SSL Set_Thms.thy

(*
  File: Set_Thms.thy
  Author: Bohua Zhan

  Setup of proof steps related to .
*)

section ‹Setup for sets and multisets›

theory Set_Thms
imports Logic_Thms "HOL-Library.Multiset"
begin

subsection ‹add_forward_prfstep_cond @{thm Set.UnI1} [with_term "?A ∪

java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90

  ‹ {?b}"]› \inp ==> {x} ∪ p = p" by auto

  ‹

  ‹add_forward_prfstep (equiv_forward_th @{thm Set.Int_iff})›add_backward_prfstep_cond (equiv_backward_th @{thm Set.Int_iff}) [with_score 500]›
 {cfhead = @{cterm inf}, unit = SOME @{cterm inf},
 assoc_th = @{thm inf_assoc}, comm_th = @{thm inf_commute},
 unitl_th = @{thm inf_top_left}, unitr_th = @{thmop>a @{thm Set.Int_empty_right}›

 {cfhead = @{cterm sup}, unit = SOME @{cterm bot},
 assoc_th = @{thm sup_assoc}, comm_th = @{thm sup_commute},
 unitl_th = @{thm sup_bot_left}, unitr_th = @{thm sup_bot_right}}]
 ›add_rewrite_rule @{thm Set.Int_absorb}› B = {\Longrightarrow> p ∈ p \notin B" by auto

subsubsection ‹add_backward_prfstep_cond @{thm subsetI} [with_score 500]›

setup ‹add_forward_prfstep @{thm subsetD}›
lemma bball_single [rwite]: "(\<forallin{x}. P x) = P  auto

subsubsection \<open>Membership\<close>

setup \<open\openadd_resolve_prfstep @{thm Set.Un_upper1}\<close>
setup \<open>add_forward_prfstepd_thhmpty_iffff<>
lemma set_membership_distinct [forward]: "x \<in> s \<Longrightarrow> y \<notin> s \<Longrightarrow> x \<noteq> y" by auto
lemma non_empty_exist_elt [backward]: "U \<noteq> {} \<Longrightarrow> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma non_univ_exist_compl [backward]: "U \<noteq> UNIV \<Longrightarrow> \<exists>x. x \<notin> U" by blast
setup<>add_resolve_prfstepm Set.UNIV_I}\<close>

subsubsection \<open>Insert\<close>

setup \ nion_subtract_elt2]"\notin> B<ongrightarrow (B \<union> {x}) - {x} = B" by simp
setup \openadd_forward_prfstep_cond (equiv_forward_th @{thm Set.insert_iff})
  [with_score 500, with_cond "?A \<noteq> {}"]\<close>
setup \<open>add_forward_prfstep_cond (equiv_forward_th{Setsert_subsetith_cond_d?A<> {}"]\<close>
setup \<open>add_backward_prfstep_cond (equiv_backward_th @{thm Set.insert_subset})
  [with_score 500, with_cond "?A \<noteq> {}"]\<close

subsubsection  \<open>dd_forward_prfstepnite_atLeastAtMost}\<lose

lemma set_ext [forward]: "setup\<open>dd_backward1_prfstepev_finite_subsetclose>
setup \<open>add_backward_prfstep_cond @{thm set_ext} [with_score 500, with_filt (order_filter "S" "T")]\<close>

lemma set_pair_ext [forward]: "\<forall>a b. (a, b) \<in> S \<longleftrightarrow> (a, b) \<in> T \<Longrightarrow> S = T" by auto

subsubsection \<open>Union\<close>

setup \<open>add_forward_prfstep_cond (equiv_forward_th @{thm Set.Un_iff}) [with_score 500]\<close>
setup \<open>add_backward_prfstep (equiv_backward_th @{thm Set.Un_iff})\<close>

lemma UnD1 [forward]: "c \<in> A \<union> B \<Longrightarrow> c \<notin> A \<Longrightarrow> c \<in> B" by auto
lemma UnD2 [forward]: "c \<in> A \<union> B \<Longrightarrow> c \<notin> B \<Longrightarrow> c \<in> A" by auto
lemma UnD1_single [forward]: "c \<in> {a} \<union> B \<Longrightarrow> c \<noteq> a \<Longrightarrow> c \<in> B" by auto
lemma UnD2_single [forward]: "c \<in> A \<union> {b} \<Longrightarrow> c \<noteq> b \<Longrightarrow> c \<in> A" by auto
setup \<open>add_forward_prfstep_cond @{thm Set.UnI1} [with_term "?A \<union> ?B"]\<close>
setup \<open>add_forward_prfstep_cond @{thm Set.UnI2} [with_term "?A \<union> ?B"]\<close>
lemma UnI1_single: "a \<in> {a} \<union> B" by auto
lemma UnI2_single: "b \<in> A \<union> {b}" by auto
setup \<open>add_forward_prfstep_cond @{thm UnI1_single} [with_term "{?a} \<union> ?B"]\<close>
setup \<open>add_forward_prfstep_cond @{thm UnI2_single} [with_term "?A \<union> {?b}"]\<close>

