author | paulson <lp15@cam.ac.uk> |
Tue, 13 Feb 2024 17:18:50 +0000 | |
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parent 78132 | 177dae28697b |
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permissions | -rw-r--r-- |
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(* Title: HOL/Library/FSet.thy |
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Author: Ondrej Kuncar, TU Muenchen |
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Author: Cezary Kaliszyk and Christian Urban |
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Author: Andrei Popescu, TU Muenchen |
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Author: Martin Desharnais, MPI-INF Saarbruecken |
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*) |
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section \<open>Type of finite sets defined as a subtype of sets\<close> |
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theory FSet |
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imports Main Countable |
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begin |
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subsection \<open>Definition of the type\<close> |
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typedef 'a fset = "{A :: 'a set. finite A}" morphisms fset Abs_fset |
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by auto |
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setup_lifting type_definition_fset |
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subsection \<open>Basic operations and type class instantiations\<close> |
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(* FIXME transfer and right_total vs. bi_total *) |
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instantiation fset :: (finite) finite |
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begin |
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instance by (standard; transfer; simp) |
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end |
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instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}" |
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begin |
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lift_definition bot_fset :: "'a fset" is "{}" parametric empty_transfer by simp |
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lift_definition less_eq_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" is subset_eq parametric subset_transfer |
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. |
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definition less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" where "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)" |
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lemma less_fset_transfer[transfer_rule]: |
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includes lifting_syntax |
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assumes [transfer_rule]: "bi_unique A" |
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shows "((pcr_fset A) ===> (pcr_fset A) ===> (=)) (\<subset>) (<)" |
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unfolding less_fset_def[abs_def] psubset_eq[abs_def] by transfer_prover |
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lift_definition sup_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is union parametric union_transfer |
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by simp |
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lift_definition inf_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is inter parametric inter_transfer |
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by simp |
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lift_definition minus_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is minus parametric Diff_transfer |
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by simp |
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instance |
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by (standard; transfer; auto)+ |
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end |
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abbreviation fempty :: "'a fset" ("{||}") where "{||} \<equiv> bot" |
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abbreviation fsubset_eq :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50) where "xs |\<subseteq>| ys \<equiv> xs \<le> ys" |
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abbreviation fsubset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50) where "xs |\<subset>| ys \<equiv> xs < ys" |
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abbreviation funion :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65) where "xs |\<union>| ys \<equiv> sup xs ys" |
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abbreviation finter :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<inter>|" 65) where "xs |\<inter>| ys \<equiv> inf xs ys" |
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abbreviation fminus :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|-|" 65) where "xs |-| ys \<equiv> minus xs ys" |
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instantiation fset :: (equal) equal |
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begin |
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definition "HOL.equal A B \<longleftrightarrow> A |\<subseteq>| B \<and> B |\<subseteq>| A" |
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instance by intro_classes (auto simp add: equal_fset_def) |
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end |
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instantiation fset :: (type) conditionally_complete_lattice |
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begin |
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context includes lifting_syntax |
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begin |
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lemma right_total_Inf_fset_transfer: |
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assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A" |
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shows "(rel_set (rel_set A) ===> rel_set A) |
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(\<lambda>S. if finite (\<Inter>S \<inter> Collect (Domainp A)) then \<Inter>S \<inter> Collect (Domainp A) else {}) |
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(\<lambda>S. if finite (Inf S) then Inf S else {})" |
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by transfer_prover |
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lemma Inf_fset_transfer: |
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assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A" |
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shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Inf A) then Inf A else {}) |
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(\<lambda>A. if finite (Inf A) then Inf A else {})" |
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by transfer_prover |
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lift_definition Inf_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Inf A) then Inf A else {}" |
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parametric right_total_Inf_fset_transfer Inf_fset_transfer by simp |
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lemma Sup_fset_transfer: |
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assumes [transfer_rule]: "bi_unique A" |
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shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>A. if finite (Sup A) then Sup A else {}) |
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(\<lambda>A. if finite (Sup A) then Sup A else {})" by transfer_prover |
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lift_definition Sup_fset :: "'a fset set \<Rightarrow> 'a fset" is "\<lambda>A. if finite (Sup A) then Sup A else {}" |
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parametric Sup_fset_transfer by simp |
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lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)" |
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by (auto intro: finite_subset) |
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lemma transfer_bdd_below[transfer_rule]: "(rel_set (pcr_fset (=)) ===> (=)) bdd_below bdd_below" |
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by auto |
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end |
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instance |
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proof |
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fix x z :: "'a fset" |
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fix X :: "'a fset set" |
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{ |
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assume "x \<in> X" "bdd_below X" |
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then show "Inf X |\<subseteq>| x" by transfer auto |
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next |
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assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)" |
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then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast) |
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next |
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assume "x \<in> X" "bdd_above X" |
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then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)" |
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by (auto simp: bdd_above_def) |
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then show "x |\<subseteq>| Sup X" |
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by transfer (auto intro!: finite_Sup) |
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next |
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assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)" |
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then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast) |
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} |
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qed |
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end |
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instantiation fset :: (finite) complete_lattice |
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begin |
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lift_definition top_fset :: "'a fset" is UNIV parametric right_total_UNIV_transfer UNIV_transfer |
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by simp |
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instance |
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by (standard; transfer; auto) |
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end |
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instantiation fset :: (finite) complete_boolean_algebra |
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begin |
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lift_definition uminus_fset :: "'a fset \<Rightarrow> 'a fset" is uminus |
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parametric right_total_Compl_transfer Compl_transfer by simp |
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instance |
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by (standard; transfer) (simp_all add: Inf_Sup Diff_eq) |
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end |
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abbreviation fUNIV :: "'a::finite fset" where "fUNIV \<equiv> top" |
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abbreviation fuminus :: "'a::finite fset \<Rightarrow> 'a fset" ("|-| _" [81] 80) where "|-| x \<equiv> uminus x" |
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declare top_fset.rep_eq[simp] |
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subsection \<open>Other operations\<close> |
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lift_definition finsert :: "'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is insert parametric Lifting_Set.insert_transfer |
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by simp |
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syntax |
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"_insert_fset" :: "args => 'a fset" ("{|(_)|}") |
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translations |
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"{|x, xs|}" == "CONST finsert x {|xs|}" |
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"{|x|}" == "CONST finsert x {||}" |
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abbreviation fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50) where |
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"x |\<in>| X \<equiv> x \<in> fset X" |
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abbreviation not_fmember :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50) where |
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"x |\<notin>| X \<equiv> x \<notin> fset X" |
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context |
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begin |
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qualified abbreviation Ball :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where |
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"Ball X \<equiv> Set.Ball (fset X)" |
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alias fBall = FSet.Ball |
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qualified abbreviation Bex :: "'a fset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" where |
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"Bex X \<equiv> Set.Bex (fset X)" |
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alias fBex = FSet.Bex |
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end |
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context includes lifting_syntax |
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begin |
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lemma fmember_transfer0[transfer_rule]: |
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assumes [transfer_rule]: "bi_unique A" |
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shows "(A ===> pcr_fset A ===> (=)) (\<in>) (|\<in>|)" |
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by transfer_prover |
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lemma fBall_transfer0[transfer_rule]: |
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assumes [transfer_rule]: "bi_unique A" |
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shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Ball) (fBall)" |
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by transfer_prover |
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lemma fBex_transfer0[transfer_rule]: |
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assumes [transfer_rule]: "bi_unique A" |
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shows "(pcr_fset A ===> (A ===> (=)) ===> (=)) (Bex) (fBex)" |
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by transfer_prover |
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lift_definition ffilter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset" is Set.filter |
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parametric Lifting_Set.filter_transfer unfolding Set.filter_def by simp |
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lift_definition fPow :: "'a fset \<Rightarrow> 'a fset fset" is Pow parametric Pow_transfer |
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by (simp add: finite_subset) |
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lift_definition fcard :: "'a fset \<Rightarrow> nat" is card parametric card_transfer . |
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lift_definition fimage :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset" (infixr "|`|" 90) is image |
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parametric image_transfer by simp |
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lift_definition fthe_elem :: "'a fset \<Rightarrow> 'a" is the_elem . |
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lift_definition fbind :: "'a fset \<Rightarrow> ('a \<Rightarrow> 'b fset) \<Rightarrow> 'b fset" is Set.bind parametric bind_transfer |
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by (simp add: Set.bind_def) |
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lift_definition ffUnion :: "'a fset fset \<Rightarrow> 'a fset" is Union parametric Union_transfer by simp |
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lift_definition ffold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" is Finite_Set.fold . |
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lift_definition fset_of_list :: "'a list \<Rightarrow> 'a fset" is set by (rule finite_set) |
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lift_definition sorted_list_of_fset :: "'a::linorder fset \<Rightarrow> 'a list" is sorted_list_of_set . |
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subsection \<open>Transferred lemmas from Set.thy\<close> |
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lemma fset_eqI: "(\<And>x. (x |\<in>| A) = (x |\<in>| B)) \<Longrightarrow> A = B" |
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by (rule set_eqI[Transfer.transferred]) |
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lemma fset_eq_iff[no_atp]: "(A = B) = (\<forall>x. (x |\<in>| A) = (x |\<in>| B))" |
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by (rule set_eq_iff[Transfer.transferred]) |
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lemma fBallI[no_atp]: "(\<And>x. x |\<in>| A \<Longrightarrow> P x) \<Longrightarrow> fBall A P" |
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by (rule ballI[Transfer.transferred]) |
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lemma fbspec[no_atp]: "fBall A P \<Longrightarrow> x |\<in>| A \<Longrightarrow> P x" |
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by (rule bspec[Transfer.transferred]) |
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lemma fBallE[no_atp]: "fBall A P \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> (x |\<notin>| A \<Longrightarrow> Q) \<Longrightarrow> Q" |
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by (rule ballE[Transfer.transferred]) |
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lemma fBexI[no_atp]: "P x \<Longrightarrow> x |\<in>| A \<Longrightarrow> fBex A P" |
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by (rule bexI[Transfer.transferred]) |
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lemma rev_fBexI[no_atp]: "x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> fBex A P" |
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by (rule rev_bexI[Transfer.transferred]) |
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lemma fBexCI[no_atp]: "(fBall A (\<lambda>x. \<not> P x) \<Longrightarrow> P a) \<Longrightarrow> a |\<in>| A \<Longrightarrow> fBex A P" |
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by (rule bexCI[Transfer.transferred]) |
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|
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lemma fBexE[no_atp]: "fBex A P \<Longrightarrow> (\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q) \<Longrightarrow> Q" |
