(* Title: HOL/Transfer.thy
Author: Brian Huffman, TU Muenchen
Author: Ondrej Kuncar, TU Muenchen
*)
section \<open>Generic theorem transfer using relations\<close>
theory Transfer
imports Basic_BNF_LFPs Hilbert_Choice Metis
begin
subsection \<open>Relator for function space\<close>
bundle lifting_syntax
begin
notation rel_fun (infixr "===>" 55)
notation map_fun (infixr "--->" 55)
end
context includes lifting_syntax
begin
lemma rel_funD2:
assumes "rel_fun A B f g" and "A x x"
shows "B (f x) (g x)"
using assms by (rule rel_funD)
lemma rel_funE:
assumes "rel_fun A B f g" and "A x y"
obtains "B (f x) (g y)"
using assms by (simp add: rel_fun_def)
lemmas rel_fun_eq = fun.rel_eq
lemma rel_fun_eq_rel:
shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
by (simp add: rel_fun_def)
subsection \<open>Transfer method\<close>
text \<open>Explicit tag for relation membership allows for
backward proof methods.\<close>
definition Rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
where "Rel r \<equiv> r"
text \<open>Handling of equality relations\<close>
definition is_equality :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
where "is_equality R \<longleftrightarrow> R = (=)"
lemma is_equality_eq: "is_equality (=)"
unfolding is_equality_def by simp
text \<open>Reverse implication for monotonicity rules\<close>
definition rev_implies where
"rev_implies x y \<longleftrightarrow> (y \<longrightarrow> x)"
text \<open>Handling of meta-logic connectives\<close>
definition transfer_forall where
"transfer_forall \<equiv> All"
definition transfer_implies where
"transfer_implies \<equiv> (\<longrightarrow>)"
definition transfer_bforall :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
where "transfer_bforall \<equiv> (\<lambda>P Q. \<forall>x. P x \<longrightarrow> Q x)"
lemma transfer_forall_eq: "(\<And>x. P x) \<equiv> Trueprop (transfer_forall (\<lambda>x. P x))"
unfolding atomize_all transfer_forall_def ..
lemma transfer_implies_eq: "(A \<Longrightarrow> B) \<equiv> Trueprop (transfer_implies A B)"
unfolding atomize_imp transfer_implies_def ..
lemma transfer_bforall_unfold:
"Trueprop (transfer_bforall P (\<lambda>x. Q x)) \<equiv> (\<And>x. P x \<Longrightarrow> Q x)"
unfolding transfer_bforall_def atomize_imp atomize_all ..
lemma transfer_start: "\<lbrakk>P; Rel (=) P Q\<rbrakk> \<Longrightarrow> Q"
unfolding Rel_def by simp
lemma transfer_start': "\<lbrakk>P; Rel (\<longrightarrow>) P Q\<rbrakk> \<Longrightarrow> Q"
unfolding Rel_def by simp
lemma transfer_prover_start: "\<lbrakk>x = x'; Rel R x' y\<rbrakk> \<Longrightarrow> Rel R x y"
by simp
lemma untransfer_start: "\<lbrakk>Q; Rel (=) P Q\<rbrakk> \<Longrightarrow> P"
unfolding Rel_def by simp
lemma Rel_eq_refl: "Rel (=) x x"
unfolding Rel_def ..
