author | desharna |
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changeset 76574 | 7bc934b99faf |
parent 75669 | 43f5dfb7fa35 |
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(* Title: HOL/Transfer.thy |
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Author: Brian Huffman, TU Muenchen |
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Author: Ondrej Kuncar, TU Muenchen |
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*) |
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section \<open>Generic theorem transfer using relations\<close> |
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theory Transfer |
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imports Basic_BNF_LFPs Hilbert_Choice Metis |
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begin |
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subsection \<open>Relator for function space\<close> |
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bundle lifting_syntax |
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begin |
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notation rel_fun (infixr "===>" 55) |
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notation map_fun (infixr "--->" 55) |
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end |
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context includes lifting_syntax |
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begin |
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lemma rel_funD2: |
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assumes "rel_fun A B f g" and "A x x" |
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shows "B (f x) (g x)" |
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using assms by (rule rel_funD) |
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lemma rel_funE: |
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assumes "rel_fun A B f g" and "A x y" |
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obtains "B (f x) (g y)" |
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using assms by (simp add: rel_fun_def) |
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lemmas rel_fun_eq = fun.rel_eq |
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lemma rel_fun_eq_rel: |
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shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))" |
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by (simp add: rel_fun_def) |
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subsection \<open>Transfer method\<close> |
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text \<open>Explicit tag for relation membership allows for |
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backward proof methods.\<close> |
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definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
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where "Rel r \<equiv> r" |
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text \<open>Handling of equality relations\<close> |
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definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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where "is_equality R \<longleftrightarrow> R = (=)" |
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lemma is_equality_eq: "is_equality (=)" |
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unfolding is_equality_def by simp |
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text \<open>Reverse implication for monotonicity rules\<close> |
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definition rev_implies where |
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"rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)" |
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text \<open>Handling of meta-logic connectives\<close> |
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definition transfer_forall where |
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"transfer_forall \<equiv> All" |
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definition transfer_implies where |
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"transfer_implies \<equiv> (\<longrightarrow>)" |
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definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" |
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where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)" |
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lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))" |
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unfolding atomize_all transfer_forall_def .. |
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lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)" |
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unfolding atomize_imp transfer_implies_def .. |
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lemma transfer_bforall_unfold: |
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"Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)" |
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unfolding transfer_bforall_def atomize_imp atomize_all .. |
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lemma transfer_start: "\<lbrakk>P; Rel (=) P Q\<rbrakk> \<Longrightarrow> Q" |
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unfolding Rel_def by simp |
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lemma transfer_start': "\<lbrakk>P; Rel (\<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q" |
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unfolding Rel_def by simp |
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lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y" |
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by simp |
