(* Title: HOL/Lifting.thy
Author: Brian Huffman and Ondrej Kuncar
Author: Cezary Kaliszyk and Christian Urban
*)
header {* Lifting package *}
theory Lifting
imports Equiv_Relations Transfer
keywords
"parametric" and
"print_quotmaps" "print_quotients" :: diag and
"lift_definition" :: thy_goal and
"setup_lifting" :: thy_decl
begin
subsection {* Function map *}
notation map_fun (infixr "--->" 55)
lemma map_fun_id:
"(id ---> id) = id"
by (simp add: fun_eq_iff)
subsection {* Quotient Predicate *}
definition
"Quotient R Abs Rep T \<longleftrightarrow>
(\<forall>a. Abs (Rep a) = a) \<and>
(\<forall>a. R (Rep a) (Rep a)) \<and>
(\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and>
T = (\<lambda>x y. R x x \<and> Abs x = y)"
lemma QuotientI:
assumes "\<And>a. Abs (Rep a) = a"
and "\<And>a. R (Rep a) (Rep a)"
and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
and "T = (\<lambda>x y. R x x \<and> Abs x = y)"
shows "Quotient R Abs Rep T"
using assms unfolding Quotient_def by blast
context
fixes R Abs Rep T
assumes a: "Quotient R Abs Rep T"
begin
lemma Quotient_abs_rep: "Abs (Rep a) = a"
using a unfolding Quotient_def
by simp
lemma Quotient_rep_reflp: "R (Rep a) (Rep a)"
using a unfolding Quotient_def
by blast
lemma Quotient_rel:
"R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
using a unfolding Quotient_def
by blast
lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)"
using a unfolding Quotient_def
by blast
lemma Quotient_refl1: "R r s \<Longrightarrow> R r r"
using a unfolding Quotient_def
by fast
lemma Quotient_refl2: "R r s \<Longrightarrow> R s s"
using a unfolding Quotient_def
by fast
lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
using a unfolding Quotient_def
by metis
lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r"
using a unfolding Quotient_def
by blast
lemma Quotient_rep_abs_fold_unmap:
assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'"
shows "R (Rep' x') x"
proof -
have "R (Rep x') x" using assms(1-2) Quotient_rep_abs by auto
then show ?thesis using assms(3) by simp
qed
lemma Quotient_Rep_eq:
assumes "x' \<equiv> Abs x"
shows "Rep x' \<equiv> Rep x'"
by simp
lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s"
using a unfolding Quotient_def
by blast
lemma Quotient_rel_abs2:
assumes "R (Rep x) y"
shows "x = Abs y"
proof -
from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs)
then show ?thesis using assms(1) by (simp add: Quotient_abs_rep)
qed
lemma Quotient_symp: "symp R"
using a unfolding Quotient_def using sympI by (metis (full_types))
lemma Quotient_transp: "transp R"
using a unfolding Quotient_def using transpI by (metis (full_types))
lemma Quotient_part_equivp: "part_equivp R"
by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI)
end
lemma identity_quotient: "Quotient (op =) id id (op =)"
unfolding Quotient_def by simp
text {* TODO: Use one of these alternatives as the real definition. *}
lemma Quotient_alt_def:
"Quotient R Abs Rep T \<longleftrightarrow>
(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
(\<forall>b. T (Rep b) b) \<and>
(\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)"
apply safe
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (simp (no_asm_use) only: Quotient_def, fast)
apply (rule QuotientI)
apply simp
apply metis
apply simp
apply (rule ext, rule ext, metis)
done
lemma Quotient_alt_def2:
"Quotient R Abs Rep T \<longleftrightarrow>
(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and>
(\<forall>b. T (Rep b) b) \<and>
(\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))"
unfolding Quotient_alt_def by (safe, metis+)
lemma Quotient_alt_def3:
"Quotient R Abs Rep T \<longleftrightarrow>
(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and>
(\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))"
unfolding Quotient_alt_def2 by (safe, metis+)
lemma Quotient_alt_def4:
"Quotient R Abs Rep T \<longleftrightarrow>
(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T"
unfolding Quotient_alt_def3 fun_eq_iff by auto
lemma fun_quotient:
assumes 1: "Quotient R1 abs1 rep1 T1"
assumes 2: "Quotient R2 abs2 rep2 T2"
shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2) (T1 ===> T2)"
using assms unfolding Quotient_alt_def2
unfolding fun_rel_def fun_eq_iff map_fun_apply
by (safe, metis+)
lemma apply_rsp:
fixes f g::"'a \<Rightarrow> 'c"
assumes q: "Quotient R1 Abs1 Rep1 T1"
and a: "(R1 ===> R2) f g" "R1 x y"
shows "R2 (f x) (g y)"