lemma Rel_app:
assumes "Rel (A ===> B) f g" and "Rel A x y"
shows "Rel B (f x) (g y)"
using assms unfolding Rel_def rel_fun_def by fast
lemma Rel_abs:
assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)"
shows "Rel (A ===> B) (\<lambda>x. f x) (\<lambda>y. g y)"
using assms unfolding Rel_def rel_fun_def by fast
subsection \<open>Predicates on relations, i.e. ``class constraints''\<close>
definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
definition left_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "left_unique R \<longleftrightarrow> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
definition right_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "right_total R \<longleftrightarrow> (\<forall>y. \<exists>x. R x y)"
definition right_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "right_unique R \<longleftrightarrow> (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
definition bi_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "bi_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y)"
definition bi_unique :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "bi_unique R \<longleftrightarrow>
(\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and>
(\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
lemma left_unique_iff: "left_unique R \<longleftrightarrow> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R x z)"
unfolding Uniq_def left_unique_def by force
lemma left_uniqueI: "(\<And>x y z. \<lbrakk> A x z; A y z \<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> left_unique A"
unfolding left_unique_def by blast
lemma left_uniqueD: "\<lbrakk> left_unique A; A x z; A y z \<rbrakk> \<Longrightarrow> x = y"
unfolding left_unique_def by blast
lemma left_totalI:
"(\<And>x. \<exists>y. R x y) \<Longrightarrow> left_total R"
unfolding left_total_def by blast
lemma left_totalE:
assumes "left_total R"
obtains "(\<And>x. \<exists>y. R x y)"
using assms unfolding left_total_def by blast
lemma bi_uniqueDr: "\<lbrakk> bi_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
by(simp add: bi_unique_def)
lemma bi_uniqueDl: "\<lbrakk> bi_unique A; A x y; A z y \<rbrakk> \<Longrightarrow> x = z"
by(simp add: bi_unique_def)
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)"
unfolding Uniq_def bi_unique_def by force
lemma right_unique_iff: "right_unique R \<longleftrightarrow> (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1x. R z x)"
unfolding Uniq_def right_unique_def by force
lemma right_uniqueI: "(\<And>x y z. \<lbrakk> A x y; A x z \<rbrakk> \<Longrightarrow> y = z) \<Longrightarrow> right_unique A"
unfolding right_unique_def by fast
lemma right_uniqueD: "\<lbrakk> right_unique A; A x y; A x z \<rbrakk> \<Longrightarrow> y = z"
unfolding right_unique_def by fast
lemma right_totalI: "(\<And>y. \<exists>x. A x y) \<Longrightarrow> right_total A"
by(simp add: right_total_def)
lemma right_totalE:
assumes "right_total A"
obtains x where "A x y"
using assms by(auto simp add: right_total_def)
lemma right_total_alt_def2:
"right_total R \<longleftrightarrow> ((R ===> (\<longrightarrow>)) ===> (\<longrightarrow>)) All All" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
unfolding right_total_def rel_fun_def by blast
next
assume \<section>: ?rhs
show ?lhs
using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. True" "\<lambda>y. \<exists>x. R x y"]
unfolding right_total_def by blast
qed
lemma right_unique_alt_def2:
"right_unique R \<longleftrightarrow> (R ===> R ===> (\<longrightarrow>)) (=) (=)"
unfolding right_unique_def rel_fun_def by auto
lemma bi_total_alt_def2:
"bi_total R \<longleftrightarrow> ((R ===> (=)) ===> (=)) All All" (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
unfolding bi_total_def rel_fun_def by blast
next
assume \<section>: ?rhs
show ?lhs
using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. \<exists>y. R x y" "\<lambda>y. True"]
using \<section> [unfolded rel_fun_def, rule_format, of "\<lambda>x. True" "\<lambda>y. \<exists>x. R x y"]
by (auto simp: bi_total_def)
qed
lemma bi_unique_alt_def2:
"bi_unique R \<longleftrightarrow> (R ===> R ===> (=)) (=) (=)"
unfolding bi_unique_def rel_fun_def by auto
lemma [simp]:
shows left_unique_conversep: "left_unique A\<inverse>\<inverse> \<longleftrightarrow> right_unique A"
and right_unique_conversep: "right_unique A\<inverse>\<inverse> \<longleftrightarrow> left_unique A"
by(auto simp add: left_unique_def right_unique_def)
lemma [simp]:
shows left_total_conversep: "left_total A\<inverse>\<inverse> \<longleftrightarrow> right_total A"
and right_total_conversep: "right_total A\<inverse>\<inverse> \<longleftrightarrow> left_total A"
by(simp_all add: left_total_def right_total_def)
lemma bi_unique_conversep [simp]: "bi_unique R\<inverse>\<inverse> = bi_unique R"
by(auto simp add: bi_unique_def)
lemma bi_total_conversep [simp]: "bi_total R\<inverse>\<inverse> = bi_total R"
by(auto simp add: bi_total_def)
lemma right_unique_alt_def: "right_unique R = (conversep R OO R \<le> (=))" unfolding right_unique_def by blast
lemma left_unique_alt_def: "left_unique R = (R OO (conversep R) \<le> (=))" unfolding left_unique_def by blast
lemma right_total_alt_def: "right_total R = (conversep R OO R \<ge> (=))" unfolding right_total_def by blast
lemma left_total_alt_def: "left_total R = (R OO conversep R \<ge> (=))" unfolding left_total_def by blast
lemma bi_total_alt_def: "bi_total A = (left_total A \<and> right_total A)"
unfolding left_total_def right_total_def bi_total_def by blast
lemma bi_unique_alt_def: "bi_unique A = (left_unique A \<and> right_unique A)"
unfolding left_unique_def right_unique_def bi_unique_def by blast
lemma bi_totalI: "left_total R \<Longrightarrow> right_total R \<Longrightarrow> bi_total R"
unfolding bi_total_alt_def ..