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by (rule bexE[Transfer.transferred]) |
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266 |
|
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lemma fBall_triv[no_atp]: "fBall A (\<lambda>x. P) = ((\<exists>x. x |\<in>| A) \<longrightarrow> P)" |
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268 |
by (rule ball_triv[Transfer.transferred]) |
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lemma fBex_triv[no_atp]: "fBex A (\<lambda>x. P) = ((\<exists>x. x |\<in>| A) \<and> P)" |
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271 |
by (rule bex_triv[Transfer.transferred]) |
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lemma fBex_triv_one_point1[no_atp]: "fBex A (\<lambda>x. x = a) = (a |\<in>| A)" |
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274 |
by (rule bex_triv_one_point1[Transfer.transferred]) |
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lemma fBex_triv_one_point2[no_atp]: "fBex A ((=) a) = (a |\<in>| A)" |
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277 |
by (rule bex_triv_one_point2[Transfer.transferred]) |
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|
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lemma fBex_one_point1[no_atp]: "fBex A (\<lambda>x. x = a \<and> P x) = (a |\<in>| A \<and> P a)" |
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280 |
by (rule bex_one_point1[Transfer.transferred]) |
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|
281 |
|
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lemma fBex_one_point2[no_atp]: "fBex A (\<lambda>x. a = x \<and> P x) = (a |\<in>| A \<and> P a)" |
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|
283 |
by (rule bex_one_point2[Transfer.transferred]) |
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|
284 |
|
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lemma fBall_one_point1[no_atp]: "fBall A (\<lambda>x. x = a \<longrightarrow> P x) = (a |\<in>| A \<longrightarrow> P a)" |
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|
286 |
by (rule ball_one_point1[Transfer.transferred]) |
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|
287 |
|
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lemma fBall_one_point2[no_atp]: "fBall A (\<lambda>x. a = x \<longrightarrow> P x) = (a |\<in>| A \<longrightarrow> P a)" |
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|
289 |
by (rule ball_one_point2[Transfer.transferred]) |
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|
290 |
|
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lemma fBall_conj_distrib: "fBall A (\<lambda>x. P x \<and> Q x) = (fBall A P \<and> fBall A Q)" |
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292 |
by (rule ball_conj_distrib[Transfer.transferred]) |
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|
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|
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lemma fBex_disj_distrib: "fBex A (\<lambda>x. P x \<or> Q x) = (fBex A P \<or> fBex A Q)" |
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by (rule bex_disj_distrib[Transfer.transferred]) |
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|
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|
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lemma fBall_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x) \<Longrightarrow> fBall A P = fBall B Q" |
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298 |
by (rule ball_cong[Transfer.transferred]) |
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|
299 |
|
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lemma fBex_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x) \<Longrightarrow> fBex A P = fBex B Q" |
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301 |
by (rule bex_cong[Transfer.transferred]) |
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|
302 |
|
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303 |
lemma fsubsetI[intro!]: "(\<And>x. x |\<in>| A \<Longrightarrow> x |\<in>| B) \<Longrightarrow> A |\<subseteq>| B" |
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304 |
by (rule subsetI[Transfer.transferred]) |
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|
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|
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lemma fsubsetD[elim, intro?]: "A |\<subseteq>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B" |
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307 |
by (rule subsetD[Transfer.transferred]) |
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|
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|
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lemma rev_fsubsetD[no_atp,intro?]: "c |\<in>| A \<Longrightarrow> A |\<subseteq>| B \<Longrightarrow> c |\<in>| B" |
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310 |
by (rule rev_subsetD[Transfer.transferred]) |
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|
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|
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lemma fsubsetCE[no_atp,elim]: "A |\<subseteq>| B \<Longrightarrow> (c |\<notin>| A \<Longrightarrow> P) \<Longrightarrow> (c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P" |
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by (rule subsetCE[Transfer.transferred]) |
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|
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|
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315 |
lemma fsubset_eq[no_atp]: "(A |\<subseteq>| B) = fBall A (\<lambda>x. x |\<in>| B)" |
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|
316 |
by (rule subset_eq[Transfer.transferred]) |
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|
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|
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lemma contra_fsubsetD[no_atp]: "A |\<subseteq>| B \<Longrightarrow> c |\<notin>| B \<Longrightarrow> c |\<notin>| A" |
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319 |
by (rule contra_subsetD[Transfer.transferred]) |
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|
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|
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lemma fsubset_refl: "A |\<subseteq>| A" |
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322 |
by (rule subset_refl[Transfer.transferred]) |
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|
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|
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lemma fsubset_trans: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<subseteq>| C" |
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325 |
by (rule subset_trans[Transfer.transferred]) |
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|
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|
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lemma fset_rev_mp: "c |\<in>| A \<Longrightarrow> A |\<subseteq>| B \<Longrightarrow> c |\<in>| B" |
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328 |
by (rule rev_subsetD[Transfer.transferred]) |
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|
329 |
|
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330 |
lemma fset_mp: "A |\<subseteq>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B" |
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331 |
by (rule subsetD[Transfer.transferred]) |
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|
332 |
|
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333 |
lemma fsubset_not_fsubset_eq[code]: "(A |\<subset>| B) = (A |\<subseteq>| B \<and> \<not> B |\<subseteq>| A)" |
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|
334 |
by (rule subset_not_subset_eq[Transfer.transferred]) |
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|
335 |
|
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336 |
lemma eq_fmem_trans: "a = b \<Longrightarrow> b |\<in>| A \<Longrightarrow> a |\<in>| A" |
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337 |
by (rule eq_mem_trans[Transfer.transferred]) |
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|
338 |
|
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339 |
lemma fsubset_antisym[intro!]: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> A = B" |
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340 |
by (rule subset_antisym[Transfer.transferred]) |
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|
341 |
|
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342 |
lemma fequalityD1: "A = B \<Longrightarrow> A |\<subseteq>| B" |
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|
343 |
by (rule equalityD1[Transfer.transferred]) |
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|
344 |
|
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lemma fequalityD2: "A = B \<Longrightarrow> B |\<subseteq>| A" |
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|
346 |
by (rule equalityD2[Transfer.transferred]) |
f40bc75b2a3f
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desharna
parents:
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changeset
|
347 |
|
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parents:
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|
348 |
lemma fequalityE: "A = B \<Longrightarrow> (A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> P) \<Longrightarrow> P" |
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parents:
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|
349 |
by (rule equalityE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
350 |
|
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parents:
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changeset
|
351 |
lemma fequalityCE[elim]: |
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|
352 |
"A = B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> (c |\<notin>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> P) \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
353 |
by (rule equalityCE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
354 |
|
f40bc75b2a3f
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parents:
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changeset
|
355 |
lemma eqfset_imp_iff: "A = B \<Longrightarrow> (x |\<in>| A) = (x |\<in>| B)" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
356 |
by (rule eqset_imp_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
357 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
358 |
lemma eqfelem_imp_iff: "x = y \<Longrightarrow> (x |\<in>| A) = (y |\<in>| A)" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
359 |
by (rule eqelem_imp_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
360 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
361 |
lemma fempty_iff[simp]: "(c |\<in>| {||}) = False" |
f40bc75b2a3f
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parents:
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changeset
|
362 |
by (rule empty_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
363 |
|
f40bc75b2a3f
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parents:
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changeset
|
364 |
lemma fempty_fsubsetI[iff]: "{||} |\<subseteq>| x" |
f40bc75b2a3f
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parents:
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changeset
|
365 |
by (rule empty_subsetI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
366 |
|
f40bc75b2a3f
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parents:
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diff
changeset
|
367 |
lemma equalsffemptyI: "(\<And>y. y |\<in>| A \<Longrightarrow> False) \<Longrightarrow> A = {||}" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
368 |
by (rule equals0I[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
369 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
370 |
lemma equalsffemptyD: "A = {||} \<Longrightarrow> a |\<notin>| A" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
371 |
by (rule equals0D[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
372 |
|
f40bc75b2a3f
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parents:
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diff
changeset
|
373 |
lemma fBall_fempty[simp]: "fBall {||} P = True" |
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parents:
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changeset
|
374 |
by (rule ball_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
375 |
|
f40bc75b2a3f
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parents:
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changeset
|
376 |
lemma fBex_fempty[simp]: "fBex {||} P = False" |
f40bc75b2a3f
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parents:
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changeset
|
377 |
by (rule bex_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
378 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
379 |
lemma fPow_iff[iff]: "(A |\<in>| fPow B) = (A |\<subseteq>| B)" |
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
380 |
by (rule Pow_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
381 |
|
f40bc75b2a3f
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parents:
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diff
changeset
|
382 |
lemma fPowI: "A |\<subseteq>| B \<Longrightarrow> A |\<in>| fPow B" |
f40bc75b2a3f
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parents:
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changeset
|
383 |
by (rule PowI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
384 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
385 |
lemma fPowD: "A |\<in>| fPow B \<Longrightarrow> A |\<subseteq>| B" |
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
386 |
by (rule PowD[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
387 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
388 |
lemma fPow_bottom: "{||} |\<in>| fPow B" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
389 |
by (rule Pow_bottom[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
390 |
|
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
391 |
lemma fPow_top: "A |\<in>| fPow A" |
f40bc75b2a3f
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desharna
parents:
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diff
changeset
|
392 |
by (rule Pow_top[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
393 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
394 |
lemma fPow_not_fempty: "fPow A \<noteq> {||}" |
f40bc75b2a3f
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desharna
parents:
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changeset
|
395 |
by (rule Pow_not_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
396 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
397 |
lemma finter_iff[simp]: "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
398 |
by (rule Int_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
399 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
400 |
lemma finterI[intro!]: "c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> c |\<in>| A |\<inter>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
401 |
by (rule IntI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
402 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
403 |
lemma finterD1: "c |\<in>| A |\<inter>| B \<Longrightarrow> c |\<in>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
404 |
by (rule IntD1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
405 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
406 |
lemma finterD2: "c |\<in>| A |\<inter>| B \<Longrightarrow> c |\<in>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
407 |
by (rule IntD2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
408 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
409 |
lemma finterE[elim!]: "c |\<in>| A |\<inter>| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
410 |
by (rule IntE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
411 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
412 |
lemma funion_iff[simp]: "(c |\<in>| A |\<union>| B) = (c |\<in>| A \<or> c |\<in>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
413 |
by (rule Un_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
414 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
415 |
lemma funionI1[elim?]: "c |\<in>| A \<Longrightarrow> c |\<in>| A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
416 |
by (rule UnI1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
417 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
418 |
lemma funionI2[elim?]: "c |\<in>| B \<Longrightarrow> c |\<in>| A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
419 |
by (rule UnI2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
420 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
421 |
lemma funionCI[intro!]: "(c |\<notin>| B \<Longrightarrow> c |\<in>| A) \<Longrightarrow> c |\<in>| A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
422 |
by (rule UnCI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
423 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
424 |
lemma funionE[elim!]: "c |\<in>| A |\<union>| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> P) \<Longrightarrow> (c |\<in>| B \<Longrightarrow> P) \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
425 |
by (rule UnE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
426 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
427 |
lemma fminus_iff[simp]: "(c |\<in>| A |-| B) = (c |\<in>| A \<and> c |\<notin>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
428 |
by (rule Diff_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
429 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
430 |
lemma fminusI[intro!]: "c |\<in>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> c |\<in>| A |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
431 |
by (rule DiffI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
432 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
433 |
lemma fminusD1: "c |\<in>| A |-| B \<Longrightarrow> c |\<in>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
434 |
by (rule DiffD1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
435 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
436 |
lemma fminusD2: "c |\<in>| A |-| B \<Longrightarrow> c |\<in>| B \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
437 |
by (rule DiffD2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
438 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
439 |
lemma fminusE[elim!]: "c |\<in>| A |-| B \<Longrightarrow> (c |\<in>| A \<Longrightarrow> c |\<notin>| B \<Longrightarrow> P) \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
440 |
by (rule DiffE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
441 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
442 |
lemma finsert_iff[simp]: "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
443 |
by (rule insert_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
444 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
445 |
lemma finsertI1: "a |\<in>| finsert a B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
446 |
by (rule insertI1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
447 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
448 |
lemma finsertI2: "a |\<in>| B \<Longrightarrow> a |\<in>| finsert b B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
449 |
by (rule insertI2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