using a by (auto elim: fun_relE)
lemma apply_rsp':
assumes a: "(R1 ===> R2) f g" "R1 x y"
shows "R2 (f x) (g y)"
using a by (auto elim: fun_relE)
lemma apply_rsp'':
assumes "Quotient R Abs Rep T"
and "(R ===> S) f f"
shows "S (f (Rep x)) (f (Rep x))"
proof -
from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp)
then show ?thesis using assms(2) by (auto intro: apply_rsp')
qed
subsection {* Quotient composition *}
lemma Quotient_compose:
assumes 1: "Quotient R1 Abs1 Rep1 T1"
assumes 2: "Quotient R2 Abs2 Rep2 T2"
shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)"
using assms unfolding Quotient_alt_def4 by (auto intro!: ext)
lemma equivp_reflp2:
"equivp R \<Longrightarrow> reflp R"
by (erule equivpE)
subsection {* Respects predicate *}
definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
where "Respects R = {x. R x x}"
lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x"
unfolding Respects_def by simp
subsection {* Invariant *}
definition invariant :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
where "invariant R = (\<lambda>x y. R x \<and> x = y)"
lemma invariant_to_eq:
assumes "invariant P x y"
shows "x = y"
using assms by (simp add: invariant_def)
lemma fun_rel_eq_invariant:
shows "((invariant R) ===> S) = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
by (auto simp add: invariant_def fun_rel_def)
lemma invariant_same_args:
shows "invariant P x x \<equiv> P x"
using assms by (auto simp add: invariant_def)
lemma UNIV_typedef_to_Quotient:
assumes "type_definition Rep Abs UNIV"
and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
shows "Quotient (op =) Abs Rep T"
proof -
interpret type_definition Rep Abs UNIV by fact
from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
by (fastforce intro!: QuotientI fun_eq_iff)
qed
lemma UNIV_typedef_to_equivp:
fixes Abs :: "'a \<Rightarrow> 'b"
and Rep :: "'b \<Rightarrow> 'a"
assumes "type_definition Rep Abs (UNIV::'a set)"
shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)"
by (rule identity_equivp)
lemma typedef_to_Quotient:
assumes "type_definition Rep Abs S"
and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
shows "Quotient (invariant (\<lambda>x. x \<in> S)) Abs Rep T"
proof -
interpret type_definition Rep Abs S by fact
from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis
by (auto intro!: QuotientI simp: invariant_def fun_eq_iff)
qed
lemma typedef_to_part_equivp:
assumes "type_definition Rep Abs S"
shows "part_equivp (invariant (\<lambda>x. x \<in> S))"
proof (intro part_equivpI)
interpret type_definition Rep Abs S by fact
show "\<exists>x. invariant (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: invariant_def)
next
show "symp (invariant (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: invariant_def)
next
show "transp (invariant (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: invariant_def)
qed
lemma open_typedef_to_Quotient:
assumes "type_definition Rep Abs {x. P x}"
and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)"
shows "Quotient (invariant P) Abs Rep T"
using typedef_to_Quotient [OF assms] by simp
lemma open_typedef_to_part_equivp:
assumes "type_definition Rep Abs {x. P x}"
shows "part_equivp (invariant P)"
using typedef_to_part_equivp [OF assms] by simp
text {* Generating transfer rules for quotients. *}
context
fixes R Abs Rep T
assumes 1: "Quotient R Abs Rep T"
begin
lemma Quotient_right_unique: "right_unique T"
using 1 unfolding Quotient_alt_def right_unique_def by metis
lemma Quotient_right_total: "right_total T"
using 1 unfolding Quotient_alt_def right_total_def by metis
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)"
using 1 unfolding Quotient_alt_def fun_rel_def by simp
lemma Quotient_abs_induct:
assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x"
using 1 assms unfolding Quotient_def by metis
lemma Quotient_All_transfer:
"((T ===> op =) ===> op =) (Ball (Respects R)) All"
unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
by (auto, metis Quotient_abs_induct)
lemma Quotient_Ex_transfer:
"((T ===> op =) ===> op =) (Bex (Respects R)) Ex"
unfolding fun_rel_def Respects_def Quotient_cr_rel [OF 1]
by (auto, metis Quotient_rep_reflp [OF 1] Quotient_abs_rep [OF 1])
lemma Quotient_forall_transfer:
"((T ===> op =) ===> op =) (transfer_bforall (\<lambda>x. R x x)) transfer_forall"
using Quotient_All_transfer
unfolding transfer_forall_def transfer_bforall_def
Ball_def [abs_def] in_respects .