lemma bi_uniqueI: "left_unique R \<Longrightarrow> right_unique R \<Longrightarrow> bi_unique R"
unfolding bi_unique_alt_def ..
end
lemma is_equality_lemma: "(\<And>R. is_equality R \<Longrightarrow> PROP (P R)) \<equiv> PROP (P (=))"
unfolding is_equality_def
proof (rule equal_intr_rule)
show "(\<And>R. R = (=) \<Longrightarrow> PROP P R) \<Longrightarrow> PROP P (=)"
apply (drule meta_spec)
apply (erule meta_mp [OF _ refl])
done
qed simp
lemma Domainp_lemma: "(\<And>R. Domainp T = R \<Longrightarrow> PROP (P R)) \<equiv> PROP (P (Domainp T))"
proof (rule equal_intr_rule)
show "(\<And>R. Domainp T = R \<Longrightarrow> PROP P R) \<Longrightarrow> PROP P (Domainp T)"
apply (drule meta_spec)
apply (erule meta_mp [OF _ refl])
done
qed simp
ML_file \<open>Tools/Transfer/transfer.ML\<close>
declare refl [transfer_rule]
hide_const (open) Rel
context includes lifting_syntax
begin
text \<open>Handling of domains\<close>
lemma Domainp_iff: "Domainp T x \<longleftrightarrow> (\<exists>y. T x y)"
by auto
lemma Domainp_refl[transfer_domain_rule]:
"Domainp T = Domainp T" ..
lemma Domain_eq_top[transfer_domain_rule]: "Domainp (=) = top" by auto
lemma Domainp_pred_fun_eq[relator_domain]:
assumes "left_unique T"
shows "Domainp (T ===> S) = pred_fun (Domainp T) (Domainp S)" (is "?lhs = ?rhs")
proof (intro ext iffI)
fix x
assume "?lhs x"
then show "?rhs x"
using assms unfolding rel_fun_def pred_fun_def by blast
next
fix x
assume "?rhs x"
then have "\<exists>g. \<forall>y xa. T xa y \<longrightarrow> S (x xa) (g y)"
using assms unfolding Domainp_iff left_unique_def pred_fun_def
by (intro choice) blast
then show "?lhs x"
by blast
qed
text \<open>Properties are preserved by relation composition.\<close>
lemma OO_def: "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
by auto
lemma bi_total_OO: "\<lbrakk>bi_total A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A OO B)"
unfolding bi_total_def OO_def by fast
lemma bi_unique_OO: "\<lbrakk>bi_unique A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A OO B)"
unfolding bi_unique_def OO_def by blast
lemma right_total_OO:
"\<lbrakk>right_total A; right_total B\<rbrakk> \<Longrightarrow> right_total (A OO B)"
unfolding right_total_def OO_def by fast
lemma right_unique_OO:
"\<lbrakk>right_unique A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A OO B)"
unfolding right_unique_def OO_def by fast
lemma left_total_OO: "left_total R \<Longrightarrow> left_total S \<Longrightarrow> left_total (R OO S)"
unfolding left_total_def OO_def by fast
lemma left_unique_OO: "left_unique R \<Longrightarrow> left_unique S \<Longrightarrow> left_unique (R OO S)"
unfolding left_unique_def OO_def by blast
subsection \<open>Properties of relators\<close>
lemma left_total_eq[transfer_rule]: "left_total (=)"
unfolding left_total_def by blast
lemma left_unique_eq[transfer_rule]: "left_unique (=)"
unfolding left_unique_def by blast
lemma right_total_eq [transfer_rule]: "right_total (=)"
unfolding right_total_def by simp
lemma right_unique_eq [transfer_rule]: "right_unique (=)"
unfolding right_unique_def by simp
lemma bi_total_eq[transfer_rule]: "bi_total (=)"
unfolding bi_total_def by simp
lemma bi_unique_eq[transfer_rule]: "bi_unique (=)"
unfolding bi_unique_def by simp