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lemma untransfer_start: "\<lbrakk>Q; Rel (=) P Q\<rbrakk> \<Longrightarrow> P" |
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unfolding Rel_def by simp |
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lemma Rel_eq_refl: "Rel (=) x x" |
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unfolding Rel_def .. |
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lemma Rel_app: |
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assumes "Rel (A ===> B) f g" and "Rel A x y" |
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shows "Rel B (f x) (g y)" |
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using assms unfolding Rel_def rel_fun_def by fast |
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lemma Rel_abs: |
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assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)" |
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shows "Rel (A ===> B) (\<lambda>x. f x) (\<lambda>y. g y)" |
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using assms unfolding Rel_def rel_fun_def by fast |
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subsection \<open>Predicates on relations, i.e. ``class constraints''\<close> |
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definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)" |
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definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)" |
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definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)" |
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definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)" |
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definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)" |
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definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" |
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where "bi_unique R \<longleftrightarrow> |
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(\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> |
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(\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)" |
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lemma left_unique_iff: "left_unique R \<longleftrightarrow> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R x z)" |
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unfolding Uniq_def left_unique_def by force |
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lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A" |
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unfolding left_unique_def by blast |
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lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y" |
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unfolding left_unique_def by blast |
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lemma left_totalI: |
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"(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R" |
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unfolding left_total_def by blast |
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lemma left_totalE: |
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assumes "left_total R" |
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obtains "(\<And>x. \<exists>y. R x y)" |
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using assms unfolding left_total_def by blast |
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lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z" |
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by(simp add: bi_unique_def) |
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lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z" |
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by(simp add: bi_unique_def) |
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lemma bi_unique_iff: "bi_unique R \<longleftrightarrow> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R x z) \<and> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R z x)" |
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unfolding Uniq_def bi_unique_def by force |
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lemma right_unique_iff: "right_unique R \<longleftrightarrow> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R z x)" |
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unfolding Uniq_def right_unique_def by force |
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lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A" |
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unfolding right_unique_def by fast |
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lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z" |
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unfolding right_unique_def by fast |
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lemma right_totalI: "(\<And>y. \<exists>x. A x y) \<Longrightarrow> right_total A" |
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by(simp add: right_total_def) |
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lemma right_totalE: |
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assumes "right_total A" |
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obtains x where "A x y" |
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using assms by(auto simp add: right_total_def) |
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lemma right_total_alt_def2: |
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"right_total R \<longleftrightarrow> ((R ===> (\<longrightarrow>)) ===> (\<longrightarrow>)) All All" (is "?lhs = ?rhs") |
