "[x] @ rs2" in meta_spec)
+ apply(drule_tac x = "[x] @ rs3" in meta_spec)
+ apply(simp)
+
+ oops
+
+
+
+lemma grewrites_shape3_aux:
+ shows "rs @ (rdistinct rsa (insert (RALTS rs) rsc)) \<leadsto>g* rs @ rdistinct (rflts (rdistinct rsa rsc)) (set rs)"
+ apply(induct rsa arbitrary: rsc rs)
+ apply simp
+ apply(case_tac "a \<in> rsc")
+ apply simp
+ apply(case_tac "a = RALTS rs")
+ apply simp
+ apply(subgoal_tac " rdistinct (rs @ rflts (rdistinct rsa (insert (RALTS rs) rsc))) (set rs) \<leadsto>g*
+ rdistinct (rflts (rdistinct rsa (insert (RALTS rs) rsc))) (set rs)")
+ apply (metis insertI1 insert_absorb rdistinct_concat2)
+ apply (simp add: rdistinct_concat)
+
+ apply simp
+ apply(case_tac "a = RZERO")
+ apply (metis gmany_steps_later grewrite.intros(1) grewrite_append rflts.simps(2))
+ apply(case_tac "\<exists>rs1. a = RALTS rs1")
+ prefer 2
+ apply simp
+ apply(subgoal_tac "rflts (a # rdistinct rsa (insert a rsc)) = a # rflts (rdistinct rsa (insert a rsc))")
+ apply (simp only:)
+ apply(case_tac "a \<notin> set rs")
+ apply simp
+ apply(drule_tac x = "insert a rsc" in meta_spec)
+ apply(drule_tac x = "rs " in meta_spec)
+
+ apply(erule exE)
+ apply simp
+ apply(subgoal_tac "RALTS rs1 #
+ rdistinct rsa
+ (insert (RALTS rs)
+ (insert (RALTS rs1)
+ rsc)) \<leadsto>g* rs1 @
+ rdistinct rsa
+ (insert (RALTS rs)
+ (insert (RALTS rs1)
+ rsc)) ")
+ apply(subgoal_tac " rs1 @
+ rdistinct rsa
+ (insert (RALTS rs)
+ (insert (RALTS rs1)
+ rsc)) \<leadsto>g*
+ rs1 @
+ rdistinct rsa
+ (insert (RALTS rs)
+ (insert (RALTS rs1)
+ rsc))")
+
+ apply(case_tac "a \<in> set rs")
+
+
+
+ sorry
+
+
+lemma grewrites_shape3:
+ shows " RALTS rs \<notin> set rsb \<Longrightarrow>
+ rsb @
+ RALTS rs #
+ rdistinct rsa
+ (insert (RALTS rs)
+ (set rsb)) \<leadsto>g* rsb @
+ rdistinct rs (set rsb) @
+ rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)"
+ apply(subgoal_tac "rsb @ RALTS rs # rdistinct rsa (insert (RALTS rs) (set rsb)) \<leadsto>g*
+ rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb))")
+ prefer 2
+ using gr_in_rstar grewrite.intros(2) grewrites_append apply presburger
+ apply(subgoal_tac "rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb )) \<leadsto>g*
+ rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb \<union> set rs))")
+ prefer 2
+ apply (metis Un_insert_left grewrite_rdistinct_aux grewrites_append)
+
+ apply(subgoal_tac "rsb @ rs @ rdistinct rsa (insert (RALTS rs) (set rsb \<union> set rs)) \<leadsto>g*
+rsb @ rs @ rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)")
+ prefer 2
+ using grewrites_append grewrites_shape3_aux apply presburger
+ apply(subgoal_tac "rsb @ rs \<leadsto>g* rsb @ rdistinct rs (set rsb)")
+ apply (smt (verit, ccfv_SIG) append_eq_appendI greal_trans grewrites.simps grewritess_concat)
+ using gstar_rdistinct_general by blast
+
+
+lemma grewrites_shape2:
+ shows " RALTS rs \<notin> set rsb \<Longrightarrow>
+ rsb @
+ rdistinct (rs @ rsa)
+ (set rsb) \<leadsto>g* rsb @
+ rdistinct rs (set rsb) @
+ rdistinct (rflts (rdistinct rsa (set rsb \<union> set rs))) (set rs)"
+
+ (* by (smt (z3) append.assoc distinct_3list flts_gstar greal_trans grewrites_append rdistinct_concat_general same_append_eq set_append)
+*)
+ sorry
+
+
+
+
lemma frewrite_rd_grewrites:
shows "rs1 \<leadsto>f rs2 \<Longrightarrow>
-\<exists>rs3. (rs1 \<leadsto>g* rs3) \<and> (rs2 \<leadsto>g* rs3) "
- apply(induct rs1 rs2 rule: frewrite.induct)
- apply(rule_tac x = "rs" in exI)
+\<exists>rs3. (rs @ (rdistinct rs1 (set rs)) \<leadsto>g* rs3) \<and> (rs @ (rdistinct rs2 (set rs)) \<leadsto>g* rs3) "
+ apply(induct rs1 rs2 arbitrary: rs rule: frewrite.induct)
+ apply(rule_tac x = "rsa @ (rdistinct rs ({RZERO} \<union> set rsa))" in exI)
apply(rule conjI)
- apply(rule gr_in_rstar)
- apply(rule grewrite.intros)
- apply(rule grewrites.intros)
- using grewrite.intros(2) apply blast
- by (meson grewrites_cons)