lemma left_total_fun[transfer_rule]:
assumes "left_unique A" "left_total B"
shows "left_total (A ===> B)"
unfolding left_total_def
proof
fix f
show "Ex ((A ===> B) f)"
unfolding rel_fun_def
proof (intro exI strip)
fix x y
assume A: "A x y"
have "(THE x. A x y) = x"
using A assms by (simp add: left_unique_def the_equality)
then show "B (f x) (SOME z. B (f (THE x. A x y)) z)"
using assms by (force simp: left_total_def intro: someI_ex)
qed
qed
lemma left_unique_fun[transfer_rule]:
"\<lbrakk>left_total A; left_unique B\<rbrakk> \<Longrightarrow> left_unique (A ===> B)"
unfolding left_total_def left_unique_def rel_fun_def
by (clarify, rule ext, fast)
lemma right_total_fun [transfer_rule]:
assumes "right_unique A" "right_total B"
shows "right_total (A ===> B)"
unfolding right_total_def
proof
fix g
show "\<exists>x. (A ===> B) x g"
unfolding rel_fun_def
proof (intro exI strip)
fix x y
assume A: "A x y"
have "(THE y. A x y) = y"
using A assms by (simp add: right_unique_def the_equality)
then show "B (SOME z. B z (g (THE y. A x y))) (g y)"
using assms by (force simp: right_total_def intro: someI_ex)
qed
qed
lemma right_unique_fun [transfer_rule]:
"\<lbrakk>right_total A; right_unique B\<rbrakk> \<Longrightarrow> right_unique (A ===> B)"
unfolding right_total_def right_unique_def rel_fun_def
by (clarify, rule ext, fast)
lemma bi_total_fun[transfer_rule]:
"\<lbrakk>bi_unique A; bi_total B\<rbrakk> \<Longrightarrow> bi_total (A ===> B)"
unfolding bi_unique_alt_def bi_total_alt_def
by (blast intro: right_total_fun left_total_fun)
lemma bi_unique_fun[transfer_rule]:
"\<lbrakk>bi_total A; bi_unique B\<rbrakk> \<Longrightarrow> bi_unique (A ===> B)"
unfolding bi_unique_alt_def bi_total_alt_def
by (blast intro: right_unique_fun left_unique_fun)
end
lemma if_conn:
"(if P \<and> Q then t else e) = (if P then if Q then t else e else e)"
"(if P \<or> Q then t else e) = (if P then t else if Q then t else e)"
"(if P \<longrightarrow> Q then t else e) = (if P then if Q then t else e else t)"
"(if \<not> P then t else e) = (if P then e else t)"
by auto
ML_file \<open>Tools/Transfer/transfer_bnf.ML\<close>
ML_file \<open>Tools/BNF/bnf_fp_rec_sugar_transfer.ML\<close>
declare pred_fun_def [simp]
declare rel_fun_eq [relator_eq]
(* Delete the automated generated rule from the bnf command;
we have a more general rule (Domainp_pred_fun_eq) that subsumes it. *)
declare fun.Domainp_rel[relator_domain del]
subsection \<open>Transfer rules\<close>
context includes lifting_syntax
begin
lemma Domainp_forall_transfer [transfer_rule]:
assumes "right_total A"
shows "((A ===> (=)) ===> (=))
(transfer_bforall (Domainp A)) transfer_forall"
using assms unfolding right_total_def
unfolding transfer_forall_def transfer_bforall_def rel_fun_def Domainp_iff
by fast
text \<open>Transfer rules using implication instead of equality on booleans.\<close>
lemma transfer_forall_transfer [transfer_rule]:
"bi_total A \<Longrightarrow> ((A ===> (=)) ===> (=)) transfer_forall transfer_forall"
"right_total A \<Longrightarrow> ((A ===> (=)) ===> implies) transfer_forall transfer_forall"
"right_total A \<Longrightarrow> ((A ===> implies) ===> implies) transfer_forall transfer_forall"
"bi_total A \<Longrightarrow> ((A ===> (=)) ===> rev_implies) transfer_forall transfer_forall"