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proof |
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assume ?lhs then show ?rhs |
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unfolding right_total_def rel_fun_def by blast |
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next |
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assume \<section>: ?rhs |
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show ?lhs |
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using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. True" "\<lambda>y. \<exists>x. R x y"] |
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unfolding right_total_def by blast |
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qed |
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lemma right_unique_alt_def2: |
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"right_unique R \<longleftrightarrow> (R ===> R ===> (\<longrightarrow>)) (=) (=)" |
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unfolding right_unique_def rel_fun_def by auto |
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lemma bi_total_alt_def2: |
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"bi_total R \<longleftrightarrow> ((R ===> (=)) ===> (=)) All All" (is "?lhs = ?rhs") |
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proof |
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assume ?lhs then show ?rhs |
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unfolding bi_total_def rel_fun_def by blast |
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next |
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assume \<section>: ?rhs |
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show ?lhs |
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using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. \<exists>y. R x y" "\<lambda>y. True"] |
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using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. True" "\<lambda>y. \<exists>x. R x y"] |
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by (auto simp: bi_total_def) |
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qed |
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lemma bi_unique_alt_def2: |
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"bi_unique R \<longleftrightarrow> (R ===> R ===> (=)) (=) (=)" |
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unfolding bi_unique_def rel_fun_def by auto |
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|
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lemma [simp]: |
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shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A" |
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and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A" |
209 |
by(auto simp add: left_unique_def right_unique_def) |
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|
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lemma [simp]: |
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shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A" |
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and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A" |
214 |
by(simp_all add: left_total_def right_total_def) |
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|
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lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R" |
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by(auto simp add: bi_unique_def) |
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219 |
lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R" |
|
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by(auto simp add: bi_total_def) |
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|
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lemma right_unique_alt_def: "right_unique R = (conversep R OO R \<le> (=))" unfolding right_unique_def by blast |
223 |
lemma left_unique_alt_def: "left_unique R = (R OO (conversep R) \<le> (=))" unfolding left_unique_def by blast |
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|
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lemma right_total_alt_def: "right_total R = (conversep R OO R \<ge> (=))" unfolding right_total_def by blast |
226 |
lemma left_total_alt_def: "left_total R = (R OO conversep R \<ge> (=))" unfolding left_total_def by blast |
|
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|
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lemma bi_total_alt_def: "bi_total A = (left_total A \<and> right_total A)" |
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unfolding left_total_def right_total_def bi_total_def by blast |
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|
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lemma bi_unique_alt_def: "bi_unique A = (left_unique A \<and> right_unique A)" |
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unfolding left_unique_def right_unique_def bi_unique_def by blast |
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|
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lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R" |
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unfolding bi_total_alt_def .. |
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|
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lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R" |
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unfolding bi_unique_alt_def .. |
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|
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end |
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|
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|