+ apply(case_tac "RZERO \<in> set rsa")
+ apply simp+
+ using gstar0 apply fastforce
+ apply (simp add: gr_in_rstar grewrite.intros(1) grewrites_append)
+ apply (simp add: gstar0)
+ prefer 2
+ apply(case_tac "r \<in> set rs")
+ apply simp
+ apply(drule_tac x = "rs @ [r]" in meta_spec)
+ apply(erule exE)
+ apply(rule_tac x = "rs3" in exI)
+ apply simp
+ apply(case_tac "RALTS rs \<in> set rsb")
+ apply simp
+ apply(rule_tac x = "rflts rsb @ rdistinct rsa (set rsb)" in exI)
+ apply(rule conjI)
+ apply (simp add: flts_gstar grewritess_concat)
+ apply (meson flts_gstar greal_trans grewrites.intros(1) grewrites_middle_distinct grewritess_concat)
+ apply(simp)
+ apply(rule_tac x =
+"rsb @ (rdistinct rs (set rsb)) @
+ (rdistinct (rflts (rdistinct rsa ( (set rsb \<union> set rs)) ) ) (set rs))" in exI)
+ apply(rule conjI)
+ prefer 2
+ using grewrites_shape2 apply force
+ using grewrites_shape3 by auto
+
+
+
+lemma frewrite_simprd:
+ shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS rs1) = rsimp (RALTS rs2)"
+ by (meson frewrite_simpeq)
lemma frewrites_rd_grewrites:
shows "rs1 \<leadsto>f* rs2 \<Longrightarrow>
-\<exists>rs3. (rs1 \<leadsto>g* rs3) \<and> (rs2 \<leadsto>g* rs3)"
+rsimp (RALTS rs1) = rsimp (RALTS rs2)"
apply(induct rs1 rs2 rule: frewrites.induct)
apply simp
- apply(rule exI)
- apply(rule grewrites.intros)
- by (metis frewrite_simpeq grewrites_equal_rsimp grewrites_equal_simp_2)
-
+ using frewrite_simprd by presburger
lemma frewrite_simpeq2:
shows "rs1 \<leadsto>f rs2 \<Longrightarrow> rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS (rdistinct rs2 {}))"
- apply(induct rs1 rs2 rule: frewrite.induct)
- apply simp
- apply (simp add: distinct_flts_no0)
- apply simp
+ apply(subgoal_tac "\<exists> rs3. (rdistinct rs1 {} \<leadsto>g* rs3) \<and> (rdistinct rs2 {} \<leadsto>g* rs3)")
+ using grewrites_equal_rsimp apply fastforce
+ using frewrite_rd_grewrites by presburger
+
(*a more refined notion of \<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
@@ -757,9 +858,7 @@
rsimp (RALTS (rdistinct rs1 {})) = rsimp (RALTS ( rdistinct rs2 {})) "
apply(induct rs1 rs2 rule: frewrites.induct)
apply simp
-
- sorry
-
+ using frewrite_simpeq2 by presburger
lemma frewrite_single_step:
@@ -814,13 +913,10 @@
shows "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {})) =
rsimp (RALTS (rdistinct (rflts (map (rder x) rs)) {}))"
apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>f* rflts (map (rder x) rs)")
- apply(subgoal_tac "rdistinct (map (rder x) (rflts rs)) {RZERO}
- \<leadsto>g* rdistinct ( rflts (map (rder x) rs)) {RZERO}")
- apply(subgoal_tac "rsimp (RALTS (rdistinct (map (rder x) (rflts rs)) {}))
- = rsimp (RALTS ( rdistinct ( rflts (map (rder x) rs)) {}))")
- apply meson
- apply (metis distinct_flts_no0 grewrites_equal_rsimp rsimp.simps(2))
- sorry
+ using frewrites_simpeq apply presburger
+ using early_late_der_frewrites by auto
+
+
@@ -832,11 +928,7 @@
lemma simp_der_pierce_flts_prelim:
shows "rsimp (rsimp_ALTs (rdistinct (map (rder x) (rflts rs)) {}))
= rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}))"
- apply(subgoal_tac "map (rder x) (rflts rs) \<leadsto>g* rflts (map (rder x) rs)")
- apply(subgoal_tac "rdistinct (map (rder x) (rflts rs)) {RZERO} \<leadsto>g* rdistinct (rflts (map (rder x) rs)) {RZERO}")
- using grewrites_equal_simp_2 grewrites_simpalts simp_der_flts apply blast
- apply (simp add: early_late_der_frewrites frewrites_with_distinct2_grewrites)
- using early_late_der_frewrites frewrites_equivalent_simp grewrites_equal_simp_2 by blast
+ by (metis append.right_neutral grewrite.intros(2) grewrite_simpalts rsimp_ALTs.simps(2) simp_der_flts)
lemma simp_der_pierce_flts:
@@ -1170,7 +1262,8 @@
lemma alts_closed_form_variant: shows
"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
- sorry
+ by (metis alts_closed_form comp_apply rders_simp_nonempty_simped)
+