"bi_total A \<Longrightarrow> ((A ===> rev_implies) ===> rev_implies) transfer_forall transfer_forall"
unfolding transfer_forall_def rev_implies_def rel_fun_def right_total_def bi_total_def
by fast+
lemma transfer_implies_transfer [transfer_rule]:
"((=) ===> (=) ===> (=) ) transfer_implies transfer_implies"
"(rev_implies ===> implies ===> implies ) transfer_implies transfer_implies"
"(rev_implies ===> (=) ===> implies ) transfer_implies transfer_implies"
"((=) ===> implies ===> implies ) transfer_implies transfer_implies"
"((=) ===> (=) ===> implies ) transfer_implies transfer_implies"
"(implies ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
"(implies ===> (=) ===> rev_implies) transfer_implies transfer_implies"
"((=) ===> rev_implies ===> rev_implies) transfer_implies transfer_implies"
"((=) ===> (=) ===> rev_implies) transfer_implies transfer_implies"
unfolding transfer_implies_def rev_implies_def rel_fun_def by auto
lemma eq_imp_transfer [transfer_rule]:
"right_unique A \<Longrightarrow> (A ===> A ===> (\<longrightarrow>)) (=) (=)"
unfolding right_unique_alt_def2 .
text \<open>Transfer rules using equality.\<close>
lemma left_unique_transfer [transfer_rule]:
assumes "right_total A"
assumes "right_total B"
assumes "bi_unique A"
shows "((A ===> B ===> (=)) ===> implies) left_unique left_unique"
using assms unfolding left_unique_def right_total_def bi_unique_def rel_fun_def
by metis
lemma eq_transfer [transfer_rule]:
assumes "bi_unique A"
shows "(A ===> A ===> (=)) (=) (=)"
using assms unfolding bi_unique_def rel_fun_def by auto
lemma right_total_Ex_transfer[transfer_rule]:
assumes "right_total A"
shows "((A ===> (=)) ===> (=)) (Bex (Collect (Domainp A))) Ex"
using assms unfolding right_total_def Bex_def rel_fun_def Domainp_iff
by fast
lemma right_total_All_transfer[transfer_rule]:
assumes "right_total A"
shows "((A ===> (=)) ===> (=)) (Ball (Collect (Domainp A))) All"
using assms unfolding right_total_def Ball_def rel_fun_def Domainp_iff
by fast
context
includes lifting_syntax
begin
lemma right_total_fun_eq_transfer:
assumes [transfer_rule]: "right_total A" "bi_unique B"
shows "((A ===> B) ===> (A ===> B) ===> (=)) (\<lambda>f g. \<forall>x\<in>Collect(Domainp A). f x = g x) (=)"
unfolding fun_eq_iff
by transfer_prover
end
lemma All_transfer [transfer_rule]:
assumes "bi_total A"
shows "((A ===> (=)) ===> (=)) All All"
using assms unfolding bi_total_def rel_fun_def by fast
lemma Ex_transfer [transfer_rule]:
assumes "bi_total A"
shows "((A ===> (=)) ===> (=)) Ex Ex"
using assms unfolding bi_total_def rel_fun_def by fast
lemma Ex1_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" "bi_total A"
shows "((A ===> (=)) ===> (=)) Ex1 Ex1"
unfolding Ex1_def by transfer_prover
declare If_transfer [transfer_rule]
lemma Let_transfer [transfer_rule]: "(A ===> (A ===> B) ===> B) Let Let"
unfolding rel_fun_def by simp
declare id_transfer [transfer_rule]
declare comp_transfer [transfer_rule]
lemma curry_transfer [transfer_rule]:
"((rel_prod A B ===> C) ===> A ===> B ===> C) curry curry"
unfolding curry_def by transfer_prover
lemma fun_upd_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "((A ===> B) ===> A ===> B ===> A ===> B) fun_upd fun_upd"
unfolding fun_upd_def by transfer_prover