450 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
451 |
lemma finsertE[elim!]: "a |\<in>| finsert b A \<Longrightarrow> (a = b \<Longrightarrow> P) \<Longrightarrow> (a |\<in>| A \<Longrightarrow> P) \<Longrightarrow> P" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
452 |
by (rule insertE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
453 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
454 |
lemma finsertCI[intro!]: "(a |\<notin>| B \<Longrightarrow> a = b) \<Longrightarrow> a |\<in>| finsert b B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
455 |
by (rule insertCI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
456 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
457 |
lemma fsubset_finsert_iff: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
458 |
"(A |\<subseteq>| finsert x B) = (if x |\<in>| A then A |-| {|x|} |\<subseteq>| B else A |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
459 |
by (rule subset_insert_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
460 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
461 |
lemma finsert_ident: "x |\<notin>| A \<Longrightarrow> x |\<notin>| B \<Longrightarrow> (finsert x A = finsert x B) = (A = B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
462 |
by (rule insert_ident[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
463 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
464 |
lemma fsingletonI[intro!,no_atp]: "a |\<in>| {|a|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
465 |
by (rule singletonI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
466 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
467 |
lemma fsingletonD[dest!,no_atp]: "b |\<in>| {|a|} \<Longrightarrow> b = a" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
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diff
changeset
|
468 |
by (rule singletonD[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
469 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
470 |
lemma fsingleton_iff: "(b |\<in>| {|a|}) = (b = a)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
471 |
by (rule singleton_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
472 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
473 |
lemma fsingleton_inject[dest!]: "{|a|} = {|b|} \<Longrightarrow> a = b" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
474 |
by (rule singleton_inject[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
475 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
476 |
lemma fsingleton_finsert_inj_eq[iff,no_atp]: "({|b|} = finsert a A) = (a = b \<and> A |\<subseteq>| {|b|})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
477 |
by (rule singleton_insert_inj_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
478 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
479 |
lemma fsingleton_finsert_inj_eq'[iff,no_atp]: "(finsert a A = {|b|}) = (a = b \<and> A |\<subseteq>| {|b|})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
480 |
by (rule singleton_insert_inj_eq'[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
481 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
482 |
lemma fsubset_fsingletonD: "A |\<subseteq>| {|x|} \<Longrightarrow> A = {||} \<or> A = {|x|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
483 |
by (rule subset_singletonD[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
484 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
485 |
lemma fminus_single_finsert: "A |-| {|x|} |\<subseteq>| B \<Longrightarrow> A |\<subseteq>| finsert x B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
486 |
by (rule Diff_single_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
487 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
488 |
lemma fdoubleton_eq_iff: "({|a, b|} = {|c, d|}) = (a = c \<and> b = d \<or> a = d \<and> b = c)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
489 |
by (rule doubleton_eq_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
490 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
491 |
lemma funion_fsingleton_iff: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
492 |
"(A |\<union>| B = {|x|}) = (A = {||} \<and> B = {|x|} \<or> A = {|x|} \<and> B = {||} \<or> A = {|x|} \<and> B = {|x|})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
493 |
by (rule Un_singleton_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
494 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
495 |
lemma fsingleton_funion_iff: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
496 |
"({|x|} = A |\<union>| B) = (A = {||} \<and> B = {|x|} \<or> A = {|x|} \<and> B = {||} \<or> A = {|x|} \<and> B = {|x|})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
497 |
by (rule singleton_Un_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
498 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
499 |
lemma fimage_eqI[simp, intro]: "b = f x \<Longrightarrow> x |\<in>| A \<Longrightarrow> b |\<in>| f |`| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
500 |
by (rule image_eqI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
501 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
502 |
lemma fimageI: "x |\<in>| A \<Longrightarrow> f x |\<in>| f |`| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
503 |
by (rule imageI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
504 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
505 |
lemma rev_fimage_eqI: "x |\<in>| A \<Longrightarrow> b = f x \<Longrightarrow> b |\<in>| f |`| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
506 |
by (rule rev_image_eqI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
507 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
508 |
lemma fimageE[elim!]: "b |\<in>| f |`| A \<Longrightarrow> (\<And>x. b = f x \<Longrightarrow> x |\<in>| A \<Longrightarrow> thesis) \<Longrightarrow> thesis" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
509 |
by (rule imageE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
510 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
511 |
lemma Compr_fimage_eq: "{x. x |\<in>| f |`| A \<and> P x} = f ` {x. x |\<in>| A \<and> P (f x)}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
512 |
by (rule Compr_image_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
513 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
514 |
lemma fimage_funion: "f |`| (A |\<union>| B) = f |`| A |\<union>| f |`| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
515 |
by (rule image_Un[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
516 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
517 |
lemma fimage_iff: "(z |\<in>| f |`| A) = fBex A (\<lambda>x. z = f x)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
518 |
by (rule image_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
519 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
520 |
lemma fimage_fsubset_iff[no_atp]: "(f |`| A |\<subseteq>| B) = fBall A (\<lambda>x. f x |\<in>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
521 |
by (rule image_subset_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
522 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
523 |
lemma fimage_fsubsetI: "(\<And>x. x |\<in>| A \<Longrightarrow> f x |\<in>| B) \<Longrightarrow> f |`| A |\<subseteq>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
524 |
by (rule image_subsetI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
525 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
526 |
lemma fimage_ident[simp]: "(\<lambda>x. x) |`| Y = Y" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
527 |
by (rule image_ident[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
528 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
529 |
lemma if_split_fmem1: "((if Q then x else y) |\<in>| b) = ((Q \<longrightarrow> x |\<in>| b) \<and> (\<not> Q \<longrightarrow> y |\<in>| b))" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
530 |
by (rule if_split_mem1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
531 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
532 |
lemma if_split_fmem2: "(a |\<in>| (if Q then x else y)) = ((Q \<longrightarrow> a |\<in>| x) \<and> (\<not> Q \<longrightarrow> a |\<in>| y))" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
533 |
by (rule if_split_mem2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
534 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
535 |
lemma pfsubsetI[intro!,no_atp]: "A |\<subseteq>| B \<Longrightarrow> A \<noteq> B \<Longrightarrow> A |\<subset>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
536 |
by (rule psubsetI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
537 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
538 |
lemma pfsubsetE[elim!,no_atp]: "A |\<subset>| B \<Longrightarrow> (A |\<subseteq>| B \<Longrightarrow> \<not> B |\<subseteq>| A \<Longrightarrow> R) \<Longrightarrow> R" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
539 |
by (rule psubsetE[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
540 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
541 |
lemma pfsubset_finsert_iff: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
542 |
"(A |\<subset>| finsert x B) = |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
543 |
(if x |\<in>| B then A |\<subset>| B else if x |\<in>| A then A |-| {|x|} |\<subset>| B else A |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
544 |
by (rule psubset_insert_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
545 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
546 |
lemma pfsubset_eq: "(A |\<subset>| B) = (A |\<subseteq>| B \<and> A \<noteq> B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
547 |
by (rule psubset_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
548 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
549 |
lemma pfsubset_imp_fsubset: "A |\<subset>| B \<Longrightarrow> A |\<subseteq>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
550 |
by (rule psubset_imp_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
551 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
552 |
lemma pfsubset_trans: "A |\<subset>| B \<Longrightarrow> B |\<subset>| C \<Longrightarrow> A |\<subset>| C" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
553 |
by (rule psubset_trans[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
554 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
555 |
lemma pfsubsetD: "A |\<subset>| B \<Longrightarrow> c |\<in>| A \<Longrightarrow> c |\<in>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
556 |
by (rule psubsetD[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
557 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
558 |
lemma pfsubset_fsubset_trans: "A |\<subset>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<subset>| C" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
559 |
by (rule psubset_subset_trans[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
560 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
561 |
lemma fsubset_pfsubset_trans: "A |\<subseteq>| B \<Longrightarrow> B |\<subset>| C \<Longrightarrow> A |\<subset>| C" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
562 |
by (rule subset_psubset_trans[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
563 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
564 |
lemma pfsubset_imp_ex_fmem: "A |\<subset>| B \<Longrightarrow> \<exists>b. b |\<in>| B |-| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
565 |
by (rule psubset_imp_ex_mem[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
566 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
567 |
lemma fimage_fPow_mono: "f |`| A |\<subseteq>| B \<Longrightarrow> (|`|) f |`| fPow A |\<subseteq>| fPow B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
568 |
by (rule image_Pow_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
569 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
570 |
lemma fimage_fPow_surj: "f |`| A = B \<Longrightarrow> (|`|) f |`| fPow A = fPow B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
571 |
by (rule image_Pow_surj[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
572 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
573 |
lemma fsubset_finsertI: "B |\<subseteq>| finsert a B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
574 |
by (rule subset_insertI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
575 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
576 |
lemma fsubset_finsertI2: "A |\<subseteq>| B \<Longrightarrow> A |\<subseteq>| finsert b B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
577 |
by (rule subset_insertI2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
578 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
579 |
lemma fsubset_finsert: "x |\<notin>| A \<Longrightarrow> (A |\<subseteq>| finsert x B) = (A |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
580 |
by (rule subset_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
581 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
582 |
lemma funion_upper1: "A |\<subseteq>| A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
583 |
by (rule Un_upper1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
584 |
|
78104 | 585 |
lemma funion_upper2: "B |\<subseteq>| A |\<union>| B" |
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
586 |
by (rule Un_upper2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
587 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
588 |
lemma funion_least: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> A |\<union>| B |\<subseteq>| C" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
589 |
by (rule Un_least[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
590 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
591 |
lemma finter_lower1: "A |\<inter>| B |\<subseteq>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
592 |
by (rule Int_lower1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
593 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
594 |
lemma finter_lower2: "A |\<inter>| B |\<subseteq>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
595 |
by (rule Int_lower2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
596 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
597 |
lemma finter_greatest: "C |\<subseteq>| A \<Longrightarrow> C |\<subseteq>| B \<Longrightarrow> C |\<subseteq>| A |\<inter>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
598 |
by (rule Int_greatest[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
599 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
600 |
lemma fminus_fsubset: "A |-| B |\<subseteq>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
601 |
by (rule Diff_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
602 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
603 |
lemma fminus_fsubset_conv: "(A |-| B |\<subseteq>| C) = (A |\<subseteq>| B |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
604 |
by (rule Diff_subset_conv[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
605 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
606 |
lemma fsubset_fempty[simp]: "(A |\<subseteq>| {||}) = (A = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
607 |
by (rule subset_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
608 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
609 |
lemma not_pfsubset_fempty[iff]: "\<not> A |\<subset>| {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
610 |
by (rule not_psubset_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
611 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
612 |
lemma finsert_is_funion: "finsert a A = {|a|} |\<union>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
613 |
by (rule insert_is_Un[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
614 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
615 |
lemma finsert_not_fempty[simp]: "finsert a A \<noteq> {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
616 |
by (rule insert_not_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
617 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
618 |
lemma fempty_not_finsert: "{||} \<noteq> finsert a A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
619 |
by (rule empty_not_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
620 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
621 |
lemma finsert_absorb: "a |\<in>| A \<Longrightarrow> finsert a A = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
622 |
by (rule insert_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
623 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
624 |
lemma finsert_absorb2[simp]: "finsert x (finsert x A) = finsert x A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
625 |
by (rule insert_absorb2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
626 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
627 |
lemma finsert_commute: "finsert x (finsert y A) = finsert y (finsert x A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
628 |
by (rule insert_commute[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
629 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
630 |
lemma finsert_fsubset[simp]: "(finsert x A |\<subseteq>| B) = (x |\<in>| B \<and> A |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
631 |
by (rule insert_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
632 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
633 |
lemma finsert_inter_finsert[simp]: "finsert a A |\<inter>| finsert a B = finsert a (A |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
634 |
by (rule insert_inter_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
635 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
636 |
lemma finsert_disjoint[simp,no_atp]: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
637 |
"(finsert a A |\<inter>| B = {||}) = (a |\<notin>| B \<and> A |\<inter>| B = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
638 |
"({||} = finsert a A |\<inter>| B) = (a |\<notin>| B \<and> {||} = A |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
639 |
by (rule insert_disjoint[Transfer.transferred])+ |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
640 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
641 |
lemma disjoint_finsert[simp,no_atp]: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
642 |
"(B |\<inter>| finsert a A = {||}) = (a |\<notin>| B \<and> B |\<inter>| A = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
643 |
"({||} = A |\<inter>| finsert b B) = (b |\<notin>| A \<and> {||} = A |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
644 |
by (rule disjoint_insert[Transfer.transferred])+ |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
645 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
646 |
lemma fimage_fempty[simp]: "f |`| {||} = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
647 |
by (rule image_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
648 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
649 |
lemma fimage_finsert[simp]: "f |`| finsert a B = finsert (f a) (f |`| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
650 |
by (rule image_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
651 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
652 |
lemma fimage_constant: "x |\<in>| A \<Longrightarrow> (\<lambda>x. c) |`| A = {|c|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
653 |
by (rule image_constant[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
654 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
