--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys3/RfltsRdistinctProps.thy Wed Jun 29 12:38:05 2022 +0100
@@ -0,0 +1,355 @@
+theory RfltsRdistinctProps imports "Rsimp"
+begin
+
+
+
+lemma all_that_same_elem:
+ shows "\<lbrakk> a \<in> rset; rdistinct rs {a} = []\<rbrakk>
+ \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct rsb rset"
+ apply(induct rs)
+ apply simp
+ apply(subgoal_tac "aa = a")
+ apply simp
+ by (metis empty_iff insert_iff list.discI rdistinct.simps(2))
+
+
+lemma rdistinct1:
+ assumes "a \<in> acc"
+ shows "a \<notin> set (rdistinct rs acc)"
+ using assms
+ apply(induct rs arbitrary: acc a)
+ apply(auto)
+ done
+
+
+lemma rdistinct_does_the_job:
+ shows "distinct (rdistinct rs s)"
+ apply(induct rs s rule: rdistinct.induct)
+ apply(auto simp add: rdistinct1)
+ done
+
+
+
+lemma rdistinct_concat:
+ assumes "set rs \<subseteq> rset"
+ shows "rdistinct (rs @ rsa) rset = rdistinct rsa rset"
+ using assms
+ apply(induct rs)
+ apply simp+
+ done
+
+lemma distinct_not_exist:
+ assumes "a \<notin> set rs"
+ shows "rdistinct rs rset = rdistinct rs (insert a rset)"
+ using assms
+ apply(induct rs arbitrary: rset)
+ apply(auto)
+ done
+
+lemma rdistinct_on_distinct:
+ shows "distinct rs \<Longrightarrow> rdistinct rs {} = rs"
+ apply(induct rs)
+ apply simp
+ using distinct_not_exist by fastforce
+
+lemma distinct_rdistinct_append:
+ assumes "distinct rs1" "\<forall>r \<in> set rs1. r \<notin> acc"
+ shows "rdistinct (rs1 @ rsa) acc = rs1 @ (rdistinct rsa (acc \<union> set rs1))"
+ using assms
+ apply(induct rs1 arbitrary: rsa acc)
+ apply(auto)[1]
+ apply(auto)[1]
+ apply(drule_tac x="rsa" in meta_spec)
+ apply(drule_tac x="{a} \<union> acc" in meta_spec)
+ apply(simp)
+ apply(drule meta_mp)
+ apply(auto)[1]
+ apply(simp)
+ done
+
+
+lemma rdistinct_set_equality1:
+ shows "set (rdistinct rs acc) = set rs - acc"
+ apply(induct rs acc rule: rdistinct.induct)
+ apply(auto)
+ done
+
+
+lemma rdistinct_set_equality:
+ shows "set (rdistinct rs {}) = set rs"
+ by (simp add: rdistinct_set_equality1)
+
+
+
+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 k0b:
+ assumes "nonalt r" "r \<noteq> RZERO"
+ shows "rflts [r] = [r]"
+ using assms
+ apply(case_tac r)
+ apply(simp_all)
+ done
+
+lemma rflts_def_idiot:
+ shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk> \<Longrightarrow> rflts (a # rs) = a # rflts rs"
+ apply(case_tac a)
+ apply simp_all
+ done
+
+lemma flts_middle0:
+ shows "rflts (rsa @ RZERO # rsb) = rflts (rsa @ rsb)"
+ apply(induct rsa)
+ apply simp
+ by (metis append_Cons rflts.simps(2) rflts.simps(3) rflts_def_idiot)
+
+
+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 rflts_spills_last:
+ shows "rflts (rs1 @ [RALTS rs]) = rflts rs1 @ rs"
+ apply (induct rs1 rule: rflts.induct)
+ apply(auto)
+ done
+
+lemma flts_keeps1:
+ shows "rflts (rs @ [RONE]) = rflts rs @ [RONE]"
+ apply (induct rs rule: rflts.induct)
+ apply(auto)
+ done
+
+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 rule: rflts.induct)
+ apply(auto)
+ by (meson k0b nonalt.elims(3))
+
+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 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 rule: rflts.induct)