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lemma is_equality_lemma: "(\<And>R. is_equality R \<Longrightarrow> PROP (P R)) \<equiv> PROP (P (=))" |
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unfolding is_equality_def |
245 |
proof (rule equal_intr_rule) |
|
246 |
show "(\<And>R. R = (=) \<Longrightarrow> PROP P R) \<Longrightarrow> PROP P (=)" |
|
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apply (drule meta_spec) |
|
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apply (erule meta_mp [OF _ refl]) |
|
249 |
done |
|
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qed simp |
|
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|
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lemma Domainp_lemma: "(\<And>R. Domainp T = R \<Longrightarrow> PROP (P R)) \<equiv> PROP (P (Domainp T))" |
|
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proof (rule equal_intr_rule) |
254 |
show "(\<And>R. Domainp T = R \<Longrightarrow> PROP P R) \<Longrightarrow> PROP P (Domainp T)" |
|
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apply (drule meta_spec) |
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apply (erule meta_mp [OF _ refl]) |
|
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done |
|
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qed simp |
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|
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ML_file \<open>Tools/Transfer/transfer.ML\<close> |
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declare refl [transfer_rule] |
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|
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hide_const (open) Rel |
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|
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context includes lifting_syntax |
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begin |
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|
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text \<open>Handling of domains\<close> |
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|
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lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)" |
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by auto |
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|
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lemma Domainp_refl[transfer_domain_rule]: |
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"Domainp T = Domainp T" .. |
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|
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lemma Domain_eq_top[transfer_domain_rule]: "Domainp (=) = top" by auto |
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|
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lemma Domainp_pred_fun_eq[relator_domain]: |
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assumes "left_unique T" |
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shows "Domainp (T ===> S) = pred_fun (Domainp T) (Domainp S)" (is "?lhs = ?rhs") |
281 |
proof (intro ext iffI) |
|
282 |
fix x |
|
283 |
assume "?lhs x" |
|
284 |
then show "?rhs x" |
|
285 |
using assms unfolding rel_fun_def pred_fun_def by blast |
|
286 |
next |
|
287 |
fix x |
|
288 |
assume "?rhs x" |
|
289 |
then have "\<exists>g. \<forall>y xa. T xa y \<longrightarrow> S (x xa) (g y)" |
|
290 |
using assms unfolding Domainp_iff left_unique_def pred_fun_def |
|
291 |
by (intro choice) blast |
|
292 |
then show "?lhs x" |
|
293 |
by blast |
|
294 |
qed |
|
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295 |
|
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text \<open>Properties are preserved by relation composition.\<close> |
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|
298 |
lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)" |
|
299 |
by auto |
|
300 |
||
301 |
lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)" |
|
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unfolding bi_total_def OO_def by fast |
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|
304 |
lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)" |
|
56085 | 305 |
unfolding bi_unique_def OO_def by blast |
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|
307 |
lemma right_total_OO: |
|
308 |
"\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)" |
|
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unfolding right_total_def OO_def by fast |
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|
311 |
lemma right_unique_OO: |
|
312 |
"\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)" |
|
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unfolding right_unique_def OO_def by fast |
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|
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lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)" |
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unfolding left_total_def OO_def by fast |
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317 |
|
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lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)" |
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unfolding left_unique_def OO_def by blast |
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320 |
|
47325 | 321 |
|
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subsection \<open>Properties of relators\<close> |
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|
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lemma left_total_eq[transfer_rule]: "left_total (=)" |
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325 |
unfolding left_total_def by blast |
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326 |
|
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lemma left_unique_eq[transfer_rule]: "left_unique (=)" |