lemma case_nat_transfer [transfer_rule]:
"(A ===> ((=) ===> A) ===> (=) ===> A) case_nat case_nat"
unfolding rel_fun_def by (simp split: nat.split)
lemma rec_nat_transfer [transfer_rule]:
"(A ===> ((=) ===> A ===> A) ===> (=) ===> A) rec_nat rec_nat"
unfolding rel_fun_def by (clarsimp, rename_tac n, induct_tac n, simp_all)
lemma funpow_transfer [transfer_rule]:
"((=) ===> (A ===> A) ===> (A ===> A)) compow compow"
unfolding funpow_def by transfer_prover
lemma mono_transfer[transfer_rule]:
assumes [transfer_rule]: "bi_total A"
assumes [transfer_rule]: "(A ===> A ===> (=)) (\<le>) (\<le>)"
assumes [transfer_rule]: "(B ===> B ===> (=)) (\<le>) (\<le>)"
shows "((A ===> B) ===> (=)) mono mono"
unfolding mono_def by transfer_prover
lemma right_total_relcompp_transfer[transfer_rule]:
assumes [transfer_rule]: "right_total B"
shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=))
(\<lambda>R S x z. \<exists>y\<in>Collect (Domainp B). R x y \<and> S y z) (OO)"
unfolding OO_def by transfer_prover
lemma relcompp_transfer[transfer_rule]:
assumes [transfer_rule]: "bi_total B"
shows "((A ===> B ===> (=)) ===> (B ===> C ===> (=)) ===> A ===> C ===> (=)) (OO) (OO)"
unfolding OO_def by transfer_prover
lemma right_total_Domainp_transfer[transfer_rule]:
assumes [transfer_rule]: "right_total B"
shows "((A ===> B ===> (=)) ===> A ===> (=)) (\<lambda>T x. \<exists>y\<in>Collect(Domainp B). T x y) Domainp"
apply(subst(2) Domainp_iff[abs_def]) by transfer_prover
lemma Domainp_transfer[transfer_rule]:
assumes [transfer_rule]: "bi_total B"
shows "((A ===> B ===> (=)) ===> A ===> (=)) Domainp Domainp"
unfolding Domainp_iff by transfer_prover
lemma reflp_transfer[transfer_rule]:
"bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> (=)) reflp reflp"
"right_total A \<Longrightarrow> ((A ===> A ===> implies) ===> implies) reflp reflp"
"right_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> implies) reflp reflp"
"bi_total A \<Longrightarrow> ((A ===> A ===> rev_implies) ===> rev_implies) reflp reflp"
"bi_total A \<Longrightarrow> ((A ===> A ===> (=)) ===> rev_implies) reflp reflp"
unfolding reflp_def rev_implies_def bi_total_def right_total_def rel_fun_def
by fast+
lemma right_unique_transfer [transfer_rule]:
"\<lbrakk> right_total A; right_total B; bi_unique B \<rbrakk>
\<Longrightarrow> ((A ===> B ===> (=)) ===> implies) right_unique right_unique"
unfolding right_unique_def right_total_def bi_unique_def rel_fun_def
by metis
lemma left_total_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_total A" "bi_total B"
shows "((A ===> B ===> (=)) ===> (=)) left_total left_total"
unfolding left_total_def by transfer_prover
lemma right_total_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_total A" "bi_total B"
shows "((A ===> B ===> (=)) ===> (=)) right_total right_total"
unfolding right_total_def by transfer_prover
lemma left_unique_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_unique A" "bi_total A" "bi_total B"
shows "((A ===> B ===> (=)) ===> (=)) left_unique left_unique"
unfolding left_unique_def by transfer_prover
lemma prod_pred_parametric [transfer_rule]:
"((A ===> (=)) ===> (B ===> (=)) ===> rel_prod A B ===> (=)) pred_prod pred_prod"
unfolding prod.pred_set Basic_BNFs.fsts_def Basic_BNFs.snds_def fstsp.simps sndsp.simps