655 |
lemma fimage_constant_conv: "(\<lambda>x. c) |`| A = (if A = {||} then {||} else {|c|})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
656 |
by (rule image_constant_conv[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
657 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
658 |
lemma fimage_fimage: "f |`| g |`| A = (\<lambda>x. f (g x)) |`| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
659 |
by (rule image_image[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
660 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
661 |
lemma finsert_fimage[simp]: "x |\<in>| A \<Longrightarrow> finsert (f x) (f |`| A) = f |`| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
662 |
by (rule insert_image[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
663 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
664 |
lemma fimage_is_fempty[iff]: "(f |`| A = {||}) = (A = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
665 |
by (rule image_is_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
666 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
667 |
lemma fempty_is_fimage[iff]: "({||} = f |`| A) = (A = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
668 |
by (rule empty_is_image[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
669 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
670 |
lemma fimage_cong: "M = N \<Longrightarrow> (\<And>x. x |\<in>| N \<Longrightarrow> f x = g x) \<Longrightarrow> f |`| M = g |`| N" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
671 |
by (rule image_cong[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
672 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
673 |
lemma fimage_finter_fsubset: "f |`| (A |\<inter>| B) |\<subseteq>| f |`| A |\<inter>| f |`| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
674 |
by (rule image_Int_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
675 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
676 |
lemma fimage_fminus_fsubset: "f |`| A |-| f |`| B |\<subseteq>| f |`| (A |-| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
677 |
by (rule image_diff_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
678 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
679 |
lemma finter_absorb: "A |\<inter>| A = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
680 |
by (rule Int_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
681 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
682 |
lemma finter_left_absorb: "A |\<inter>| (A |\<inter>| B) = A |\<inter>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
683 |
by (rule Int_left_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
684 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
685 |
lemma finter_commute: "A |\<inter>| B = B |\<inter>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
686 |
by (rule Int_commute[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
687 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
688 |
lemma finter_left_commute: "A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
689 |
by (rule Int_left_commute[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
690 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
691 |
lemma finter_assoc: "A |\<inter>| B |\<inter>| C = A |\<inter>| (B |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
692 |
by (rule Int_assoc[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
693 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
694 |
lemma finter_ac: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
695 |
"A |\<inter>| B |\<inter>| C = A |\<inter>| (B |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
696 |
"A |\<inter>| (A |\<inter>| B) = A |\<inter>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
697 |
"A |\<inter>| B = B |\<inter>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
698 |
"A |\<inter>| (B |\<inter>| C) = B |\<inter>| (A |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
699 |
by (rule Int_ac[Transfer.transferred])+ |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
700 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
701 |
lemma finter_absorb1: "B |\<subseteq>| A \<Longrightarrow> A |\<inter>| B = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
702 |
by (rule Int_absorb1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
703 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
704 |
lemma finter_absorb2: "A |\<subseteq>| B \<Longrightarrow> A |\<inter>| B = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
705 |
by (rule Int_absorb2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
706 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
707 |
lemma finter_fempty_left: "{||} |\<inter>| B = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
708 |
by (rule Int_empty_left[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
709 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
710 |
lemma finter_fempty_right: "A |\<inter>| {||} = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
711 |
by (rule Int_empty_right[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
712 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
713 |
lemma disjoint_iff_fnot_equal: "(A |\<inter>| B = {||}) = fBall A (\<lambda>x. fBall B ((\<noteq>) x))" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
714 |
by (rule disjoint_iff_not_equal[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
715 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
716 |
lemma finter_funion_distrib: "A |\<inter>| (B |\<union>| C) = A |\<inter>| B |\<union>| (A |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
717 |
by (rule Int_Un_distrib[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
718 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
719 |
lemma finter_funion_distrib2: "B |\<union>| C |\<inter>| A = B |\<inter>| A |\<union>| (C |\<inter>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
720 |
by (rule Int_Un_distrib2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
721 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
722 |
lemma finter_fsubset_iff[no_atp, simp]: "(C |\<subseteq>| A |\<inter>| B) = (C |\<subseteq>| A \<and> C |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
723 |
by (rule Int_subset_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
724 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
725 |
lemma funion_absorb: "A |\<union>| A = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
726 |
by (rule Un_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
727 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
728 |
lemma funion_left_absorb: "A |\<union>| (A |\<union>| B) = A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
729 |
by (rule Un_left_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
730 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
731 |
lemma funion_commute: "A |\<union>| B = B |\<union>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
732 |
by (rule Un_commute[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
733 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
734 |
lemma funion_left_commute: "A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
735 |
by (rule Un_left_commute[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
736 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
737 |
lemma funion_assoc: "A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
738 |
by (rule Un_assoc[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
739 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
740 |
lemma funion_ac: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
741 |
"A |\<union>| B |\<union>| C = A |\<union>| (B |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
742 |
"A |\<union>| (A |\<union>| B) = A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
743 |
"A |\<union>| B = B |\<union>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
744 |
"A |\<union>| (B |\<union>| C) = B |\<union>| (A |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
745 |
by (rule Un_ac[Transfer.transferred])+ |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
746 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
747 |
lemma funion_absorb1: "A |\<subseteq>| B \<Longrightarrow> A |\<union>| B = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
748 |
by (rule Un_absorb1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
749 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
750 |
lemma funion_absorb2: "B |\<subseteq>| A \<Longrightarrow> A |\<union>| B = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
751 |
by (rule Un_absorb2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
752 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
753 |
lemma funion_fempty_left: "{||} |\<union>| B = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
754 |
by (rule Un_empty_left[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
755 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
756 |
lemma funion_fempty_right: "A |\<union>| {||} = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
757 |
by (rule Un_empty_right[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
758 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
759 |
lemma funion_finsert_left[simp]: "finsert a B |\<union>| C = finsert a (B |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
760 |
by (rule Un_insert_left[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
761 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
762 |
lemma funion_finsert_right[simp]: "A |\<union>| finsert a B = finsert a (A |\<union>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
763 |
by (rule Un_insert_right[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
764 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
765 |
lemma finter_finsert_left: "finsert a B |\<inter>| C = (if a |\<in>| C then finsert a (B |\<inter>| C) else B |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
766 |
by (rule Int_insert_left[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
767 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
768 |
lemma finter_finsert_left_ifffempty[simp]: "a |\<notin>| C \<Longrightarrow> finsert a B |\<inter>| C = B |\<inter>| C" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
769 |
by (rule Int_insert_left_if0[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
770 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
771 |
lemma finter_finsert_left_if1[simp]: "a |\<in>| C \<Longrightarrow> finsert a B |\<inter>| C = finsert a (B |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
772 |
by (rule Int_insert_left_if1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
773 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
774 |
lemma finter_finsert_right: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
775 |
"A |\<inter>| finsert a B = (if a |\<in>| A then finsert a (A |\<inter>| B) else A |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
776 |
by (rule Int_insert_right[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
777 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
778 |
lemma finter_finsert_right_ifffempty[simp]: "a |\<notin>| A \<Longrightarrow> A |\<inter>| finsert a B = A |\<inter>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
779 |
by (rule Int_insert_right_if0[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
780 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
781 |
lemma finter_finsert_right_if1[simp]: "a |\<in>| A \<Longrightarrow> A |\<inter>| finsert a B = finsert a (A |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
782 |
by (rule Int_insert_right_if1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
783 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
784 |
lemma funion_finter_distrib: "A |\<union>| (B |\<inter>| C) = A |\<union>| B |\<inter>| (A |\<union>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
785 |
by (rule Un_Int_distrib[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
786 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
787 |
lemma funion_finter_distrib2: "B |\<inter>| C |\<union>| A = B |\<union>| A |\<inter>| (C |\<union>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
788 |
by (rule Un_Int_distrib2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
789 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
790 |
lemma funion_finter_crazy: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
791 |
"A |\<inter>| B |\<union>| (B |\<inter>| C) |\<union>| (C |\<inter>| A) = A |\<union>| B |\<inter>| (B |\<union>| C) |\<inter>| (C |\<union>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
792 |
by (rule Un_Int_crazy[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
793 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
794 |
lemma fsubset_funion_eq: "(A |\<subseteq>| B) = (A |\<union>| B = B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
795 |
by (rule subset_Un_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
796 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
797 |
lemma funion_fempty[iff]: "(A |\<union>| B = {||}) = (A = {||} \<and> B = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
798 |
by (rule Un_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
799 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
800 |
lemma funion_fsubset_iff[no_atp, simp]: "(A |\<union>| B |\<subseteq>| C) = (A |\<subseteq>| C \<and> B |\<subseteq>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
801 |
by (rule Un_subset_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
802 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
803 |
lemma funion_fminus_finter: "A |-| B |\<union>| (A |\<inter>| B) = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
804 |
by (rule Un_Diff_Int[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
805 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
806 |
lemma ffunion_empty[simp]: "ffUnion {||} = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
807 |
by (rule Union_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
808 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
809 |
lemma ffunion_mono: "A |\<subseteq>| B \<Longrightarrow> ffUnion A |\<subseteq>| ffUnion B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
810 |
by (rule Union_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
811 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
812 |
lemma ffunion_insert[simp]: "ffUnion (finsert a B) = a |\<union>| ffUnion B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
813 |
by (rule Union_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
814 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
815 |
lemma fminus_finter2: "A |\<inter>| C |-| (B |\<inter>| C) = A |\<inter>| C |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
816 |
by (rule Diff_Int2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
817 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
818 |
lemma funion_finter_assoc_eq: "(A |\<inter>| B |\<union>| C = A |\<inter>| (B |\<union>| C)) = (C |\<subseteq>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
819 |
by (rule Un_Int_assoc_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
820 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
821 |
lemma fBall_funion: "fBall (A |\<union>| B) P = (fBall A P \<and> fBall B P)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
822 |
by (rule ball_Un[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
823 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
824 |
lemma fBex_funion: "fBex (A |\<union>| B) P = (fBex A P \<or> fBex B P)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
825 |
by (rule bex_Un[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
826 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
827 |
lemma fminus_eq_fempty_iff[simp,no_atp]: "(A |-| B = {||}) = (A |\<subseteq>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
828 |
by (rule Diff_eq_empty_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
829 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
830 |
lemma fminus_cancel[simp]: "A |-| A = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
831 |
by (rule Diff_cancel[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
832 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
833 |
lemma fminus_idemp[simp]: "A |-| B |-| B = A |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
834 |
by (rule Diff_idemp[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
835 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
836 |
lemma fminus_triv: "A |\<inter>| B = {||} \<Longrightarrow> A |-| B = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
837 |
by (rule Diff_triv[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
838 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
839 |
lemma fempty_fminus[simp]: "{||} |-| A = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
840 |
by (rule empty_Diff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
841 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
842 |
lemma fminus_fempty[simp]: "A |-| {||} = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
843 |
by (rule Diff_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
844 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
845 |
lemma fminus_finsertffempty[simp,no_atp]: "x |\<notin>| A \<Longrightarrow> A |-| finsert x B = A |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
846 |
by (rule Diff_insert0[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
847 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
848 |
lemma fminus_finsert: "A |-| finsert a B = A |-| B |-| {|a|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
849 |
by (rule Diff_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
850 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
851 |
lemma fminus_finsert2: "A |-| finsert a B = A |-| {|a|} |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
852 |
by (rule Diff_insert2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
853 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
854 |
lemma finsert_fminus_if: "finsert x A |-| B = (if x |\<in>| B then A |-| B else finsert x (A |-| B))" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
855 |
by (rule insert_Diff_if[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
856 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
857 |
lemma finsert_fminus1[simp]: "x |\<in>| B \<Longrightarrow> finsert x A |-| B = A |-| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
858 |
by (rule insert_Diff1[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
859 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
860 |
lemma finsert_fminus_single[simp]: "finsert a (A |-| {|a|}) = finsert a A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
861 |
by (rule insert_Diff_single[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
862 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
863 |
lemma finsert_fminus: "a |\<in>| A \<Longrightarrow> finsert a (A |-| {|a|}) = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
864 |
by (rule insert_Diff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
865 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
866 |
lemma fminus_finsert_absorb: "x |\<notin>| A \<Longrightarrow> finsert x A |-| {|x|} = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
867 |
by (rule Diff_insert_absorb[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
868 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
869 |
lemma fminus_disjoint[simp]: "A |\<inter>| (B |-| A) = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
870 |