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328 |
unfolding left_unique_def by blast |
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329 |
|
67399 | 330 |
lemma right_total_eq [transfer_rule]: "right_total (=)" |
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unfolding right_total_def by simp |
332 |
||
67399 | 333 |
lemma right_unique_eq [transfer_rule]: "right_unique (=)" |
47325 | 334 |
unfolding right_unique_def by simp |
335 |
||
67399 | 336 |
lemma bi_total_eq[transfer_rule]: "bi_total (=)" |
47325 | 337 |
unfolding bi_total_def by simp |
338 |
||
67399 | 339 |
lemma bi_unique_eq[transfer_rule]: "bi_unique (=)" |
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unfolding bi_unique_def by simp |
341 |
||
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lemma left_total_fun[transfer_rule]: |
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assumes "left_unique A" "left_total B" |
344 |
shows "left_total (A ===> B)" |
|
345 |
unfolding left_total_def |
|
346 |
proof |
|
347 |
fix f |
|
348 |
show "Ex ((A ===> B) f)" |
|
349 |
unfolding rel_fun_def |
|
350 |
proof (intro exI strip) |
|
351 |
fix x y |
|
352 |
assume A: "A x y" |
|
353 |
have "(THE x. A x y) = x" |
|
354 |
using A assms by (simp add: left_unique_def the_equality) |
|
355 |
then show "B (f x) (SOME z. B (f (THE x. A x y)) z)" |
|
356 |
using assms by (force simp: left_total_def intro: someI_ex) |
|
357 |
qed |
|
358 |
qed |
|
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359 |
|
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360 |
lemma left_unique_fun[transfer_rule]: |
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"\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)" |
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|
362 |
unfolding left_total_def left_unique_def rel_fun_def |
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363 |
by (clarify, rule ext, fast) |
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364 |
|
47325 | 365 |
lemma right_total_fun [transfer_rule]: |
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assumes "right_unique A" "right_total B" |
367 |
shows "right_total (A ===> B)" |
|
368 |
unfolding right_total_def |
|
369 |
proof |
|
370 |
fix g |
|
371 |
show "\<exists>x. (A ===> B) x g" |
|
372 |
unfolding rel_fun_def |
|
373 |
proof (intro exI strip) |
|
374 |
fix x y |
|
375 |
assume A: "A x y" |
|
376 |
have "(THE y. A x y) = y" |
|
377 |
using A assms by (simp add: right_unique_def the_equality) |
|
378 |
then show "B (SOME z. B z (g (THE y. A x y))) (g y)" |
|
379 |
using assms by (force simp: right_total_def intro: someI_ex) |
|
380 |
qed |
|
381 |
qed |
|
47325 | 382 |
|
383 |
lemma right_unique_fun [transfer_rule]: |
|
384 |
"\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)" |
|
55945 | 385 |
unfolding right_total_def right_unique_def rel_fun_def |
47325 | 386 |
by (clarify, rule ext, fast) |
387 |
||
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388 |
lemma bi_total_fun[transfer_rule]: |
47325 | 389 |
"\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)" |
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|
390 |
unfolding bi_unique_alt_def bi_total_alt_def |
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|
391 |
by (blast intro: right_total_fun left_total_fun) |
47325 | 392 |
|
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393 |
lemma bi_unique_fun[transfer_rule]: |
47325 | 394 |
"\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)" |
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395 |
unfolding bi_unique_alt_def bi_total_alt_def |
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396 |
by (blast intro: right_unique_fun left_unique_fun) |
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|
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398 |
end |
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399 |
|
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lemma if_conn: |
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"(if P \<and> Q then t else e) = (if P then if Q then t else e else e)" |
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"(if P \<or> Q then t else e) = (if P then t else if Q then t else e)" |
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"(if P \<longrightarrow> Q then t else e) = (if P then if Q then t else e else t)" |
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"(if \<not> P then t else e) = (if P then e else t)" |
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|
405 |
by auto |
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Add plugin to generate transfer theorem for primrec and primcorec
desharna
parents:
59141
diff
changeset
|
406 |
|
69605 | 407 |
ML_file \<open>Tools/Transfer/transfer_bnf.ML\<close> |
408 |
ML_file \<open>Tools/BNF/bnf_fp_rec_sugar_transfer.ML\<close> |
|
59275
77cd4992edcd
Add plugin to generate transfer theorem for primrec and primcorec
desharna
parents:
59141
diff
changeset
|
409 |
|
56543
9bd56f2e4c10
bi_unique and co. rules from the BNF hook must be introduced after bi_unique op= and co. rules are introduced
kuncar
parents:
56524
diff
changeset
|
410 |
declare pred_fun_def [simp] |
9bd56f2e4c10
bi_unique and co. rules from the BNF hook must be introduced after bi_unique op= and co. rules are introduced
kuncar
parents:
56524
diff
changeset
|
411 |
declare rel_fun_eq [relator_eq] |
9bd56f2e4c10
bi_unique and co. rules from the BNF hook must be introduced after bi_unique op= and co. rules are introduced
kuncar
parents:
56524
diff
changeset
|
412 |
|
64425
b17acc1834e3
a more general relator domain rule for the function type
kuncar
parents:
64014
diff
changeset
|
413 |
(* Delete the automated generated rule from the bnf command; |
b17acc1834e3
a more general relator domain rule for the function type