+ apply(auto)
+ 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 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_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 last_elem_out:
+ shows "\<lbrakk>x \<notin> set xs; x \<notin> rset \<rbrakk> \<Longrightarrow> rdistinct (xs @ [x]) rset = rdistinct xs rset @ [x]"
+ apply(induct xs arbitrary: rset)
+ apply simp+
+ done
+
+
+lemma rdistinct_concat_general:
+ shows "rdistinct (rs1 @ rs2) {} = (rdistinct rs1 {}) @ (rdistinct rs2 (set rs1))"
+ apply(induct rs1 arbitrary: rs2 rule: rev_induct)
+ apply simp
+ apply(drule_tac x = "x # rs2" in meta_spec)
+ apply simp
+ apply(case_tac "x \<in> set xs")
+ apply simp
+
+ apply (simp add: distinct_removes_middle3 insert_absorb)
+ apply simp
+ by (simp add: last_elem_out)
+
+
+
+
+lemma distinct_once_enough:
+ shows "rdistinct (rs @ rsa) {} = rdistinct (rdistinct rs {} @ rsa) {}"
+ apply(subgoal_tac "distinct (rdistinct rs {})")
+ apply(subgoal_tac
+" rdistinct (rdistinct rs {} @ rsa) {} = rdistinct rs {} @ (rdistinct rsa (set rs))")
+ apply(simp only:)
+ using rdistinct_concat_general apply blast
+ apply (simp add: distinct_rdistinct_append rdistinct_set_equality1)
+ by (simp add: rdistinct_does_the_job)
+
+
+
+
+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_last(1) 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_last(1) rflts_def_idiot2 rrexp.distinct(21) rrexp.distinct(3))
+
+ apply (metis distinct_removes_last(1) flts_keeps_others rflts_def_idiot2 rrexp.distinct(25) rrexp.distinct(5))
+ prefer 2
+ apply (metis distinct_removes_last(1) 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(subgoal_tac "\<forall> r \<in> set x. r \<in> set (rflts (map rsimp rsa))")
+ prefer 2
+ apply (metis (mono_tags, lifting) image_iff image_set spilled_alts_contained)
+ apply (metis rflts_spills_last)
+ by (metis distinct_removes_list spilled_alts_contained)
+
+
+
+
+
+lemma distinct_early_app1:
+ shows "rset1 \<subseteq> rset \<Longrightarrow> rdistinct rs rset = rdistinct (rdistinct rs rset1) rset"
+ apply(induct rs arbitrary: rset rset1)
+ apply simp
+ apply simp
+ apply(case_tac "a \<in> rset1")
+ apply simp
+ apply(case_tac "a \<in> rset")
+ apply simp+
+
+ apply blast
+ apply(case_tac "a \<in> rset1")
+ apply simp+
+ apply(case_tac "a \<in> rset")
+ apply simp
+ apply (metis insert_subsetI)
+ apply simp
+ by (meson insert_mono)
+
+
+lemma distinct_early_app:
+ shows " rdistinct (rs @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset"
+ apply(induct rsb)
+ apply simp
+ using distinct_early_app1 apply blast
+ by (metis distinct_early_app1 distinct_once_enough empty_subsetI)
+
+
+lemma distinct_eq_interesting1:
+ shows "a \<in> rset \<Longrightarrow> rdistinct (rs @ rsb) rset = rdistinct (rdistinct (a # rs) {} @ rsb) rset"
+ apply(subgoal_tac "rdistinct (rdistinct (a # rs) {} @ rsb) rset = rdistinct (rdistinct rs {} @ rsb) rset")
+ apply(simp only:)
+ using distinct_early_app apply blast
+ by (metis append_Cons distinct_early_app rdistinct.simps(2))
+
+
+
+
+end
\ No newline at end of file