by (rule Diff_disjoint[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
871 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
872 |
lemma fminus_partition: "A |\<subseteq>| B \<Longrightarrow> A |\<union>| (B |-| A) = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
873 |
by (rule Diff_partition[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
874 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
875 |
lemma double_fminus: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| C \<Longrightarrow> B |-| (C |-| A) = A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
876 |
by (rule double_diff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
877 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
878 |
lemma funion_fminus_cancel[simp]: "A |\<union>| (B |-| A) = A |\<union>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
879 |
by (rule Un_Diff_cancel[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
880 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
881 |
lemma funion_fminus_cancel2[simp]: "B |-| A |\<union>| A = B |\<union>| A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
882 |
by (rule Un_Diff_cancel2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
883 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
884 |
lemma fminus_funion: "A |-| (B |\<union>| C) = A |-| B |\<inter>| (A |-| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
885 |
by (rule Diff_Un[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
886 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
887 |
lemma fminus_finter: "A |-| (B |\<inter>| C) = A |-| B |\<union>| (A |-| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
888 |
by (rule Diff_Int[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
889 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
890 |
lemma funion_fminus: "A |\<union>| B |-| C = A |-| C |\<union>| (B |-| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
891 |
by (rule Un_Diff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
892 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
893 |
lemma finter_fminus: "A |\<inter>| B |-| C = A |\<inter>| (B |-| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
894 |
by (rule Int_Diff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
895 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
896 |
lemma fminus_finter_distrib: "C |\<inter>| (A |-| B) = C |\<inter>| A |-| (C |\<inter>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
897 |
by (rule Diff_Int_distrib[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
898 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
899 |
lemma fminus_finter_distrib2: "A |-| B |\<inter>| C = A |\<inter>| C |-| (B |\<inter>| C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
900 |
by (rule Diff_Int_distrib2[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
901 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
902 |
lemma fUNIV_bool[no_atp]: "fUNIV = {|False, True|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
903 |
by (rule UNIV_bool[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
904 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
905 |
lemma fPow_fempty[simp]: "fPow {||} = {|{||}|}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
906 |
by (rule Pow_empty[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
907 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
908 |
lemma fPow_finsert: "fPow (finsert a A) = fPow A |\<union>| finsert a |`| fPow A" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
909 |
by (rule Pow_insert[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
910 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
911 |
lemma funion_fPow_fsubset: "fPow A |\<union>| fPow B |\<subseteq>| fPow (A |\<union>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
912 |
by (rule Un_Pow_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
913 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
914 |
lemma fPow_finter_eq[simp]: "fPow (A |\<inter>| B) = fPow A |\<inter>| fPow B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
915 |
by (rule Pow_Int_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
916 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
917 |
lemma fset_eq_fsubset: "(A = B) = (A |\<subseteq>| B \<and> B |\<subseteq>| A)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
918 |
by (rule set_eq_subset[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
919 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
920 |
lemma fsubset_iff[no_atp]: "(A |\<subseteq>| B) = (\<forall>t. t |\<in>| A \<longrightarrow> t |\<in>| B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
921 |
by (rule subset_iff[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
922 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
923 |
lemma fsubset_iff_pfsubset_eq: "(A |\<subseteq>| B) = (A |\<subset>| B \<or> A = B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
924 |
by (rule subset_iff_psubset_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
925 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
926 |
lemma all_not_fin_conv[simp]: "(\<forall>x. x |\<notin>| A) = (A = {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
927 |
by (rule all_not_in_conv[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
928 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
929 |
lemma ex_fin_conv: "(\<exists>x. x |\<in>| A) = (A \<noteq> {||})" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
930 |
by (rule ex_in_conv[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
931 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
932 |
lemma fimage_mono: "A |\<subseteq>| B \<Longrightarrow> f |`| A |\<subseteq>| f |`| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
933 |
by (rule image_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
934 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
935 |
lemma fPow_mono: "A |\<subseteq>| B \<Longrightarrow> fPow A |\<subseteq>| fPow B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
936 |
by (rule Pow_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
937 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
938 |
lemma finsert_mono: "C |\<subseteq>| D \<Longrightarrow> finsert a C |\<subseteq>| finsert a D" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
939 |
by (rule insert_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
940 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
941 |
lemma funion_mono: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| D \<Longrightarrow> A |\<union>| B |\<subseteq>| C |\<union>| D" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
942 |
by (rule Un_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
943 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
944 |
lemma finter_mono: "A |\<subseteq>| C \<Longrightarrow> B |\<subseteq>| D \<Longrightarrow> A |\<inter>| B |\<subseteq>| C |\<inter>| D" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
945 |
by (rule Int_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
946 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
947 |
lemma fminus_mono: "A |\<subseteq>| C \<Longrightarrow> D |\<subseteq>| B \<Longrightarrow> A |-| B |\<subseteq>| C |-| D" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
948 |
by (rule Diff_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
949 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
950 |
lemma fin_mono: "A |\<subseteq>| B \<Longrightarrow> x |\<in>| A \<longrightarrow> x |\<in>| B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
951 |
by (rule in_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
952 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
953 |
lemma fthe_felem_eq[simp]: "fthe_elem {|x|} = x" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
954 |
by (rule the_elem_eq[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
955 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
956 |
lemma fLeast_mono: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
957 |
"mono f \<Longrightarrow> fBex S (\<lambda>x. fBall S ((\<le>) x)) \<Longrightarrow> (LEAST y. y |\<in>| f |`| S) = f (LEAST x. x |\<in>| S)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
958 |
by (rule Least_mono[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
959 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
960 |
lemma fbind_fbind: "fbind (fbind A B) C = fbind A (\<lambda>x. fbind (B x) C)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
961 |
by (rule Set.bind_bind[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
962 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
963 |
lemma fempty_fbind[simp]: "fbind {||} f = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
964 |
by (rule empty_bind[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
965 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
966 |
lemma nonfempty_fbind_const: "A \<noteq> {||} \<Longrightarrow> fbind A (\<lambda>_. B) = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
967 |
by (rule nonempty_bind_const[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
968 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
969 |
lemma fbind_const: "fbind A (\<lambda>_. B) = (if A = {||} then {||} else B)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
970 |
by (rule bind_const[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
971 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
972 |
lemma ffmember_filter[simp]: "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
973 |
by (rule member_filter[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
974 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
975 |
lemma fequalityI: "A |\<subseteq>| B \<Longrightarrow> B |\<subseteq>| A \<Longrightarrow> A = B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
976 |
by (rule equalityI[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
977 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
978 |
lemma fset_of_list_simps[simp]: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
979 |
"fset_of_list [] = {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
980 |
"fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
981 |
by (rule set_simps[Transfer.transferred])+ |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
982 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
983 |
lemma fset_of_list_append[simp]: "fset_of_list (xs @ ys) = fset_of_list xs |\<union>| fset_of_list ys" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
984 |
by (rule set_append[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
985 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
986 |
lemma fset_of_list_rev[simp]: "fset_of_list (rev xs) = fset_of_list xs" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
987 |
by (rule set_rev[Transfer.transferred]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
988 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
989 |
lemma fset_of_list_map[simp]: "fset_of_list (map f xs) = f |`| fset_of_list xs" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
990 |
by (rule set_map[Transfer.transferred]) |
53953 | 991 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
992 |
|
60500 | 993 |
subsection \<open>Additional lemmas\<close> |
53953 | 994 |
|
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
995 |
subsubsection \<open>\<open>ffUnion\<close>\<close> |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
996 |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
997 |
lemma ffUnion_funion_distrib[simp]: "ffUnion (A |\<union>| B) = ffUnion A |\<union>| ffUnion B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
998 |
by (rule Union_Un_distrib[Transfer.transferred]) |
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
999 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1000 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1001 |
subsubsection \<open>\<open>fbind\<close>\<close> |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1002 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1003 |
lemma fbind_cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> f x = g x) \<Longrightarrow> fbind A f = fbind B g" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1004 |
by transfer force |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1005 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1006 |
|
61585 | 1007 |
subsubsection \<open>\<open>fsingleton\<close>\<close> |
53953 | 1008 |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
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parents:
76305
diff
changeset
|
1009 |
lemma fsingletonE: " b |\<in>| {|a|} \<Longrightarrow> (b = a \<Longrightarrow> thesis) \<Longrightarrow> thesis" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1010 |
by (rule fsingletonD [elim_format]) |
53953 | 1011 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1012 |
|
61585 | 1013 |
subsubsection \<open>\<open>femepty\<close>\<close> |
53953 | 1014 |
|
1015 |
lemma fempty_ffilter[simp]: "ffilter (\<lambda>_. False) A = {||}" |
|
1016 |
by transfer auto |
|
1017 |
||
1018 |
(* FIXME, transferred doesn't work here *) |
|
1019 |
lemma femptyE [elim!]: "a |\<in>| {||} \<Longrightarrow> P" |
|
1020 |
by simp |
|
1021 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
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parents:
54258
diff
changeset
|
1022 |
|
61585 | 1023 |
subsubsection \<open>\<open>fset\<close>\<close> |
53953 | 1024 |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1025 |
lemma fset_simps[simp]: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1026 |
"fset {||} = {}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1027 |
"fset (finsert x X) = insert x (fset X)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1028 |
by (rule bot_fset.rep_eq finsert.rep_eq)+ |
53953 | 1029 |
|
63331 | 1030 |
lemma finite_fset [simp]: |
53953 | 1031 |
shows "finite (fset S)" |
1032 |
by transfer simp |
|
1033 |
||
53963 | 1034 |
lemmas fset_cong = fset_inject |
53953 | 1035 |
|
1036 |
lemma filter_fset [simp]: |
|
1037 |
shows "fset (ffilter P xs) = Collect P \<inter> fset xs" |
|
1038 |
by transfer auto |
|
1039 |
||
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1040 |
lemma inter_fset[simp]: "fset (A |\<inter>| B) = fset A \<inter> fset B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1041 |
by (rule inf_fset.rep_eq) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1042 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1043 |
lemma union_fset[simp]: "fset (A |\<union>| B) = fset A \<union> fset B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1044 |
by (rule sup_fset.rep_eq) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1045 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1046 |
lemma minus_fset[simp]: "fset (A |-| B) = fset A - fset B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1047 |
by (rule minus_fset.rep_eq) |
53953 | 1048 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1049 |
|
63622 | 1050 |
subsubsection \<open>\<open>ffilter\<close>\<close> |
53953 | 1051 |
|
63331 | 1052 |
lemma subset_ffilter: |
53953 | 1053 |
"ffilter P A |\<subseteq>| ffilter Q A = (\<forall> x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)" |
1054 |
by transfer auto |
|
1055 |
||
63331 | 1056 |
lemma eq_ffilter: |
53953 | 1057 |
"(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)" |
1058 |
by transfer auto |
|
1059 |
||
53964 | 1060 |
lemma pfsubset_ffilter: |
67091 | 1061 |
"(\<And>x. x |\<in>| A \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| A \<and> \<not> P x \<and> Q x) \<Longrightarrow> |
53953 | 1062 |
ffilter P A |\<subset>| ffilter Q A" |
1063 |
unfolding less_fset_def by (auto simp add: subset_ffilter eq_ffilter) |
|
1064 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1065 |
|
63622 | 1066 |
subsubsection \<open>\<open>fset_of_list\<close>\<close> |
1067 |
||
1068 |
lemma fset_of_list_filter[simp]: |
|
1069 |
"fset_of_list (filter P xs) = ffilter P (fset_of_list xs)" |
|
1070 |
by transfer (auto simp: Set.filter_def) |
|
1071 |
||
1072 |
lemma fset_of_list_subset[intro]: |
|
1073 |
"set xs \<subseteq> set ys \<Longrightarrow> fset_of_list xs |\<subseteq>| fset_of_list ys" |
|
1074 |
by transfer simp |
|
1075 |
||
1076 |
lemma fset_of_list_elem: "(x |\<in>| fset_of_list xs) \<longleftrightarrow> (x \<in> set xs)" |
|
1077 |
by transfer simp |
|
1078 |
||
1079 |
||
61585 | 1080 |
subsubsection \<open>\<open>finsert\<close>\<close> |
53953 | 1081 |
|
1082 |
(* FIXME, transferred doesn't work here *) |
|
1083 |
lemma set_finsert: |
|
1084 |
assumes "x |\<in>| A" |
|
1085 |
obtains B where "A = finsert x B" and "x |\<notin>| B" |
|
1086 |
using assms by transfer (metis Set.set_insert finite_insert) |
|
1087 |
||
1088 |
lemma mk_disjoint_finsert: "a |\<in>| A \<Longrightarrow> \<exists>B. A = finsert a B \<and> a |\<notin>| B" |
|
63649 | 1089 |
by (rule exI [where x = "A |-| {|a|}"]) blast |
53953 | 1090 |
|
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1091 |
lemma finsert_eq_iff: |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1092 |
assumes "a |\<notin>| A" and "b |\<notin>| B" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1093 |
shows "(finsert a A = finsert b B) = |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1094 |
(if a = b then A = B else \<exists>C. A = finsert b C \<and> b |\<notin>| C \<and> B = finsert a C \<and> a |\<notin>| C)" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1095 |
using assms by transfer (force simp: insert_eq_iff) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1096 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1097 |
|
61585 | 1098 |
subsubsection \<open>\<open>fimage\<close>\<close> |
53953 | 1099 |
|
1100 |
lemma subset_fimage_iff: "(B |\<subseteq>| f|`|A) = (\<exists> AA. AA |\<subseteq>| A \<and> B = f|`|AA)" |
|
1101 |
by transfer (metis mem_Collect_eq rev_finite_subset subset_image_iff) |
|
1102 |
||
76269 | 1103 |
lemma fimage_strict_mono: |
1104 |
assumes "inj_on f (fset B)" and "A |\<subset>| B" |
|
1105 |
shows "f |`| A |\<subset>| f |`| B" |
|
76281
457f1cba78fb
renamed lemma inj_on_strict_subset to image_strict_mono for symmetry with image_mono and to distinguish from inj_on_subset
desharna
parents:
76269
diff
changeset
|
1106 |
\<comment> \<open>TODO: Configure transfer framework to lift @{thm Fun.image_strict_mono}.\<close> |
76269 | 1107 |
proof (rule pfsubsetI) |
1108 |
from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B" |
|
1109 |
by (rule pfsubset_imp_fsubset) |
|
1110 |
thus "f |`| A |\<subseteq>| f |`| B" |
|
1111 |
by (rule fimage_mono) |
|
1112 |
next |
|
1113 |
from \<open>A |\<subset>| B\<close> have "A |\<subseteq>| B" and "A \<noteq> B" |
|
1114 |
by (simp_all add: pfsubset_eq) |
|
1115 |
||
1116 |
have "fset A \<noteq> fset B" |
|
1117 |
using \<open>A \<noteq> B\<close> |
|
1118 |
by (simp add: fset_cong) |
|
1119 |
hence "f ` fset A \<noteq> f ` fset B" |
|
1120 |
using \<open>A |\<subseteq>| B\<close> |
|
1121 |
by (simp add: inj_on_image_eq_iff[OF \<open>inj_on f (fset B)\<close>] less_eq_fset.rep_eq) |
|
1122 |
hence "fset (f |`| A) \<noteq> fset (f |`| B)" |
|
1123 |
by (simp add: fimage.rep_eq) |
|
1124 |
thus "f |`| A \<noteq> f |`| B" |
|
1125 |
by (simp add: fset_cong) |
|
1126 |
qed |
|
1127 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1128 |
|
60500 | 1129 |
subsubsection \<open>bounded quantification\<close> |
53953 | 1130 |
|
1131 |
lemma bex_simps [simp, no_atp]: |
|
63331 | 1132 |
"\<And>A P Q. fBex A (\<lambda>x. P x \<and> Q) = (fBex A P \<and> Q)" |