kuncar
parents:
64014
diff
changeset
|
414 |
we have a more general rule (Domainp_pred_fun_eq) that subsumes it. *) |
b17acc1834e3
a more general relator domain rule for the function type
kuncar
parents:
64014
diff
changeset
|
415 |
declare fun.Domainp_rel[relator_domain del] |
b17acc1834e3
a more general relator domain rule for the function type
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parents:
64014
diff
changeset
|
416 |
|
60758 | 417 |
subsection \<open>Transfer rules\<close> |
47325 | 418 |
|
63343 | 419 |
context includes lifting_syntax |
56543
9bd56f2e4c10
bi_unique and co. rules from the BNF hook must be introduced after bi_unique op= and co. rules are introduced
kuncar
parents:
56524
diff
changeset
|
420 |
begin |
9bd56f2e4c10
bi_unique and co. rules from the BNF hook must be introduced after bi_unique op= and co. rules are introduced
kuncar
parents:
56524
diff
changeset
|
421 |
|
53952 | 422 |
lemma Domainp_forall_transfer [transfer_rule]: |
423 |
assumes "right_total A" |
|
67399 | 424 |
shows "((A ===> (=)) ===> (=)) |
53952 | 425 |
(transfer_bforall (Domainp A)) transfer_forall" |
426 |
using assms unfolding right_total_def |
|
55945 | 427 |
unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff |
56085 | 428 |
by fast |
53952 | 429 |
|
60758 | 430 |
text \<open>Transfer rules using implication instead of equality on booleans.\<close> |
47684 | 431 |
|
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
432 |
lemma transfer_forall_transfer [transfer_rule]: |
67399 | 433 |
"bi_total A \<Longrightarrow> ((A ===> (=)) ===> (=)) transfer_forall transfer_forall" |
434 |
"right_total A \<Longrightarrow> ((A ===> (=)) ===> implies) transfer_forall transfer_forall" |
|
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
435 |
"right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall" |
67399 | 436 |
"bi_total A \<Longrightarrow> ((A ===> (=)) ===> rev_implies) transfer_forall transfer_forall" |
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
437 |
"bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall" |
55945 | 438 |
unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def |
56085 | 439 |
by fast+ |
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
440 |
|
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
441 |
lemma transfer_implies_transfer [transfer_rule]: |
67399 | 442 |
"((=) ===> (=) ===> (=) ) transfer_implies transfer_implies" |
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
443 |
"(rev_implies ===> implies ===> implies ) transfer_implies transfer_implies" |
67399 | 444 |
"(rev_implies ===> (=) ===> implies ) transfer_implies transfer_implies" |
445 |
"((=) ===> implies ===> implies ) transfer_implies transfer_implies" |
|
446 |
"((=) ===> (=) ===> implies ) transfer_implies transfer_implies" |
|
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
447 |
"(implies ===> rev_implies ===> rev_implies) transfer_implies transfer_implies" |
67399 | 448 |
"(implies ===> (=) ===> rev_implies) transfer_implies transfer_implies" |
449 |
"((=) ===> rev_implies ===> rev_implies) transfer_implies transfer_implies" |
|
450 |
"((=) ===> (=) ===> rev_implies) transfer_implies transfer_implies" |
|
55945 | 451 |
unfolding transfer_implies_def rev_implies_def rel_fun_def by auto |
52354
acb4f932dd24
implement 'transferred' attribute for transfer package, with support for monotonicity of !!/==>
huffman
parents:
51956
diff
changeset
|
452 |
|
47684 | 453 |
lemma eq_imp_transfer [transfer_rule]: |
67399 | 454 |
"right_unique A \<Longrightarrow> (A ===> A ===> (\<longrightarrow>)) (=) (=)" |
56524
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
455 |
unfolding right_unique_alt_def2 . |
47684 | 456 |
|
60758 | 457 |
text \<open>Transfer rules using equality.\<close> |
56518
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
458 |
|
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
459 |
lemma left_unique_transfer [transfer_rule]: |
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
460 |
assumes "right_total A" |
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
461 |
assumes "right_total B" |
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
462 |
assumes "bi_unique A" |
67399 | 463 |
shows "((A ===> B ===> (=)) ===> implies) left_unique left_unique" |
71697 | 464 |
using assms unfolding left_unique_def right_total_def bi_unique_def rel_fun_def |
465 |
by metis |
|
56518
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56085
diff
changeset
|
466 |
|
47636 | 467 |
lemma eq_transfer [transfer_rule]: |
47325 | 468 |
assumes "bi_unique A" |
67399 | 469 |
shows "(A ===> A ===> (=)) (=) (=)" |
55945 | 470 |
using assms unfolding bi_unique_def rel_fun_def by auto |
47325 | 471 |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
472 |
lemma right_total_Ex_transfer[transfer_rule]: |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
473 |
assumes "right_total A" |
67399 | 474 |
shows "((A ===> (=)) ===> (=)) (Bex (Collect (Domainp A))) Ex" |
71697 | 475 |
using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff |
476 |
by fast |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
477 |
|
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
478 |
lemma right_total_All_transfer[transfer_rule]: |
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
479 |
assumes "right_total A" |
67399 | 480 |
shows "((A ===> (=)) ===> (=)) (Ball (Collect (Domainp A))) All" |
71697 | 481 |
using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff |
482 |
by fast |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51955
diff
changeset
|
483 |
|
70927 | 484 |
context |
485 |
includes lifting_syntax |
|
486 |
begin |
|
487 |
||
68521 | 488 |
lemma right_total_fun_eq_transfer: |
489 |
assumes [transfer_rule]: "right_total A" "bi_unique B" |
|
490 |
