SSL Set_Thms.thy

(*
  File: Set_Thms.thy
  Author: Bohua Zhan

  Setup of proof steps related to sets.
*)

section ‹Setup for sets and multisets›

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

subsection ‹Set›

subsubsection ‹Injective functions›

setup ‹add_backward_prfstep @{thm injI}›

subsubsection ‹AC property of intersection and union›

setup ‹fold Auto2_HOL_Extra_Setup.ACUtil.add_ac_data [
 {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 = @{thm inf_top_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}}]
 ›

subsubsection ‹Collection and bounded quantification›

setup ‹add_rewrite_rule @{thm Set.mem_Collect_eq}›
lemma ball_single [rewrite]: "(∀x∈{x}. P x) = P x" by auto

subsubsection ‹Membership›

setup ‹add_rewrite_rule @{thm Set.singleton_iff}›
setup ‹add_forward_prfstep (equiv_forward_th @{thm Set.empty_iff})›
lemma set_membership_distinct [forward]: "x ∈ s ==> y ∉ s ==> x ≠ y" by auto
lemma non_empty_exist_elt [backward]: "U ≠ {} ==> ∃x. x ∈ U" by blast
lemma non_univ_exist_compl [backward]: "U ≠ UNIV ==> ∃x. x ∉ U" by blast
setup ‹add_resolve_prfstep @{thm Set.UNIV_I}›

subsubsection ‹Insert›

setup ‹add_backward_prfstep_cond (equiv_backward_th @{thm Set.insert_iff}) [with_cond "?A ≠ {}"]›
setup ‹add_forward_prfstep_cond (equiv_forward_th @{thm Set.insert_iff})
 [with_score 500, with_cond "?A ≠ {}"]›
setup ‹add_forward_prfstep_cond (equiv_forward_th @{thm Set.insert_subset}) [with_cond "?A ≠ {}"]›
setup ‹add_backward_prfstep_cond (equiv_backward_th @{thm Set.insert_subset})
 [with_score 500, with_cond "?A ≠ {}"]›

subsubsection ‹Extensionality›

lemma set_ext [forward]: "∀a. a ∈ S ⟷ a ∈ T ==> S = T" by auto
setup ‹add_backward_prfstep_cond @{thm set_ext} [with_score 500, with_filt (order_filter "S" "T")]›

lemma set_pair_ext [forward]: "∀a b. (a, b) ∈ S ⟷ (a, b) ∈ T ==> S = T" by auto

subsubsection ‹Union›

setup ‹add_forward_prfstep_cond (equiv_forward_th @{thm Set.Un_iff}) [with_score 500]›
setup ‹add_backward_prfstep (equiv_backward_th @{thm Set.Un_iff})›

lemma UnD1 [forward]: "c ∈ A ∪ B ==> c ∉ A ==> c ∈ B" by auto
lemma UnD2   sets
lemma UnD1_single [forward]: "c ∈ {a} ∪ B ==> c ≠ a ==> c ∈ B")
lemma UnD2_single [forward]: "c ∈ A ∪
setup ‹ ?B"]›
setup ‹add_forward_prfstep_cond @{thm Set.UnI2}} [wit_erm "? \union> ?B"]🚫
lemma UnI1_single: "  ∈ {a} ∪ B" by auto
  UnI2_single: "b ∈ A ∪ {b}" by auto
  ‹add_forward_prfstep_cond @{thm UnI1_single} [with_ter
  ‹add_forward_prfstep_cond @{thm UnI2_single} [with_term "?A ∪

  union_single_eq [rewrite, backward]: "x <> 

  ‹AC property of intersection and union›

  ‹
  ‹

  ‹
  \<penadd_rewrite_rule
  ‹
  set_disjoint_mp [forward, backward2]: "A ∩} \Longrightarrow> p ∈ A ==>∉
  set_disjoint_single [rewrite]: "{x} ∩ B = {} ⟷ x ∉ B" by simp

  ‹subset›

 Collection and bounded quantification›
  ‹

  ‹add_resolve_prfstep @{thm emp
  ‹
  subset_single [rewrite]: "{a} ⊆ B \lemma[rw>x∈x" by
  ‹add_resolve_prfstep @{t
  ‹
  ‹add_resolve_prfstep @{thm Set.Un_upper2}› (equiv_forward_th @{thm Set.empty_iff})›
  union_is_subset [forward]: "A ∪ B ⊆ C ==> A ⊆ C ∧ B ⊆ C" by simp
  ‹add_backward1_prfstep @{thm Set.Un_least}›
  ‹add_backward2_prfstep @{thm Set.Un_least}›
  subset_union_same1 [backward]: "B ⊆ C ==> A ∪ B ⊆ A ∪ C" by auto
  subset_union_same2 [backward]: "A ⊆ B ==> A ∪ C ⊆ B ∪ C" by auto