53953 | 1133 |
"\<And>A P Q. fBex A (\<lambda>x. P \<and> Q x) = (P \<and> fBex A Q)" |
63331 | 1134 |
"\<And>P. fBex {||} P = False" |
53953 | 1135 |
"\<And>a B P. fBex (finsert a B) P = (P a \<or> fBex B P)" |
1136 |
"\<And>A P f. fBex (f |`| A) P = fBex A (\<lambda>x. P (f x))" |
|
1137 |
"\<And>A P. (\<not> fBex A P) = fBall A (\<lambda>x. \<not> P x)" |
|
1138 |
by auto |
|
1139 |
||
1140 |
lemma ball_simps [simp, no_atp]: |
|
1141 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<or> Q) = (fBall A P \<or> Q)" |
|
1142 |
"\<And>A P Q. fBall A (\<lambda>x. P \<or> Q x) = (P \<or> fBall A Q)" |
|
1143 |
"\<And>A P Q. fBall A (\<lambda>x. P \<longrightarrow> Q x) = (P \<longrightarrow> fBall A Q)" |
|
1144 |
"\<And>A P Q. fBall A (\<lambda>x. P x \<longrightarrow> Q) = (fBex A P \<longrightarrow> Q)" |
|
1145 |
"\<And>P. fBall {||} P = True" |
|
1146 |
"\<And>a B P. fBall (finsert a B) P = (P a \<and> fBall B P)" |
|
1147 |
"\<And>A P f. fBall (f |`| A) P = fBall A (\<lambda>x. P (f x))" |
|
1148 |
"\<And>A P. (\<not> fBall A P) = fBex A (\<lambda>x. \<not> P x)" |
|
1149 |
by auto |
|
1150 |
||
1151 |
lemma atomize_fBall: |
|
1152 |
"(\<And>x. x |\<in>| A ==> P x) == Trueprop (fBall A (\<lambda>x. P x))" |
|
1153 |
apply (simp only: atomize_all atomize_imp) |
|
1154 |
apply (rule equal_intr_rule) |
|
63622 | 1155 |
by (transfer, simp)+ |
1156 |
||
1157 |
lemma fBall_mono[mono]: "P \<le> Q \<Longrightarrow> fBall S P \<le> fBall S Q" |
|
1158 |
by auto |
|
1159 |
||
68463
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1160 |
lemma fBex_mono[mono]: "P \<le> Q \<Longrightarrow> fBex S P \<le> fBex S Q" |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1161 |
by auto |
53953 | 1162 |
|
53963 | 1163 |
end |
1164 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1165 |
|
61585 | 1166 |
subsubsection \<open>\<open>fcard\<close>\<close> |
53963 | 1167 |
|
53964 | 1168 |
(* FIXME: improve transferred to handle bounded meta quantification *) |
1169 |
||
53963 | 1170 |
lemma fcard_fempty: |
1171 |
"fcard {||} = 0" |
|
72302
d7d90ed4c74e
fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1172 |
by transfer (rule card.empty) |
53963 | 1173 |
|
1174 |
lemma fcard_finsert_disjoint: |
|
1175 |
"x |\<notin>| A \<Longrightarrow> fcard (finsert x A) = Suc (fcard A)" |
|
1176 |
by transfer (rule card_insert_disjoint) |
|
1177 |
||
1178 |
lemma fcard_finsert_if: |
|
1179 |
"fcard (finsert x A) = (if x |\<in>| A then fcard A else Suc (fcard A))" |
|
1180 |
by transfer (rule card_insert_if) |
|
1181 |
||
66265 | 1182 |
lemma fcard_0_eq [simp, no_atp]: |
53963 | 1183 |
"fcard A = 0 \<longleftrightarrow> A = {||}" |
1184 |
by transfer (rule card_0_eq) |
|
1185 |
||
1186 |
lemma fcard_Suc_fminus1: |
|
1187 |
"x |\<in>| A \<Longrightarrow> Suc (fcard (A |-| {|x|})) = fcard A" |
|
1188 |
by transfer (rule card_Suc_Diff1) |
|
1189 |
||
1190 |
lemma fcard_fminus_fsingleton: |
|
1191 |
"x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) = fcard A - 1" |
|
1192 |
by transfer (rule card_Diff_singleton) |
|
1193 |
||
1194 |
lemma fcard_fminus_fsingleton_if: |
|
1195 |
"fcard (A |-| {|x|}) = (if x |\<in>| A then fcard A - 1 else fcard A)" |
|
1196 |
by transfer (rule card_Diff_singleton_if) |
|
1197 |
||
1198 |
lemma fcard_fminus_finsert[simp]: |
|
1199 |
assumes "a |\<in>| A" and "a |\<notin>| B" |
|
1200 |
shows "fcard (A |-| finsert a B) = fcard (A |-| B) - 1" |
|
1201 |
using assms by transfer (rule card_Diff_insert) |
|
1202 |
||
1203 |
lemma fcard_finsert: "fcard (finsert x A) = Suc (fcard (A |-| {|x|}))" |
|
72302
d7d90ed4c74e
fixed some remarkably ugly proofs
paulson <lp15@cam.ac.uk>
parents:
69712
diff
changeset
|
1204 |
by transfer (rule card.insert_remove) |
53963 | 1205 |
|
1206 |
lemma fcard_finsert_le: "fcard A \<le> fcard (finsert x A)" |
|
1207 |
by transfer (rule card_insert_le) |
|
1208 |
||
1209 |
lemma fcard_mono: |
|
1210 |
"A |\<subseteq>| B \<Longrightarrow> fcard A \<le> fcard B" |
|
1211 |
by transfer (rule card_mono) |
|
1212 |
||
1213 |
lemma fcard_seteq: "A |\<subseteq>| B \<Longrightarrow> fcard B \<le> fcard A \<Longrightarrow> A = B" |
|
1214 |
by transfer (rule card_seteq) |
|
1215 |
||
1216 |
lemma pfsubset_fcard_mono: "A |\<subset>| B \<Longrightarrow> fcard A < fcard B" |
|
1217 |
by transfer (rule psubset_card_mono) |
|
1218 |
||
63331 | 1219 |
lemma fcard_funion_finter: |
53963 | 1220 |
"fcard A + fcard B = fcard (A |\<union>| B) + fcard (A |\<inter>| B)" |
1221 |
by transfer (rule card_Un_Int) |
|
1222 |
||
1223 |
lemma fcard_funion_disjoint: |
|
1224 |
"A |\<inter>| B = {||} \<Longrightarrow> fcard (A |\<union>| B) = fcard A + fcard B" |
|
1225 |
by transfer (rule card_Un_disjoint) |
|
1226 |
||
1227 |
lemma fcard_funion_fsubset: |
|
1228 |
"B |\<subseteq>| A \<Longrightarrow> fcard (A |-| B) = fcard A - fcard B" |
|
1229 |
by transfer (rule card_Diff_subset) |
|
1230 |
||
1231 |
lemma diff_fcard_le_fcard_fminus: |
|
1232 |
"fcard A - fcard B \<le> fcard(A |-| B)" |
|
1233 |
by transfer (rule diff_card_le_card_Diff) |
|
1234 |
||
1235 |
lemma fcard_fminus1_less: "x |\<in>| A \<Longrightarrow> fcard (A |-| {|x|}) < fcard A" |
|
1236 |
by transfer (rule card_Diff1_less) |
|
1237 |
||
1238 |
lemma fcard_fminus2_less: |
|
1239 |
"x |\<in>| A \<Longrightarrow> y |\<in>| A \<Longrightarrow> fcard (A |-| {|x|} |-| {|y|}) < fcard A" |
|
1240 |
by transfer (rule card_Diff2_less) |
|
1241 |
||
1242 |
lemma fcard_fminus1_le: "fcard (A |-| {|x|}) \<le> fcard A" |
|
1243 |
by transfer (rule card_Diff1_le) |
|
1244 |
||
1245 |
lemma fcard_pfsubset: "A |\<subseteq>| B \<Longrightarrow> fcard A < fcard B \<Longrightarrow> A < B" |
|
1246 |
by transfer (rule card_psubset) |
|
1247 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1248 |
|
68463
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1249 |
subsubsection \<open>\<open>sorted_list_of_fset\<close>\<close> |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1250 |
|
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1251 |
lemma sorted_list_of_fset_simps[simp]: |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1252 |
"set (sorted_list_of_fset S) = fset S" |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1253 |
"fset_of_list (sorted_list_of_fset S) = S" |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1254 |
by (transfer, simp)+ |
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1255 |
|
410818a69ee3
material on finite sets and maps
Lars Hupel <lars.hupel@mytum.de>
parents:
67829
diff
changeset
|
1256 |
|
61585 | 1257 |
subsubsection \<open>\<open>ffold\<close>\<close> |
53963 | 1258 |
|
1259 |
(* FIXME: improve transferred to handle bounded meta quantification *) |
|
1260 |
||
1261 |
context comp_fun_commute |
|
1262 |
begin |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1263 |
lemma ffold_empty[simp]: "ffold f z {||} = z" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1264 |
by (rule fold_empty[Transfer.transferred]) |
53963 | 1265 |
|
1266 |
lemma ffold_finsert [simp]: |
|
1267 |
assumes "x |\<notin>| A" |
|
1268 |
shows "ffold f z (finsert x A) = f x (ffold f z A)" |
|
1269 |
using assms by (transfer fixing: f) (rule fold_insert) |
|
1270 |
||
1271 |
lemma ffold_fun_left_comm: |
|
1272 |
"f x (ffold f z A) = ffold f (f x z) A" |
|
1273 |
by (transfer fixing: f) (rule fold_fun_left_comm) |
|
1274 |
||
1275 |
lemma ffold_finsert2: |
|
56646 | 1276 |
"x |\<notin>| A \<Longrightarrow> ffold f z (finsert x A) = ffold f (f x z) A" |
53963 | 1277 |
by (transfer fixing: f) (rule fold_insert2) |
1278 |
||
1279 |
lemma ffold_rec: |
|
1280 |
assumes "x |\<in>| A" |
|
1281 |
shows "ffold f z A = f x (ffold f z (A |-| {|x|}))" |
|
1282 |
using assms by (transfer fixing: f) (rule fold_rec) |
|
63331 | 1283 |
|
53963 | 1284 |
lemma ffold_finsert_fremove: |
1285 |
"ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))" |
|
1286 |
by (transfer fixing: f) (rule fold_insert_remove) |
|
1287 |
end |
|
1288 |
||
1289 |
lemma ffold_fimage: |
|
1290 |
assumes "inj_on g (fset A)" |
|
1291 |
shows "ffold f z (g |`| A) = ffold (f \<circ> g) z A" |
|
1292 |
using assms by transfer' (rule fold_image) |
|
1293 |
||
1294 |
lemma ffold_cong: |
|
1295 |
assumes "comp_fun_commute f" "comp_fun_commute g" |
|
1296 |
"\<And>x. x |\<in>| A \<Longrightarrow> f x = g x" |
|
1297 |
and "s = t" and "A = B" |
|
1298 |
shows "ffold f s A = ffold g t B" |
|
73832 | 1299 |
using assms[unfolded comp_fun_commute_def'] |
1300 |
by transfer (meson Finite_Set.fold_cong subset_UNIV) |
|
53963 | 1301 |
|
1302 |
context comp_fun_idem |
|
1303 |
begin |
|
1304 |
||
1305 |
lemma ffold_finsert_idem: |
|
56646 | 1306 |
"ffold f z (finsert x A) = f x (ffold f z A)" |
53963 | 1307 |
by (transfer fixing: f) (rule fold_insert_idem) |
63331 | 1308 |
|
53963 | 1309 |
declare ffold_finsert [simp del] ffold_finsert_idem [simp] |
63331 | 1310 |
|
53963 | 1311 |
lemma ffold_finsert_idem2: |
1312 |
"ffold f z (finsert x A) = ffold f (f x z) A" |
|
1313 |
by (transfer fixing: f) (rule fold_insert_idem2) |
|
1314 |
||
1315 |
end |
|
1316 |
||
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1317 |
|
76268 | 1318 |
subsubsection \<open>@{term fsubset}\<close> |
1319 |
||
1320 |
lemma wfP_pfsubset: "wfP (|\<subset>|)" |
|
1321 |
proof (rule wfP_if_convertible_to_nat) |
|
1322 |
show "\<And>x y. x |\<subset>| y \<Longrightarrow> fcard x < fcard y" |
|
1323 |
by (rule pfsubset_fcard_mono) |
|
1324 |
qed |
|
1325 |
||
1326 |
||
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1327 |
subsubsection \<open>Group operations\<close> |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1328 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1329 |
locale comm_monoid_fset = comm_monoid |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1330 |
begin |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1331 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1332 |
sublocale set: comm_monoid_set .. |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1333 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1334 |
lift_definition F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b fset \<Rightarrow> 'a" is set.F . |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1335 |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1336 |
lemma cong[fundef_cong]: "A = B \<Longrightarrow> (\<And>x. x |\<in>| B \<Longrightarrow> g x = h x) \<Longrightarrow> F g A = F h B" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1337 |
by (rule set.cong[Transfer.transferred]) |
66261 | 1338 |
|
69654 | 1339 |
lemma cong_simp[cong]: |
69164 | 1340 |
"\<lbrakk> A = B; \<And>x. x |\<in>| B =simp=> g x = h x \<rbrakk> \<Longrightarrow> F g A = F h B" |
1341 |
unfolding simp_implies_def by (auto cong: cong) |
|
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1342 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1343 |
end |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1344 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1345 |
context comm_monoid_add begin |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1346 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1347 |
sublocale fsum: comm_monoid_fset plus 0 |
67764 | 1348 |
rewrites "comm_monoid_set.F plus 0 = sum" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1349 |
defines fsum = fsum.F |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1350 |
proof - |
67399 | 1351 |
show "comm_monoid_fset (+) 0" by standard |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1352 |
|
67399 | 1353 |
show "comm_monoid_set.F (+) 0 = sum" unfolding sum_def .. |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1354 |
qed |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1355 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1356 |
end |
66261 | 1357 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1358 |
|
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1359 |
subsubsection \<open>Semilattice operations\<close> |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1360 |
|
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1361 |
locale semilattice_fset = semilattice |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1362 |
begin |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1363 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1364 |
sublocale set: semilattice_set .. |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1365 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1366 |
lift_definition F :: "'a fset \<Rightarrow> 'a" is set.F . |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1367 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1368 |
lemma eq_fold: "F (finsert x A) = ffold f x A" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1369 |
by transfer (rule set.eq_fold) |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1370 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1371 |
lemma singleton [simp]: "F {|x|} = x" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1372 |
by transfer (rule set.singleton) |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1373 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1374 |
lemma insert_not_elem: "x |\<notin>| A \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1375 |
by transfer (rule set.insert_not_elem) |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1376 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1377 |
lemma in_idem: "x |\<in>| A \<Longrightarrow> x \<^bold>* F A = F A" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1378 |
by transfer (rule set.in_idem) |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1379 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1380 |
lemma insert [simp]: "A \<noteq> {||} \<Longrightarrow> F (finsert x A) = x \<^bold>* F A" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1381 |
by transfer (rule set.insert) |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1382 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1383 |
end |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1384 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1385 |
locale semilattice_order_fset = binary?: semilattice_order + semilattice_fset |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1386 |
begin |
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1387 |
|
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1388 |
end |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1389 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1390 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1391 |
context linorder begin |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1392 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1393 |
sublocale fMin: semilattice_order_fset min less_eq less |
67764 | 1394 |
rewrites "semilattice_set.F min = Min" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1395 |
defines fMin = fMin.F |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1396 |
proof - |
67399 | 1397 |
show "semilattice_order_fset min (\<le>) (<)" by standard |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1398 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1399 |
show "semilattice_set.F min = Min" unfolding Min_def .. |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1400 |
qed |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1401 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1402 |
sublocale fMax: semilattice_order_fset max greater_eq greater |
67764 | 1403 |
rewrites "semilattice_set.F max = Max" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1404 |
defines fMax = fMax.F |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1405 |
proof - |
67399 | 1406 |
show "semilattice_order_fset max (\<ge>) (>)" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1407 |
by standard |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1408 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1409 |
show "semilattice_set.F max = Max" |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1410 |
unfolding Max_def .. |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1411 |
qed |
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1412 |
|
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1413 |
end |
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1414 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1415 |
lemma mono_fMax_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMax A) = fMax (f |`| A)" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1416 |
by transfer (rule mono_Max_commute) |
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1417 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1418 |
lemma mono_fMin_commute: "mono f \<Longrightarrow> A \<noteq> {||} \<Longrightarrow> f (fMin A) = fMin (f |`| A)" |
66292
9930f4cf6c7a
improve setup for fMin/fMax/fsum; courtesy of Ondřej Kunčar & Florian Haftmann
Lars Hupel <lars.hupel@mytum.de>
parents:
66265
diff
changeset
|
1419 |
by transfer (rule mono_Min_commute) |
66264
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1420 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1421 |
lemma fMax_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMax A |\<in>| A" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1422 |
by transfer (rule Max_in) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1423 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1424 |
lemma fMin_in[simp]: "A \<noteq> {||} \<Longrightarrow> fMin A |\<in>| A" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1425 |
by transfer (rule Min_in) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1426 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1427 |
lemma fMax_ge[simp]: "x |\<in>| A \<Longrightarrow> x \<le> fMax A" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1428 |
by transfer (rule Max_ge) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1429 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1430 |
lemma fMin_le[simp]: "x |\<in>| A \<Longrightarrow> fMin A \<le> x" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1431 |
by transfer (rule Min_le) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1432 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1433 |
lemma fMax_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> y \<le> x) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMax A = x" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1434 |
by transfer (rule Max_eqI) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1435 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1436 |
lemma fMin_eqI: "(\<And>y. y |\<in>| A \<Longrightarrow> x \<le> y) \<Longrightarrow> x |\<in>| A \<Longrightarrow> fMin A = x" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1437 |
by transfer (rule Min_eqI) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1438 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1439 |
lemma fMax_finsert[simp]: "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1440 |
by transfer simp |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1441 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1442 |
lemma fMin_finsert[simp]: "fMin (finsert x A) = (if A = {||} then x else min x (fMin A))" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1443 |
by transfer simp |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1444 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1445 |
context linorder begin |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1446 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1447 |
lemma fset_linorder_max_induct[case_names fempty finsert]: |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1448 |