shows "((A ===> B) ===> (A ===> B) ===> (=)) (\<lambda>f g. \<forall>x\<in>Collect(Domainp A). f x = g x) (=)" |
|
491 |
unfolding fun_eq_iff |
|
492 |
by transfer_prover |
|
493 |
||
70927 | 494 |
end |
495 |
||
47636 | 496 |
lemma All_transfer [transfer_rule]: |
47325 | 497 |
assumes "bi_total A" |
67399 | 498 |
shows "((A ===> (=)) ===> (=)) All All" |
55945 | 499 |
using assms unfolding bi_total_def rel_fun_def by fast |
47325 | 500 |
|
47636 | 501 |
lemma Ex_transfer [transfer_rule]: |
47325 | 502 |
assumes "bi_total A" |
67399 | 503 |
shows "((A ===> (=)) ===> (=)) Ex Ex" |
55945 | 504 |
using assms unfolding bi_total_def rel_fun_def by fast |
47325 | 505 |
|
59515 | 506 |
lemma Ex1_parametric [transfer_rule]: |
507 |
assumes [transfer_rule]: "bi_unique A" "bi_total A" |
|
67399 | 508 |
shows "((A ===> (=)) ===> (=)) Ex1 Ex1" |
71697 | 509 |
unfolding Ex1_def by transfer_prover |
59515 | 510 |
|
58448 | 511 |
declare If_transfer [transfer_rule] |
47325 | 512 |
|
47636 | 513 |
lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let" |
55945 | 514 |
unfolding rel_fun_def by simp |
47612 | 515 |
|
58916 | 516 |
declare id_transfer [transfer_rule] |
47625 | 517 |
|
58444 | 518 |
declare comp_transfer [transfer_rule] |
47325 | 519 |
|
58916 | 520 |
lemma curry_transfer [transfer_rule]: |
521 |
"((rel_prod A B ===> C) ===> A ===> B ===> C) curry curry" |
|
522 |
unfolding curry_def by transfer_prover |
|
523 |
||
47636 | 524 |
lemma fun_upd_transfer [transfer_rule]: |
47325 | 525 |
assumes [transfer_rule]: "bi_unique A" |
526 |
shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd" |
|
71697 | 527 |
unfolding fun_upd_def by transfer_prover |
47325 | 528 |
|
55415 | 529 |
lemma case_nat_transfer [transfer_rule]: |
67399 | 530 |
"(A ===> ((=) ===> A) ===> (=) ===> A) case_nat case_nat" |
55945 | 531 |
unfolding rel_fun_def by (simp split: nat.split) |
47627
2b1d3eda59eb
add secondary transfer rule for universal quantifiers on non-bi-total relations
huffman
parents:
47625
diff
changeset
|
532 |
|
55415 | 533 |
lemma rec_nat_transfer [transfer_rule]: |
67399 | 534 |
"(A ===> ((=) ===> A ===> A) ===> (=) ===> A) rec_nat rec_nat" |
75669
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
535 |
unfolding rel_fun_def |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
536 |
apply safe |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
537 |
subgoal for _ _ _ _ _ n |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
538 |
by (induction n) simp_all |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
539 |
done |
43f5dfb7fa35
tuned (some HOL lints, by Yecine Megdiche);
Fabian Huch <huch@in.tum.de>
parents:
71827
diff
changeset
|
540 |
|
47924 | 541 |
|
542 |
lemma funpow_transfer [transfer_rule]: |
|
67399 | 543 |
"((=) ===> (A ===> A) ===> (A ===> A)) compow compow" |
47924 | 544 |
unfolding funpow_def by transfer_prover |
545 |
||
53952 | 546 |
lemma mono_transfer[transfer_rule]: |
547 |
assumes [transfer_rule]: "bi_total A" |
|
67399 | 548 |
assumes [transfer_rule]: "(A ===> A ===> (=)) (\<le>) (\<le>)" |
549 |
assumes [transfer_rule]: "(B ===> B ===> (=)) (\<le>) (\<le>)" |
|
550 |
shows "((A ===> B) ===> (=)) mono mono" |
|
71697 | 551 |
unfolding mono_def by transfer_prover |
53952 | 552 |
|
58182 | 553 |
lemma right_total_relcompp_transfer[transfer_rule]: |
53952 | 554 |
assumes [transfer_rule]: "right_total B" |
67399 | 555 |
shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=)) |
556 |
(\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) (OO)" |
|
71697 | 557 |
unfolding OO_def by transfer_prover |
53952 | 558 |
|
58182 | 559 |
lemma relcompp_transfer[transfer_rule]: |
53952 | 560 |
assumes [transfer_rule]: "bi_total B" |
67399 | 561 |
shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=)) (OO) (OO)" |
71697 | 562 |
unfolding OO_def by transfer_prover |
47627
2b1d3eda59eb
add secondary transfer rule for universal quantifiers on non-bi-total relations
huffman
parents:
47625
diff
changeset
|
563 |
|
53952 | 564 |
lemma right_total_Domainp_transfer[transfer_rule]: |
565 |
assumes [transfer_rule]: "right_total B" |
|
67399 | 566 |
shows "((A ===> B ===> (=)) ===> A ===> (=)) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp" |
53952 | 567 |
apply(subst(2) Domainp_iff[abs_def]) by transfer_prover |
568 |
||
569 |
lemma Domainp_transfer[transfer_rule]: |
|
570 |
assumes [transfer_rule]: "bi_total B" |
|
67399 | 571 |
shows "((A ===> B ===> (=)) ===> A ===> (=)) Domainp Domainp" |
71697 | 572 |
unfolding Domainp_iff by transfer_prover |
53952 | 573 |
|
58182 | 574 |
lemma reflp_transfer[transfer_rule]: |
67399 | 575 |
"bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> (=)) reflp reflp" |
53952 | 576 |
"right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp" |
67399 | 577 |
"right_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> implies) reflp reflp" |
53952 | 578 |
"bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp" |
67399 | 579 |
"bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> rev_implies) reflp reflp" |
71697 | 580 |
unfolding reflp_def rev_implies_def bi_total_def right_total_def rel_fun_def |
53952 | 581 |
by fast+ |
582 |
||
583 |
lemma right_unique_transfer [transfer_rule]: |
|
59523 | 584 |
"\<lbrakk> right_total A; right_total B; bi_unique B \<rbrakk> |
67399 | 585 |
\<Longrightarrow> ((A ===> B ===> (=)) ===> implies) right_unique right_unique" |
71697 | 586 |
unfolding right_unique_def right_total_def bi_unique_def rel_fun_def |
53952 | 587 |
by metis |
47325 | 588 |
|
59523 | 589 |
lemma left_total_parametric [transfer_rule]: |
590 |
assumes [transfer_rule]: "bi_total A" "bi_total B" |
|
67399 | 591 |
shows "((A ===> B ===> (=)) ===> (=)) left_total left_total" |
71697 | 592 |
unfolding left_total_def by transfer_prover |
59523 | 593 |
|
594 |