lemma union_single_eq [rewrite, backward]: "x \<in> p \<Longrightarrow> {x} \<union> p = p" by auto

subsubsection \<open>Intersection\<close>

setup \<open>add_forward_prfstep (equiv_forward_th @{thm Set.Int_iff})\<close>
setup \<open>add_backward_prfstep_cond (equiv_backward_th @{thm Set.Int_iff}) [with_score 500]\<close>

setup \<open>add_rewrite_rule @{thm Set.Int_empty_left}\<close>
setup \<open>add_rewrite_rule @{thm Set.Int_empty_right}\<close>
setup \<open>add_rewrite_rule @{thm Set.Int_absorb}\<close>
lemma set_disjoint_mp [forward, backward2]: "A \<inter> B = {} \<Longrightarrow> p \<in> A \<Longrightarrow> p \<notin> B" by auto
lemma set_disjoint_single [rewrite]: "{x} \<inter> B = {} \<longleftrightarrow> x \<notin> B" by simp

subsubsection \<open>subset\<close>

setup \<open>add_forward_prfstep @{thm subsetI}\<close>
setup \<open>add_backward_prfstep_cond @{thm subsetI} [with_score 500]\<close>

setup \<open>add_resolve_prfstep @{thm empty_subsetI}\<close>
setup \<open>add_forward_prfstep @{thm subsetD}\<close>
lemma subset_single [rewrite]: "{a} \<subseteq> B \<longleftrightarrow> a \<in> B" by simp
setup \<open>add_resolve_prfstep @{thm Set.basic_monos(1)}\<close>
setup \<open>add_resolve_prfstep @{thm Set.Un_upper1}\<close>
setup \<open>add_resolve_prfstep @{thm Set.Un_upper2}\<close>
lemma union_is_subset [forward]: "A \<union> B \<subseteq> C \<Longrightarrow> A \<subseteq> C \<and> B \<subseteq> C" by simp
setup \<open>add_backward1_prfstep @{thm Set.Un_least}\<close>
setup \<open>add_backward2_prfstep @{thm Set.Un_least}\<close>
lemma subset_union_same1 [backward]: "B \<subseteq> C \<Longrightarrow> A \<union> B \<subseteq> A \<union> C" by auto
lemma subset_union_same2 [backward]: "A \<subseteq> B \<Longrightarrow> A \<union> C \<subseteq> B \<union> C" by auto

subsubsection \<open>Diff\<close>

setup \<open>add_forward_prfstep (equiv_forward_th @{thm Set.Diff_iff})\<close>
setup \<open>add_backward_prfstep_cond (equiv_backward_th @{thm Set.Diff_iff}) [with_score 500]\<close>

setup \<open>add_rewrite_rule @{thm Set.empty_Diff}\<close>
lemma mem_diff [rewrite]: "x \<in> A - B \<longleftrightarrow> x \<in> A \<and> x \<notin> B" by simp
lemma set_union_minus_same1 [rewrite]: "(A \<union> B) - B = A - B" by auto
lemma set_union_minus_same2 [rewrite]: "(B \<union> A) - B = A - B" by auto
lemma set_union_minus_distinct [rewrite]: "a \<noteq> c \<Longrightarrow> {a} \<union> (B - {c}) = {a} \<union> B - {c}" by auto
setup \<open>add_forward_prfstep_cond @{thm Set.Diff_subset} [with_term "?A - ?B"]\<close>
lemma union_subtract_elt1 [rewrite]: "x \<notin> B \<Longrightarrow> ({x} \<union> B) - {x} = B" by simp
lemma union_subtract_elt2 [rewrite]: "x \<notin> B \<Longrightarrow> (B \<union> {x}) - {x} = B" by simp
lemma subset_sub1 [backward]: "x \<in> A \<Longrightarrow> A - {x} \<subset> A" by auto
lemma member_notin [forward]: "x \<in> S - {y} \<Longrightarrow> x \<noteq> y" by simp
lemma member_notin_contra: "x \<in> S \<Longrightarrow> x \<noteq> y \<Longrightarrow> x \<in> S - {y}" by simp
setup \<open>add_forward_prfstep_cond @{thm member_notin_contra} [with_term "?S - {?y}"]\<close>