assumes "P {||}" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1449 |
and "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y < x; P S\<rbrakk> \<Longrightarrow> P (finsert x S)" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1450 |
shows "P S" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1451 |
proof - |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1452 |
(* FIXME transfer and right_total vs. bi_total *) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1453 |
note Domainp_forall_transfer[transfer_rule] |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1454 |
show ?thesis |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1455 |
using assms by (transfer fixing: less) (auto intro: finite_linorder_max_induct) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1456 |
qed |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1457 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1458 |
lemma fset_linorder_min_induct[case_names fempty finsert]: |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1459 |
assumes "P {||}" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1460 |
and "\<And>x S. \<lbrakk>\<forall>y. y |\<in>| S \<longrightarrow> y > x; P S\<rbrakk> \<Longrightarrow> P (finsert x S)" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1461 |
shows "P S" |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1462 |
proof - |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1463 |
(* FIXME transfer and right_total vs. bi_total *) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1464 |
note Domainp_forall_transfer[transfer_rule] |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1465 |
show ?thesis |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1466 |
using assms by (transfer fixing: less) (auto intro: finite_linorder_min_induct) |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1467 |
qed |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1468 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1469 |
end |
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1470 |
|
d516da3e7c42
material from $AFP/Formula_Derivatives/FSet_More
Lars Hupel <lars.hupel@mytum.de>
parents:
66262
diff
changeset
|
1471 |
|
60500 | 1472 |
subsection \<open>Choice in fsets\<close> |
53953 | 1473 |
|
63331 | 1474 |
lemma fset_choice: |
53953 | 1475 |
assumes "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)" |
1476 |
shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)" |
|
1477 |
using assms by transfer metis |
|
1478 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1479 |
|
60500 | 1480 |
subsection \<open>Induction and Cases rules for fsets\<close> |
53953 | 1481 |
|
1482 |
lemma fset_exhaust [case_names empty insert, cases type: fset]: |
|
63331 | 1483 |
assumes fempty_case: "S = {||} \<Longrightarrow> P" |
53953 | 1484 |
and finsert_case: "\<And>x S'. S = finsert x S' \<Longrightarrow> P" |
1485 |
shows "P" |
|
1486 |
using assms by transfer blast |
|
1487 |
||
1488 |
lemma fset_induct [case_names empty insert]: |
|
1489 |
assumes fempty_case: "P {||}" |
|
1490 |
and finsert_case: "\<And>x S. P S \<Longrightarrow> P (finsert x S)" |
|
1491 |
shows "P S" |
|
1492 |
proof - |
|
1493 |
(* FIXME transfer and right_total vs. bi_total *) |
|
1494 |
note Domainp_forall_transfer[transfer_rule] |
|
1495 |
show ?thesis |
|
1496 |
using assms by transfer (auto intro: finite_induct) |
|
1497 |
qed |
|
1498 |
||
1499 |
lemma fset_induct_stronger [case_names empty insert, induct type: fset]: |
|
1500 |
assumes empty_fset_case: "P {||}" |
|
1501 |
and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)" |
|
1502 |
shows "P S" |
|
1503 |
proof - |
|
1504 |
(* FIXME transfer and right_total vs. bi_total *) |
|
1505 |
note Domainp_forall_transfer[transfer_rule] |
|
1506 |
show ?thesis |
|
1507 |
using assms by transfer (auto intro: finite_induct) |
|
1508 |
qed |
|
1509 |
||
1510 |
lemma fset_card_induct: |
|
1511 |
assumes empty_fset_case: "P {||}" |
|
1512 |
and card_fset_Suc_case: "\<And>S T. Suc (fcard S) = (fcard T) \<Longrightarrow> P S \<Longrightarrow> P T" |
|
1513 |
shows "P S" |
|
1514 |
proof (induct S) |
|
1515 |
case empty |
|
1516 |
show "P {||}" by (rule empty_fset_case) |
|
1517 |
next |
|
1518 |
case (insert x S) |
|
1519 |
have h: "P S" by fact |
|
1520 |
have "x |\<notin>| S" by fact |
|
63331 | 1521 |
then have "Suc (fcard S) = fcard (finsert x S)" |
53953 | 1522 |
by transfer auto |
63331 | 1523 |
then show "P (finsert x S)" |
53953 | 1524 |
using h card_fset_Suc_case by simp |
1525 |
qed |
|
1526 |
||
1527 |
lemma fset_strong_cases: |
|
1528 |
obtains "xs = {||}" |
|
1529 |
| ys x where "x |\<notin>| ys" and "xs = finsert x ys" |
|
1530 |
by transfer blast |
|
1531 |
||
1532 |
lemma fset_induct2: |
|
1533 |
"P {||} {||} \<Longrightarrow> |
|
1534 |
(\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (finsert x xs) {||}) \<Longrightarrow> |
|
1535 |
(\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (finsert y ys)) \<Longrightarrow> |
|
1536 |
(\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (finsert x xs) (finsert y ys)) \<Longrightarrow> |
|
1537 |
P xsa ysa" |
|
1538 |
apply (induct xsa arbitrary: ysa) |
|
1539 |
apply (induct_tac x rule: fset_induct_stronger) |
|
1540 |
apply simp_all |
|
1541 |
apply (induct_tac xa rule: fset_induct_stronger) |
|
1542 |
apply simp_all |
|
1543 |
done |
|
1544 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1545 |
|
78132 | 1546 |
subsection \<open>Lemmas depending on induction\<close> |
1547 |
||
1548 |
lemma ffUnion_fsubset_iff: "ffUnion A |\<subseteq>| B \<longleftrightarrow> fBall A (\<lambda>x. x |\<subseteq>| B)" |
|
1549 |
by (induction A) simp_all |
|
1550 |
||
1551 |
||
60500 | 1552 |
subsection \<open>Setup for Lifting/Transfer\<close> |
53953 | 1553 |
|
60500 | 1554 |
subsubsection \<open>Relator and predicator properties\<close> |
53953 | 1555 |
|
55938 | 1556 |
lift_definition rel_fset :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'b fset \<Rightarrow> bool" is rel_set |
1557 |
parametric rel_set_transfer . |
|
53953 | 1558 |
|
63331 | 1559 |
lemma rel_fset_alt_def: "rel_fset R = (\<lambda>A B. (\<forall>x.\<exists>y. x|\<in>|A \<longrightarrow> y|\<in>|B \<and> R x y) |
53953 | 1560 |
\<and> (\<forall>y. \<exists>x. y|\<in>|B \<longrightarrow> x|\<in>|A \<and> R x y))" |
1561 |
apply (rule ext)+ |
|
1562 |
apply transfer' |
|
63331 | 1563 |
apply (subst rel_set_def[unfolded fun_eq_iff]) |
53953 | 1564 |
by blast |
1565 |
||
55938 | 1566 |
lemma finite_rel_set: |
53953 | 1567 |
assumes fin: "finite X" "finite Z" |
55938 | 1568 |
assumes R_S: "rel_set (R OO S) X Z" |
1569 |
shows "\<exists>Y. finite Y \<and> rel_set R X Y \<and> rel_set S Y Z" |
|
53953 | 1570 |
proof - |
1571 |
obtain f where f: "\<forall>x\<in>X. R x (f x) \<and> (\<exists>z\<in>Z. S (f x) z)" |
|
1572 |
apply atomize_elim |
|
1573 |
apply (subst bchoice_iff[symmetric]) |
|
55938 | 1574 |
using R_S[unfolded rel_set_def OO_def] by blast |
63331 | 1575 |
|
56646 | 1576 |
obtain g where g: "\<forall>z\<in>Z. S (g z) z \<and> (\<exists>x\<in>X. R x (g z))" |
53953 | 1577 |
apply atomize_elim |
1578 |
apply (subst bchoice_iff[symmetric]) |
|
55938 | 1579 |
using R_S[unfolded rel_set_def OO_def] by blast |
63331 | 1580 |
|
53953 | 1581 |
let ?Y = "f ` X \<union> g ` Z" |
1582 |
have "finite ?Y" by (simp add: fin) |
|
55938 | 1583 |
moreover have "rel_set R X ?Y" |
1584 |
unfolding rel_set_def |
|
53953 | 1585 |
using f g by clarsimp blast |
55938 | 1586 |
moreover have "rel_set S ?Y Z" |
1587 |
unfolding rel_set_def |
|
53953 | 1588 |
using f g by clarsimp blast |
1589 |
ultimately show ?thesis by metis |
|
1590 |
qed |
|
1591 |
||
60500 | 1592 |
subsubsection \<open>Transfer rules for the Transfer package\<close> |
53953 | 1593 |
|
60500 | 1594 |
text \<open>Unconditional transfer rules\<close> |
53953 | 1595 |
|
63343 | 1596 |
context includes lifting_syntax |
53963 | 1597 |
begin |
1598 |
||
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1599 |
lemma fempty_transfer [transfer_rule]: |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1600 |
"rel_fset A {||} {||}" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1601 |
by (rule empty_transfer[Transfer.transferred]) |
53953 | 1602 |
|
1603 |
lemma finsert_transfer [transfer_rule]: |
|
55933 | 1604 |
"(A ===> rel_fset A ===> rel_fset A) finsert finsert" |
55945 | 1605 |
unfolding rel_fun_def rel_fset_alt_def by blast |
53953 | 1606 |
|
1607 |
lemma funion_transfer [transfer_rule]: |
|
55933 | 1608 |
"(rel_fset A ===> rel_fset A ===> rel_fset A) funion funion" |
55945 | 1609 |
unfolding rel_fun_def rel_fset_alt_def by blast |
53953 | 1610 |
|
1611 |
lemma ffUnion_transfer [transfer_rule]: |
|
55933 | 1612 |
"(rel_fset (rel_fset A) ===> rel_fset A) ffUnion ffUnion" |
55945 | 1613 |
unfolding rel_fun_def rel_fset_alt_def by transfer (simp, fast) |
53953 | 1614 |
|
1615 |
lemma fimage_transfer [transfer_rule]: |
|
55933 | 1616 |
"((A ===> B) ===> rel_fset A ===> rel_fset B) fimage fimage" |
55945 | 1617 |
unfolding rel_fun_def rel_fset_alt_def by simp blast |
53953 | 1618 |
|
1619 |
lemma fBall_transfer [transfer_rule]: |
|
67399 | 1620 |
"(rel_fset A ===> (A ===> (=)) ===> (=)) fBall fBall" |
55945 | 1621 |
unfolding rel_fset_alt_def rel_fun_def by blast |
53953 | 1622 |
|
1623 |
lemma fBex_transfer [transfer_rule]: |
|
67399 | 1624 |
"(rel_fset A ===> (A ===> (=)) ===> (=)) fBex fBex" |
55945 | 1625 |
unfolding rel_fset_alt_def rel_fun_def by blast |
53953 | 1626 |
|
1627 |
(* FIXME transfer doesn't work here *) |
|
1628 |
lemma fPow_transfer [transfer_rule]: |
|
55933 | 1629 |
"(rel_fset A ===> rel_fset (rel_fset A)) fPow fPow" |
55945 | 1630 |
unfolding rel_fun_def |
1631 |
using Pow_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] |
|
53953 | 1632 |
by blast |
1633 |
||
55933 | 1634 |
lemma rel_fset_transfer [transfer_rule]: |
67399 | 1635 |
"((A ===> B ===> (=)) ===> rel_fset A ===> rel_fset B ===> (=)) |
55933 | 1636 |
rel_fset rel_fset" |
55945 | 1637 |
unfolding rel_fun_def |
1638 |
using rel_set_transfer[unfolded rel_fun_def,rule_format, Transfer.transferred, where A = A and B = B] |
|
53953 | 1639 |
by simp |
1640 |
||
1641 |
lemma bind_transfer [transfer_rule]: |
|
55933 | 1642 |
"(rel_fset A ===> (A ===> rel_fset B) ===> rel_fset B) fbind fbind" |
63092 | 1643 |
unfolding rel_fun_def |
55945 | 1644 |
using bind_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
53953 | 1645 |
|
60500 | 1646 |
text \<open>Rules requiring bi-unique, bi-total or right-total relations\<close> |
53953 | 1647 |
|
1648 |
lemma fmember_transfer [transfer_rule]: |
|
1649 |
assumes "bi_unique A" |
|
67399 | 1650 |
shows "(A ===> rel_fset A ===> (=)) (|\<in>|) (|\<in>|)" |
55945 | 1651 |
using assms unfolding rel_fun_def rel_fset_alt_def bi_unique_def by metis |
53953 | 1652 |
|
1653 |
lemma finter_transfer [transfer_rule]: |
|
1654 |
assumes "bi_unique A" |
|
55933 | 1655 |
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) finter finter" |
55945 | 1656 |
using assms unfolding rel_fun_def |
1657 |
using inter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
|
53953 | 1658 |
|
53963 | 1659 |
lemma fminus_transfer [transfer_rule]: |
53953 | 1660 |
assumes "bi_unique A" |
67399 | 1661 |
shows "(rel_fset A ===> rel_fset A ===> rel_fset A) (|-|) (|-|)" |
55945 | 1662 |
using assms unfolding rel_fun_def |
1663 |
using Diff_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
|
53953 | 1664 |
|
1665 |
lemma fsubset_transfer [transfer_rule]: |
|
1666 |
assumes "bi_unique A" |
|
67399 | 1667 |
shows "(rel_fset A ===> rel_fset A ===> (=)) (|\<subseteq>|) (|\<subseteq>|)" |
55945 | 1668 |
using assms unfolding rel_fun_def |
1669 |
using subset_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
|
53953 | 1670 |
|
1671 |
lemma fSup_transfer [transfer_rule]: |
|
55938 | 1672 |
"bi_unique A \<Longrightarrow> (rel_set (rel_fset A) ===> rel_fset A) Sup Sup" |
63092 | 1673 |
unfolding rel_fun_def |
53953 | 1674 |
apply clarify |
1675 |
apply transfer' |
|
55945 | 1676 |
using Sup_fset_transfer[unfolded rel_fun_def] by blast |
53953 | 1677 |
|
1678 |
(* FIXME: add right_total_fInf_transfer *) |
|
1679 |
||
1680 |
lemma fInf_transfer [transfer_rule]: |
|
1681 |
assumes "bi_unique A" and "bi_total A" |
|
55938 | 1682 |
shows "(rel_set (rel_fset A) ===> rel_fset A) Inf Inf" |
55945 | 1683 |
using assms unfolding rel_fun_def |
53953 | 1684 |
apply clarify |
1685 |
apply transfer' |
|
55945 | 1686 |
using Inf_fset_transfer[unfolded rel_fun_def] by blast |
53953 | 1687 |
|
1688 |
lemma ffilter_transfer [transfer_rule]: |
|
1689 |
assumes "bi_unique A" |
|
67399 | 1690 |
shows "((A ===> (=)) ===> rel_fset A ===> rel_fset A) ffilter ffilter" |
55945 | 1691 |
using assms unfolding rel_fun_def |
1692 |
using Lifting_Set.filter_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
|
53953 | 1693 |
|
1694 |
lemma card_transfer [transfer_rule]: |
|
67399 | 1695 |
"bi_unique A \<Longrightarrow> (rel_fset A ===> (=)) fcard fcard" |
63092 | 1696 |
unfolding rel_fun_def |
55945 | 1697 |
using card_transfer[unfolded rel_fun_def, rule_format, Transfer.transferred] by blast |
53953 | 1698 |
|
1699 |
end |
|
1700 |
||
1701 |
lifting_update fset.lifting |
|
1702 |
lifting_forget fset.lifting |
|
1703 |
||
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1704 |
|
60500 | 1705 |
subsection \<open>BNF setup\<close> |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1706 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1707 |
context |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1708 |
includes fset.lifting |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1709 |
begin |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1710 |
|
55933 | 1711 |
lemma rel_fset_alt: |
1712 |
"rel_fset R a b \<longleftrightarrow> (\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>t \<in> fset b. \<exists>u \<in> fset a. R u t)" |
|
55938 | 1713 |
by transfer (simp add: rel_set_def) |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1714 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1715 |
lemma fset_to_fset: "finite A \<Longrightarrow> fset (the_inv fset A) = A" |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1716 |
apply (rule f_the_inv_into_f[unfolded inj_on_def]) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1717 |
apply (simp add: fset_inject) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1718 |
apply (rule range_eqI Abs_fset_inverse[symmetric] CollectI)+ |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1719 |
. |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1720 |
|
55933 | 1721 |
lemma rel_fset_aux: |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1722 |
"(\<forall>t \<in> fset a. \<exists>u \<in> fset b. R t u) \<and> (\<forall>u \<in> fset b. \<exists>t \<in> fset a. R t u) \<longleftrightarrow> |
57398 | 1723 |
((BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage fst))\<inverse>\<inverse> OO |
1724 |
BNF_Def.Grp {a. fset a \<subseteq> {(a, b). R a b}} (fimage snd)) a b" (is "?L = ?R") |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1725 |
proof |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1726 |
assume ?L |
63040 | 1727 |
define R' where "R' = |
1728 |
the_inv fset (Collect (case_prod R) \<inter> (fset a \<times> fset b))" (is "_ = the_inv fset ?L'") |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1729 |
have "finite ?L'" by (intro finite_Int[OF disjI2] finite_cartesian_product) (transfer, simp)+ |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1730 |
hence *: "fset R' = ?L'" unfolding R'_def by (intro fset_to_fset) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1731 |
show ?R unfolding Grp_def relcompp.simps conversep.simps |
55414
eab03e9cee8a
renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents:
55129
diff
changeset
|
1732 |
proof (intro CollectI case_prodI exI[of _ a] exI[of _ b] exI[of _ R'] conjI refl) |
60500 | 1733 |
from * show "a = fimage fst R'" using conjunct1[OF \<open>?L\<close>] |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1734 |
by (transfer, auto simp add: image_def Int_def split: prod.splits) |
60500 | 1735 |
from * show "b = fimage snd R'" using conjunct2[OF \<open>?L\<close>] |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1736 |
by (transfer, auto simp add: image_def Int_def split: prod.splits) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1737 |
qed (auto simp add: *) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1738 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1739 |
assume ?R thus ?L unfolding Grp_def relcompp.simps conversep.simps |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1740 |
apply (simp add: subset_eq Ball_def) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1741 |
apply (rule conjI) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1742 |
apply (transfer, clarsimp, metis snd_conv) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1743 |
by (transfer, clarsimp, metis fst_conv) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1744 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1745 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1746 |
bnf "'a fset" |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1747 |
map: fimage |
63331 | 1748 |
sets: fset |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1749 |
bd: natLeq |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1750 |
wits: "{||}" |
55933 | 1751 |
rel: rel_fset |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1752 |
apply - |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1753 |
apply transfer' apply simp |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1754 |
apply transfer' apply force |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1755 |
apply transfer apply force |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1756 |
apply transfer' apply force |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1757 |
apply (rule natLeq_card_order) |
75624 | 1758 |
apply (rule natLeq_cinfinite) |
1759 |
apply (rule regularCard_natLeq) |
|
1760 |
apply transfer apply (metis finite_iff_ordLess_natLeq) |
|
55933 | 1761 |
apply (fastforce simp: rel_fset_alt) |
62324 | 1762 |
apply (simp add: Grp_def relcompp.simps conversep.simps fun_eq_iff rel_fset_alt |
63331 | 1763 |
rel_fset_aux[unfolded OO_Grp_alt]) |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1764 |
apply transfer apply simp |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1765 |
done |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1766 |
|
55938 | 1767 |
lemma rel_fset_fset: "rel_set \<chi> (fset A1) (fset A2) = rel_fset \<chi> A1 A2" |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1768 |
by transfer (rule refl) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1769 |
|
53953 | 1770 |
end |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1771 |
|
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1772 |
declare |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1773 |
fset.map_comp[simp] |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1774 |
fset.map_id[simp] |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1775 |
fset.set_map[simp] |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1776 |
|
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1777 |
|
60500 | 1778 |
subsection \<open>Size setup\<close> |
56646 | 1779 |
|
1780 |
context includes fset.lifting begin |
|
64267 | 1781 |
lift_definition size_fset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a fset \<Rightarrow> nat" is "\<lambda>f. sum (Suc \<circ> f)" . |
56646 | 1782 |
end |
1783 |
||
1784 |
instantiation fset :: (type) size begin |