lemma right_total_parametric [transfer_rule]: |
|
595 |
assumes [transfer_rule]: "bi_total A" "bi_total B" |
|
67399 | 596 |
shows "((A ===> B ===> (=)) ===> (=)) right_total right_total" |
71697 | 597 |
unfolding right_total_def by transfer_prover |
59523 | 598 |
|
599 |
lemma left_unique_parametric [transfer_rule]: |
|
600 |
assumes [transfer_rule]: "bi_unique A" "bi_total A" "bi_total B" |
|
67399 | 601 |
shows "((A ===> B ===> (=)) ===> (=)) left_unique left_unique" |
71697 | 602 |
unfolding left_unique_def by transfer_prover |
59523 | 603 |
|
604 |
lemma prod_pred_parametric [transfer_rule]: |
|
67399 | 605 |
"((A ===> (=)) ===> (B ===> (=)) ===> rel_prod A B ===> (=)) pred_prod pred_prod" |
71697 | 606 |
unfolding prod.pred_set Basic_BNFs.fsts_def Basic_BNFs.snds_def fstsp.simps sndsp.simps |
59523 | 607 |
by simp transfer_prover |
608 |
||
609 |
lemma apfst_parametric [transfer_rule]: |
|
610 |
"((A ===> B) ===> rel_prod A C ===> rel_prod B C) apfst apfst" |
|
71697 | 611 |
unfolding apfst_def by transfer_prover |
59523 | 612 |
|
67399 | 613 |
lemma rel_fun_eq_eq_onp: "((=) ===> eq_onp P) = eq_onp (\<lambda>f. \<forall>x. P(f x))" |
56524
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
614 |
unfolding eq_onp_def rel_fun_def by auto |
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
615 |
|
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
616 |
lemma rel_fun_eq_onp_rel: |
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
617 |
shows "((eq_onp R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))" |
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
618 |
by (auto simp add: eq_onp_def rel_fun_def) |
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
619 |
|
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
620 |
lemma eq_onp_transfer [transfer_rule]: |
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
621 |
assumes [transfer_rule]: "bi_unique A" |
67399 | 622 |
shows "((A ===> (=)) ===> A ===> A ===> (=)) eq_onp eq_onp" |
71697 | 623 |
unfolding eq_onp_def by transfer_prover |
56524
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56520
diff
changeset
|
624 |
|
57599 | 625 |
lemma rtranclp_parametric [transfer_rule]: |
626 |
assumes "bi_unique A" "bi_total A" |
|
67399 | 627 |
shows "((A ===> A ===> (=)) ===> A ===> A ===> (=)) rtranclp rtranclp" |
57599 | 628 |
proof(rule rel_funI iffI)+ |
629 |
fix R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and R' x y x' y' |
|
67399 | 630 |
assume R: "(A ===> A ===> (=)) R R'" and "A x x'" |
57599 | 631 |
{ |
632 |
assume "R\<^sup>*\<^sup>* x y" "A y y'" |
|
633 |
thus "R'\<^sup>*\<^sup>* x' y'" |
|
634 |
proof(induction arbitrary: y') |
|
635 |
case base |
|
60758 | 636 |
with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x' = y'" by(rule bi_uniqueDr) |
57599 | 637 |
thus ?case by simp |
638 |
next |
|
639 |
case (step y z z') |
|
60758 | 640 |
from \<open>bi_total A\<close> obtain y' where "A y y'" unfolding bi_total_def by blast |
57599 | 641 |
hence "R'\<^sup>*\<^sup>* x' y'" by(rule step.IH) |
60758 | 642 |
moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R y z\<close> |
57599 | 643 |
have "R' y' z'" by(auto dest: rel_funD) |
644 |
ultimately show ?case .. |
|
645 |
qed |
|
646 |
next |
|
647 |
assume "R'\<^sup>*\<^sup>* x' y'" "A y y'" |
|
648 |
thus "R\<^sup>*\<^sup>* x y" |
|
649 |
proof(induction arbitrary: y) |
|
650 |
case base |
|
60758 | 651 |
with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x = y" by(rule bi_uniqueDl) |
57599 | 652 |
thus ?case by simp |
653 |
next |
|
654 |
case (step y' z' z) |
|
60758 | 655 |
from \<open>bi_total A\<close> obtain y where "A y y'" unfolding bi_total_def by blast |
57599 | 656 |
hence "R\<^sup>*\<^sup>* x y" by(rule step.IH) |
60758 | 657 |
moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R' y' z'\<close> |
57599 | 658 |
have "R y z" by(auto dest: rel_funD) |
659 |
ultimately show ?case .. |
|
660 |
qed |
|
661 |
} |
|
662 |
qed |
|
663 |
||
59523 | 664 |
lemma right_unique_parametric [transfer_rule]: |
665 |
assumes [transfer_rule]: "bi_total A" "bi_unique B" "bi_total B" |
|
67399 | 666 |
shows "((A ===> B ===> (=)) ===> (=)) right_unique right_unique" |
71697 | 667 |
unfolding right_unique_def by transfer_prover |
59523 | 668 |
|
61630 | 669 |
lemma map_fun_parametric [transfer_rule]: |
670 |
"((A ===> B) ===> (C ===> D) ===> (B ===> C) ===> A ===> D) map_fun map_fun" |
|
71697 | 671 |
unfolding map_fun_def by transfer_prover |
61630 | 672 |
|
47325 | 673 |
end |
53011
aeee0a4be6cf
introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents:
52358
diff
changeset
|
674 |
|
64014 | 675 |
|
71182 | 676 |
subsection \<open>\<^const>\<open>of_bool\<close> and \<^const>\<open>of_nat\<close>\<close> |
677 |
||
678 |
context |
|
679 |
includes lifting_syntax |
|
680 |
begin |
|
681 |
||
682 |
lemma transfer_rule_of_bool: |
|
683 |
\<open>((\<longleftrightarrow>) ===> (\<cong>)) of_bool of_bool\<close> |
|
684 |
if [transfer_rule]: \<open>0 \<cong> 0\<close> \<open>1 \<cong> 1\<close> |
|
685 |
for R :: \<open>'a::zero_neq_one \<Rightarrow> 'b::zero_neq_one \<Rightarrow> bool\<close> (infix \<open>\<cong>\<close> 50) |
|
71697 | 686 |
unfolding of_bool_def by transfer_prover |
64014 | 687 |
|
688 |
lemma transfer_rule_of_nat: |
|
71182 | 689 |
"((=) ===> (\<cong>)) of_nat of_nat" |
690 |
if [transfer_rule]: \<open>0 \<cong> 0\<close> \<open>1 \<cong> 1\<close> |
|
691 |
\<open>((\<cong>) ===> (\<cong>) ===> (\<cong>)) (+) (+)\<close> |
|
692 |
for R :: \<open>'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool\<close> (infix \<open>\<cong>\<close> 50) |
|
71697 | 693 |
unfolding of_nat_def by transfer_prover |
64014 | 694 |
|
53011
aeee0a4be6cf
introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents:
52358
diff
changeset
|
695 |
end |
71182 | 696 |
|
697 |
end |