subsubsection \<open>Results on finite sets\<close>

setup \<open>add_resolve_prfstep @{thm Finite_Set.finite.emptyI}\<close>
lemma set_finite_single [resolve]: "finite {x}" by simp
setup \<open>add_rewrite_rule @{thm Finite_Set.finite_Un}\<close>
lemma Max_ge' [forward]: "finite A \<Longrightarrow> x > Max A \<Longrightarrow> \<not>(x \<in> A)" using Max_ge leD by auto
setup \<open>add_backward_prfstep @{thm finite_image_set}\<close>
setup \<open>add_forward_prfstep @{thm finite_atLeastAtMost}\<close>
setup \<open>add_forward_prfstep @{thm rev_finite_subset}\<close>
setup \<open>add_backward1_prfstep @{thm rev_finite_subset}\<close>

subsubsection \<open>Cardinality\<close>

setup \<open>add_rewrite_rule @{thm card.empty}\<close>
lemma card_emptyD [rewrite]: "finite S \<Longrightarrow> card S = 0 \<Longrightarrow> S = {}" by simp
lemma card_minus1 [rewrite]: "x \<in> S \<Longrightarrow> card (S - {x}) = card S - 1" by (simp add: card_Diff_subset)
setup \<open>add_forward_prfstep @{thm finite_Diff}\<close>
setup \<open>add_resolve_prfstep @{thm card_mono}\<close>

subsubsection \<open>Image set\<close>

setup \<open>add_rewrite_rule @{thm Set.image_Un}\<close>
setup \<open>add_rewrite_rule @{thm image_set_diff}\<close>

subsection \<open>Multiset\<close>

subsubsection \<open>Basic properties\<close>

lemma mset_member_empty [resolve]: "\<not>p \<in># {#}" by simp
lemma mem_multiset_single [rewrite]: "x \<in># {#y#} \<longleftrightarrow> x = y" by simp
setup \<open>add_backward2_prfstep @{thm subset_mset.antisym}\<close>
setup \<open>add_resolve_prfstep @{thm Multiset.empty_le}\<close>
setup \<open>add_forward_prfstep @{thm mset_subsetD}\<close>

lemma multi_contain_add_self1 [resolve]: "A \<subset># {#x#} + A" by simp
lemma multi_contain_add_self2 [resolve]: "A \<subset># A + {#x#}" by simp
setup \<open>add_forward_prfstep_cond @{thm Multiset.multi_member_this} [with_term "{#?x#} + ?XS"]\<close>
lemma multi_member_this2: "x \<in># XS + {#x#}" by simp
setup \<open>add_forward_prfstep_cond @{thm multi_member_this2} [with_term "?XS + {#?x#}"]\<close>
setup \<open>add_backward_prfstep @{thm Multiset.subset_mset.add_left_mono}\<close>
setup \<open>add_backward_prfstep @{thm Multiset.subset_mset.add_right_mono}\<close>

subsubsection \<open>Case checking and induction\<close>

lemma multi_nonempty_split' [resolve]: "M \<noteq> {#} \<Longrightarrow> \<exists>M' m. M = M' + {#m#}"
  using multi_nonempty_split by auto

lemma multi_member_split' [backward]: "x \<in># M \<Longrightarrow> \<exists>M'. M = M' + {#x#}"
  by (metis insert_DiffM2)

setup \<open>add_strong_induct_rule @{thm full_multiset_induct}\<close>

subsubsection \<open>Results on mset\<close>

setup \<open>add_rewrite_rule @{thm set_mset_empty}\<close>
setup \<open>add_rewrite_rule @{thm set_mset_single}\<close>
setup \<open>add_rewrite_rule @{thm set_mset_union}\<close>

setup \<open>add_rewrite_rule @{thm image_mset_empty}\<close>
setup \<open>add_rewrite_rule @{thm image_mset_single}\<close>
setup \<open>add_rewrite_rule @{thm image_mset_union}\<close>

setup \<open>add_rewrite_rule @{thm prod_mset_empty}\<close>
setup \<open>add_rewrite_rule @{thm prod_mset_singleton}\<close>
setup \<open>add_rewrite_rule @{thm prod_mset_Un}\<close>

subsubsection \<open>Set interval\<close>

setup \<open>add_rewrite_rule @{thm Set_Interval.ord_class.lessThan_iff}\<close>
setup \<open>add_rewrite_rule @{thm Set_Interval.ord_class.atLeastAtMost_iff}\<close>

end