|
1785 |
definition size_fset where |
|
1786 |
size_fset_overloaded_def: "size_fset = FSet.size_fset (\<lambda>_. 0)" |
|
1787 |
instance .. |
|
1788 |
end |
|
1789 |
||
78102
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1790 |
lemma size_fset_simps[simp]: "size_fset f X = (\<Sum>x \<in> fset X. Suc (f x))" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1791 |
by (rule size_fset_def[THEN meta_eq_to_obj_eq, THEN fun_cong, THEN fun_cong, |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1792 |
unfolded map_fun_def comp_def id_apply]) |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1793 |
|
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1794 |
lemma size_fset_overloaded_simps[simp]: "size X = (\<Sum>x \<in> fset X. Suc 0)" |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1795 |
by (rule size_fset_simps[of "\<lambda>_. 0", unfolded add_0_left add_0_right, |
f40bc75b2a3f
replaced some lemmas' implicit formulas by explicit ones to avoid silent changes
desharna
parents:
76305
diff
changeset
|
1796 |
folded size_fset_overloaded_def]) |
56646 | 1797 |
|
1798 |
lemma fset_size_o_map: "inj f \<Longrightarrow> size_fset g \<circ> fimage f = size_fset (g \<circ> f)" |
|
60228
32dd7adba5a4
tuned proof; forget the transfer rule for size_fset
kuncar
parents:
58881
diff
changeset
|
1799 |
apply (subst fun_eq_iff) |
64267 | 1800 |
including fset.lifting by transfer (auto intro: sum.reindex_cong subset_inj_on) |
63331 | 1801 |
|
60500 | 1802 |
setup \<open> |
69593 | 1803 |
BNF_LFP_Size.register_size_global \<^type_name>\<open>fset\<close> \<^const_name>\<open>size_fset\<close> |
62082 | 1804 |
@{thm size_fset_overloaded_def} @{thms size_fset_simps size_fset_overloaded_simps} |
1805 |
@{thms fset_size_o_map} |
|
60500 | 1806 |
\<close> |
56646 | 1807 |
|
60228
32dd7adba5a4
tuned proof; forget the transfer rule for size_fset
kuncar
parents:
58881
diff
changeset
|
1808 |
lifting_update fset.lifting |
32dd7adba5a4
tuned proof; forget the transfer rule for size_fset
kuncar
parents:
58881
diff
changeset
|
1809 |
lifting_forget fset.lifting |
56646 | 1810 |
|
60500 | 1811 |
subsection \<open>Advanced relator customization\<close> |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1812 |
|
67408 | 1813 |
text \<open>Set vs. sum relators:\<close> |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1814 |
|
63331 | 1815 |
lemma rel_set_rel_sum[simp]: |
1816 |
"rel_set (rel_sum \<chi> \<phi>) A1 A2 \<longleftrightarrow> |
|
55938 | 1817 |
rel_set \<chi> (Inl -` A1) (Inl -` A2) \<and> rel_set \<phi> (Inr -` A1) (Inr -` A2)" |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1818 |
(is "?L \<longleftrightarrow> ?Rl \<and> ?Rr") |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1819 |
proof safe |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1820 |
assume L: "?L" |
55938 | 1821 |
show ?Rl unfolding rel_set_def Bex_def vimage_eq proof safe |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1822 |
fix l1 assume "Inl l1 \<in> A1" |
55943 | 1823 |
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inl l1) a2" |
55938 | 1824 |
using L unfolding rel_set_def by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1825 |
then obtain l2 where "a2 = Inl l2 \<and> \<chi> l1 l2" by (cases a2, auto) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1826 |
thus "\<exists> l2. Inl l2 \<in> A2 \<and> \<chi> l1 l2" using a2 by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1827 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1828 |
fix l2 assume "Inl l2 \<in> A2" |
55943 | 1829 |
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inl l2)" |
55938 | 1830 |
using L unfolding rel_set_def by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1831 |
then obtain l1 where "a1 = Inl l1 \<and> \<chi> l1 l2" by (cases a1, auto) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1832 |
thus "\<exists> l1. Inl l1 \<in> A1 \<and> \<chi> l1 l2" using a1 by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1833 |
qed |
55938 | 1834 |
show ?Rr unfolding rel_set_def Bex_def vimage_eq proof safe |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1835 |
fix r1 assume "Inr r1 \<in> A1" |
55943 | 1836 |
then obtain a2 where a2: "a2 \<in> A2" and "rel_sum \<chi> \<phi> (Inr r1) a2" |
55938 | 1837 |
using L unfolding rel_set_def by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1838 |
then obtain r2 where "a2 = Inr r2 \<and> \<phi> r1 r2" by (cases a2, auto) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1839 |
thus "\<exists> r2. Inr r2 \<in> A2 \<and> \<phi> r1 r2" using a2 by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1840 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1841 |
fix r2 assume "Inr r2 \<in> A2" |
55943 | 1842 |
then obtain a1 where a1: "a1 \<in> A1" and "rel_sum \<chi> \<phi> a1 (Inr r2)" |
55938 | 1843 |
using L unfolding rel_set_def by auto |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1844 |
then obtain r1 where "a1 = Inr r1 \<and> \<phi> r1 r2" by (cases a1, auto) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1845 |
thus "\<exists> r1. Inr r1 \<in> A1 \<and> \<phi> r1 r2" using a1 by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1846 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1847 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1848 |
assume Rl: "?Rl" and Rr: "?Rr" |
55938 | 1849 |
show ?L unfolding rel_set_def Bex_def vimage_eq proof safe |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1850 |
fix a1 assume a1: "a1 \<in> A1" |
55943 | 1851 |
show "\<exists> a2. a2 \<in> A2 \<and> rel_sum \<chi> \<phi> a1 a2" |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1852 |
proof(cases a1) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1853 |
case (Inl l1) then obtain l2 where "Inl l2 \<in> A2 \<and> \<chi> l1 l2" |
55938 | 1854 |
using Rl a1 unfolding rel_set_def by blast |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1855 |
thus ?thesis unfolding Inl by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1856 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1857 |
case (Inr r1) then obtain r2 where "Inr r2 \<in> A2 \<and> \<phi> r1 r2" |
55938 | 1858 |
using Rr a1 unfolding rel_set_def by blast |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1859 |
thus ?thesis unfolding Inr by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1860 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1861 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1862 |
fix a2 assume a2: "a2 \<in> A2" |
55943 | 1863 |
show "\<exists> a1. a1 \<in> A1 \<and> rel_sum \<chi> \<phi> a1 a2" |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1864 |
proof(cases a2) |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1865 |
case (Inl l2) then obtain l1 where "Inl l1 \<in> A1 \<and> \<chi> l1 l2" |
55938 | 1866 |
using Rl a2 unfolding rel_set_def by blast |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1867 |
thus ?thesis unfolding Inl by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1868 |
next |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1869 |
case (Inr r2) then obtain r1 where "Inr r1 \<in> A1 \<and> \<phi> r1 r2" |
55938 | 1870 |
using Rr a2 unfolding rel_set_def by blast |
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1871 |
thus ?thesis unfolding Inr by auto |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1872 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1873 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1874 |
qed |
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
54258
diff
changeset
|
1875 |
|
60712
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1876 |
|
66262
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1877 |
subsubsection \<open>Countability\<close> |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1878 |
|
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1879 |
lemma exists_fset_of_list: "\<exists>xs. fset_of_list xs = S" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1880 |
including fset.lifting |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1881 |
by transfer (rule finite_list) |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1882 |
|
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1883 |
lemma fset_of_list_surj[simp, intro]: "surj fset_of_list" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1884 |
proof - |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1885 |
have "x \<in> range fset_of_list" for x :: "'a fset" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1886 |
unfolding image_iff |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1887 |
using exists_fset_of_list by fastforce |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1888 |
thus ?thesis by auto |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1889 |
qed |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1890 |
|
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1891 |
instance fset :: (countable) countable |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1892 |
proof |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1893 |
obtain to_nat :: "'a list \<Rightarrow> nat" where "inj to_nat" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1894 |
by (metis ex_inj) |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1895 |
moreover have "inj (inv fset_of_list)" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1896 |
using fset_of_list_surj by (rule surj_imp_inj_inv) |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1897 |
ultimately have "inj (to_nat \<circ> inv fset_of_list)" |
69700
7a92cbec7030
new material about summations and powers, along with some tweaks
paulson <lp15@cam.ac.uk>
parents:
69654
diff
changeset
|
1898 |
by (rule inj_compose) |
66262
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1899 |
thus "\<exists>to_nat::'a fset \<Rightarrow> nat. inj to_nat" |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1900 |
by auto |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1901 |
qed |
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1902 |
|
4a2c9d32e7aa
finite sets are countable
Lars Hupel <lars.hupel@mytum.de>
parents:
66261
diff
changeset
|
1903 |
|
60712
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1904 |
subsection \<open>Quickcheck setup\<close> |
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1905 |
|
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1906 |
text \<open>Setup adapted from sets.\<close> |
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1907 |
|
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1908 |
notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55) |
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1909 |
|
72607 | 1910 |
context |
1911 |
includes term_syntax |
|
1912 |
begin |
|
1913 |
||
1914 |
definition [code_unfold]: |
|
60712
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1915 |
"valterm_femptyset = Code_Evaluation.valtermify ({||} :: ('a :: typerep) fset)" |
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1916 |
|
72607 | 1917 |
definition [code_unfold]: |
60712
3ba16d28449d
Quickcheck setup for finite sets
Lars Hupel <lars.hupel@mytum.de>
parents:
60679
diff
changeset
|
1918 |
"valtermify_finsert x s = Code_Evaluation.valtermify finsert {\<cdot>} (x :: ('a :: typerep * _)) {\<cdot>} s" |
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1919 |
|
72607 | 1920 |
end |
1921 |
||
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1922 |
instantiation fset :: (exhaustive) exhaustive |
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Quickcheck setup for finite sets
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|
1923 |
begin |
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Quickcheck setup for finite sets
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1924 |
|
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Quickcheck setup for finite sets
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|
1925 |
fun exhaustive_fset where |
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Quickcheck setup for finite sets
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|
1926 |
"exhaustive_fset f i = (if i = 0 then None else (f {||} orelse exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.exhaustive (\<lambda>x. if x |\<in>| A then None else f (finsert x A)) (i - 1)) (i - 1)))" |
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|
1927 |
|
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Quickcheck setup for finite sets
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|
1928 |
instance .. |
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Quickcheck setup for finite sets
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1929 |
|
55129
26bd1cba3ab5
killed 'More_BNFs' by moving its various bits where they (now) belong
blanchet
parents:
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|
1930 |
end |
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1931 |
|
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Quickcheck setup for finite sets
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1932 |
instantiation fset :: (full_exhaustive) full_exhaustive |
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1933 |
begin |
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1934 |
|
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1935 |
fun full_exhaustive_fset where |
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1936 |
"full_exhaustive_fset f i = (if i = 0 then None else (f valterm_femptyset orelse full_exhaustive_fset (\<lambda>A. f A orelse Quickcheck_Exhaustive.full_exhaustive (\<lambda>x. if fst x |\<in>| fst A then None else f (valtermify_finsert x A)) (i - 1)) (i - 1)))" |
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1937 |
|
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Quickcheck setup for finite sets
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1938 |
instance .. |
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Quickcheck setup for finite sets
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1939 |
|
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|
1940 |
end |
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1941 |
|
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Quickcheck setup for finite sets
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1942 |
no_notation Quickcheck_Exhaustive.orelse (infixr "orelse" 55) |
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Quickcheck setup for finite sets
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1943 |
|
72581 | 1944 |
instantiation fset :: (random) random |
1945 |
begin |
|
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1946 |
|
72581 | 1947 |
context |
1948 |
includes state_combinator_syntax |
|
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1949 |
begin |
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1950 |
|
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1951 |
fun random_aux_fset :: "natural \<Rightarrow> natural \<Rightarrow> natural \<times> natural \<Rightarrow> ('a fset \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where |
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1952 |
"random_aux_fset 0 j = Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset)])" | |
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1953 |
"random_aux_fset (Code_Numeral.Suc i) j = |
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|
1954 |
Quickcheck_Random.collapse (Random.select_weight |
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|
1955 |
[(1, Pair valterm_femptyset), |
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Quickcheck setup for finite sets
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|
1956 |
(Code_Numeral.Suc i, |
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1957 |
Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset i j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])" |
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1958 |
|
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|
1959 |
lemma [code]: |
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Quickcheck setup for finite sets
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|
1960 |
"random_aux_fset i j = |
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|
1961 |
Quickcheck_Random.collapse (Random.select_weight [(1, Pair valterm_femptyset), |
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Quickcheck setup for finite sets
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|
1962 |
(i, Quickcheck_Random.random j \<circ>\<rightarrow> (\<lambda>x. random_aux_fset (i - 1) j \<circ>\<rightarrow> (\<lambda>s. Pair (valtermify_finsert x s))))])" |
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Quickcheck setup for finite sets
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|
1963 |
proof (induct i rule: natural.induct) |
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Quickcheck setup for finite sets
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|
1964 |
case zero |
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|
1965 |
show ?case by (subst select_weight_drop_zero[symmetric]) (simp add: less_natural_def) |
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|
1966 |
next |
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|
1967 |
case (Suc i) |
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Quickcheck setup for finite sets
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|
1968 |
show ?case by (simp only: random_aux_fset.simps Suc_natural_minus_one) |
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Quickcheck setup for finite sets
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|
1969 |
qed |
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Quickcheck setup for finite sets
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|
1970 |
|
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Quickcheck setup for finite sets
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|
1971 |
definition "random_fset i = random_aux_fset i i" |
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Quickcheck setup for finite sets
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|
1972 |
|
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Quickcheck setup for finite sets
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|
1973 |
instance .. |
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Quickcheck setup for finite sets
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|
1974 |
|
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Quickcheck setup for finite sets
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|
1975 |
end |
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|
1976 |
|
72581 | 1977 |
end |
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1978 |
|
78118 | 1979 |
|
1980 |
subsection \<open>Code Generation Setup\<close> |
|
1981 |
||
1982 |
text \<open>The following @{attribute code_unfold} lemmas are so the pre-processor of the code generator |
|
1983 |
will perform conversions like, e.g., |
|
1984 |
@{lemma "x |\<in>| fimage f (fset_of_list xs) \<longleftrightarrow> x \<in> f ` set xs" |
|
1985 |
by (simp only: fimage.rep_eq fset_of_list.rep_eq)}.\<close> |
|
1986 |
||
1987 |
declare |
|
1988 |
ffilter.rep_eq[code_unfold] |
|
1989 |
fimage.rep_eq[code_unfold] |
|
1990 |
finsert.rep_eq[code_unfold] |
|
1991 |
fset_of_list.rep_eq[code_unfold] |
|
1992 |
inf_fset.rep_eq[code_unfold] |
|
1993 |
minus_fset.rep_eq[code_unfold] |
|
1994 |
sup_fset.rep_eq[code_unfold] |
|
1995 |
uminus_fset.rep_eq[code_unfold] |
|
1996 |
||
67399 | 1997 |
end |