  ‹Diff›

  ‹add_forward_prfstep (equiv_forward_th @{thm Set.Diff_iff})›
  ‹add_backward_prfstep_cond (equiv_backward_th @{thm Set.Diff_iff}) [with_score 500]›

  ‹add_rewrite_rule @{thm Set.empty_Diff}›
  mem_diff [rewrite]: "x ∈ A - B ⟷ x ∈ A ∧ x ∉ B" by simp
  set_union_minus_same1 [rewrite]: "(A ∪ B) - B = A - B" by auto
  set_union_minus_same2 [rewrite]: "(B ∪ A) - B = A - B" by auto
  \opena@{thm
  ‹
  union_subtract_elt1 [rewrite]
  union_[rewrite]:" \notin B 🚫
  subset_sub1 [backward]: "x ∈ A ==> A - {x} ⊂<>add_forward_prfstep_cond
  member_notin [forward]: "x ∈ S - {y} ==> x ≠ y" by simp
  member_notin_contra: "x ∈ S ==> x ≠ y ==> x ∈ S - {y}" by simp
  ‹add_forward_prfstep_cond @{thm member_notin_contra} [with_term "?S - {?y}"]› @{hm S.inse) [with_cnd "?A ≠

  ‹

  ‹>
  set_finite_single [resolve]: "finite {x}" by simp
  ‹add_rewrite_rule @{thm Finite_Set.finite_Un}›
  Max_ge' [forward]: "finite A ==> x > Max A ==> ¬(x ∈ A)" using Max_ge leD by auto
  ‹
 open>dd @{thm ficlose>
  ‹
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

  ‹Cardinality›

  ‹add_rewrite_rule @{thm card.empty}›
  card_emptyD [rewrite]: "finite S ==> card S = 0 ==> S = {}" by simp
  card_minus1 [rewrite]: "x ∈ S ==> card (S - {x}) = card S - 1" by (simp add: card_Diff_subset)
  ‹add_forward_prfstep @{thm finite_Diff}›
  ‹add_resolve_prfstep @{thm card_mono}›

  ‹Image set›

  ‹add_rewrite_rule @{thm Set.image_Un}›
  ‹add_rewrite_rule @{thm image_set_diff}›

  ‹Multiset›

  ‹Basic properties›

  mset_member_empty [resolve]: "¬p ∈# {#}" by simp
  mem_multiset_single [rewrite]: "x ∈# {#y#} ⟷ x = y" by simp
  ‹add_backward2_prfstep @{thm subset_mset.antisym}›
  ‹add_resolve_prfstep @{thm Multiset.empty_le}›
  ‹add_forward_prfstep @{thm mset_subsetD}›

  multi_contain_add_self1 [resolve]: "A ⊂# {#x#} + A" by simp
  multi_contain_add_self2 [resolve]: "A ⊂# A + {#x#}" by simp
  ‹add_forward_prfstep_cond @{thm Multiset.multi_member_this} [with_term "{#?x#} + ?XS"]›
  multi_member_this2: "x ∈# XS + {#x#}" by simp
  ‹add_forward_prfstep_cond @{thm multi_member_this2} [with_term "?XS + {#?x#}"]›
  ‹add_backward_prfstep @{thm Multiset.subset_mset.add_left_mono}›
  ‹add_backward_prfstep @{thm Multiset.subset_mset.add_right_mono}›

  ‹Case checking and induction›

  multi_nonempty_split' [resolve]: "M ≠ {#} ==> ∃M' m. M = M' + {#m#}"
 using multi_nonempty_split by auto

  multi_member_split' [backward]: "x ∈# M ==> ∃M'. M = M' + {#x#}"
 by (metis insert_DiffM2)

  ‹add_strong_induct_rule @{thm full_multiset_induct}›

  ‹Results on mset›

  ‹add_rewrite_rule @{thm set_mset_empty}›
  ‹add_rewrite_rule @{thm set_mset_single}›
  ‹add_rewrite_rule @{thm set_mset_union}›

  ‹add_rewrite_rule @{thm image_mset_empty}›
  ‹add_rewrite_rule @{thm image_mset_single}›
  ‹add_rewrite_rule @{thm image_mset_union}›

  ‹add_rewrite_rule @{thm prod_mset_empty}›
  ‹add_rewrite_rule @{thm prod_mset_singleton}›
  ‹add_rewrite_rule @{thm prod_mset_Un}›

  ‹Set interval›

  ‹add_rewrite_rule @{thm Set_Interval.ord_class.lessThan_iff}›
  ‹add_rewrite_rule @{thm Set_Interval.ord_class.atLeastAtMost_iff}›