by simp transfer_prover
lemma apfst_parametric [transfer_rule]:
"((A ===> B) ===> rel_prod A C ===> rel_prod B C) apfst apfst"
unfolding apfst_def by transfer_prover
lemma rel_fun_eq_eq_onp: "((=) ===> eq_onp P) = eq_onp (\<lambda>f. \<forall>x. P(f x))"
unfolding eq_onp_def rel_fun_def by auto
lemma rel_fun_eq_onp_rel:
shows "((eq_onp R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
by (auto simp add: eq_onp_def rel_fun_def)
lemma eq_onp_transfer [transfer_rule]:
assumes [transfer_rule]: "bi_unique A"
shows "((A ===> (=)) ===> A ===> A ===> (=)) eq_onp eq_onp"
unfolding eq_onp_def by transfer_prover
lemma rtranclp_parametric [transfer_rule]:
assumes "bi_unique A" "bi_total A"
shows "((A ===> A ===> (=)) ===> A ===> A ===> (=)) rtranclp rtranclp"
proof(rule rel_funI iffI)+
fix R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and R' x y x' y'
assume R: "(A ===> A ===> (=)) R R'" and "A x x'"
{
assume "R\<^sup>*\<^sup>* x y" "A y y'"
thus "R'\<^sup>*\<^sup>* x' y'"
proof(induction arbitrary: y')
case base
with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x' = y'" by(rule bi_uniqueDr)
thus ?case by simp
next
case (step y z z')
from \<open>bi_total A\<close> obtain y' where "A y y'" unfolding bi_total_def by blast
hence "R'\<^sup>*\<^sup>* x' y'" by(rule step.IH)
moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R y z\<close>
have "R' y' z'" by(auto dest: rel_funD)
ultimately show ?case ..
qed
next
assume "R'\<^sup>*\<^sup>* x' y'" "A y y'"
thus "R\<^sup>*\<^sup>* x y"
proof(induction arbitrary: y)
case base
with \<open>bi_unique A\<close> \<open>A x x'\<close> have "x = y" by(rule bi_uniqueDl)
thus ?case by simp
next
case (step y' z' z)
from \<open>bi_total A\<close> obtain y where "A y y'" unfolding bi_total_def by blast
hence "R\<^sup>*\<^sup>* x y" by(rule step.IH)
moreover from R \<open>A y y'\<close> \<open>A z z'\<close> \<open>R' y' z'\<close>
have "R y z" by(auto dest: rel_funD)
ultimately show ?case ..
qed
}
qed
lemma right_unique_parametric [transfer_rule]:
assumes [transfer_rule]: "bi_total A" "bi_unique B" "bi_total B"
shows "((A ===> B ===> (=)) ===> (=)) right_unique right_unique"
unfolding right_unique_def by transfer_prover
lemma map_fun_parametric [transfer_rule]:
"((A ===> B) ===> (C ===> D) ===> (B ===> C) ===> A ===> D) map_fun map_fun"
unfolding map_fun_def by transfer_prover
end
subsection \<open>\<^const>\<open>of_bool\<close> and \<^const>\<open>of_nat\<close>\<close>
context
includes lifting_syntax
begin
lemma transfer_rule_of_bool:
\<open>((\<longleftrightarrow>) ===> (\<cong>)) of_bool of_bool\<close>
if [transfer_rule]: \<open>0 \<cong> 0\<close> \<open>1 \<cong> 1\<close>
for R :: \<open>'a::zero_neq_one \<Rightarrow> 'b::zero_neq_one \<Rightarrow> bool\<close> (infix \<open>\<cong>\<close> 50)
unfolding of_bool_def by transfer_prover
lemma transfer_rule_of_nat:
"((=) ===> (\<cong>)) of_nat of_nat"
if [transfer_rule]: \<open>0 \<cong> 0\<close> \<open>1 \<cong> 1\<close>
\<open>((\<cong>) ===> (\<cong>) ===> (\<cong>)) (+) (+)\<close>
for R :: \<open>'a::semiring_1 \<Rightarrow> 'b::semiring_1 \<Rightarrow> bool\<close> (infix \<open>\<cong>\<close> 50)
unfolding of_nat_def by transfer_prover
end
end