end
text {* Generating transfer rules for total quotients. *}
context
fixes R Abs Rep T
assumes 1: "Quotient R Abs Rep T" and 2: "reflp R"
begin
lemma Quotient_bi_total: "bi_total T"
using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs"
using 1 2 unfolding Quotient_alt_def reflp_def fun_rel_def by simp
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x"
using 1 2 assms unfolding Quotient_alt_def reflp_def by metis
lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y"
using Quotient_rel [OF 1] 2 unfolding reflp_def by simp
end
text {* Generating transfer rules for a type defined with @{text "typedef"}. *}
context
fixes Rep Abs A T
assumes type: "type_definition Rep Abs A"
assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)"
begin
lemma typedef_bi_unique: "bi_unique T"
unfolding bi_unique_def T_def
by (simp add: type_definition.Rep_inject [OF type])
(* the following two theorems are here only for convinience *)
lemma typedef_right_unique: "right_unique T"
using T_def type Quotient_right_unique typedef_to_Quotient
by blast
lemma typedef_right_total: "right_total T"
using T_def type Quotient_right_total typedef_to_Quotient
by blast
lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep"
unfolding fun_rel_def T_def by simp
lemma typedef_All_transfer: "((T ===> op =) ===> op =) (Ball A) All"
unfolding T_def fun_rel_def
by (metis type_definition.Rep [OF type]
type_definition.Abs_inverse [OF type])
lemma typedef_Ex_transfer: "((T ===> op =) ===> op =) (Bex A) Ex"
unfolding T_def fun_rel_def
by (metis type_definition.Rep [OF type]
type_definition.Abs_inverse [OF type])
lemma typedef_forall_transfer:
"((T ===> op =) ===> op =)
(transfer_bforall (\<lambda>x. x \<in> A)) transfer_forall"
unfolding transfer_bforall_def transfer_forall_def Ball_def [symmetric]
by (rule typedef_All_transfer)
end
text {* Generating the correspondence rule for a constant defined with
@{text "lift_definition"}. *}
lemma Quotient_to_transfer:
assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c"
shows "T c c'"
using assms by (auto dest: Quotient_cr_rel)
text {* Proving reflexivity *}
definition left_total :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
where "left_total R \<longleftrightarrow> (\<forall>x. \<exists>y. R x y)"
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 Quotient_to_left_total:
assumes q: "Quotient R Abs Rep T"
and r_R: "reflp R"
shows "left_total T"
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE)
lemma reflp_Quotient_composition:
assumes lt_R1: "left_total R1"
and r_R2: "reflp R2"
shows "reflp (R1 OO R2 OO R1\<inverse>\<inverse>)"
using assms
proof -
{
fix x
from lt_R1 obtain y where "R1 x y" unfolding left_total_def by blast
moreover then have "R1\<inverse>\<inverse> y x" by simp
moreover have "R2 y y" using r_R2 by (auto elim: reflpE)
ultimately have "(R1 OO R2 OO R1\<inverse>\<inverse>) x x" by auto
}
then show ?thesis by (auto intro: reflpI)
qed
lemma reflp_equality: "reflp (op =)"
by (auto intro: reflpI)
text {* Proving a parametrized correspondence relation *}
lemma eq_OO: "op= OO R = R"
unfolding OO_def by metis
definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
"POS A B \<equiv> A \<le> B"
definition NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
"NEG A B \<equiv> B \<le> A"
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)"
(*
The following two rules are here because we don't have any proper
left-unique ant left-total relations. Left-unique and left-total
assumptions show up in distributivity rules for the function type.
*)
lemma bi_unique_left_unique[transfer_rule]: "bi_unique R \<Longrightarrow> left_unique R"
unfolding bi_unique_def left_unique_def by blast
lemma bi_total_left_total[transfer_rule]: "bi_total R \<Longrightarrow> left_total R"
unfolding bi_total_def left_total_def by blast
lemma pos_OO_eq:
shows "POS (A OO op=) A"
unfolding POS_def OO_def by blast
lemma pos_eq_OO:
shows "POS (op= OO A) A"
unfolding POS_def OO_def by blast
lemma neg_OO_eq:
shows "NEG (A OO op=) A"
unfolding NEG_def OO_def by auto
lemma neg_eq_OO:
shows "NEG (op= OO A) A"
unfolding NEG_def OO_def by blast
lemma POS_trans:
assumes "POS A B"
assumes "POS B C"
shows "POS A C"
using assms unfolding POS_def by auto
lemma NEG_trans:
assumes "NEG A B"
assumes "NEG B C"
shows "NEG A C"
using assms unfolding NEG_def by auto
lemma POS_NEG:
"POS A B \<equiv> NEG B A"
unfolding POS_def NEG_def by auto
lemma NEG_POS:
"NEG A B \<equiv> POS B A"
unfolding POS_def NEG_def by auto
lemma POS_pcr_rule:
assumes "POS (A OO B) C"
shows "POS (A OO B OO X) (C OO X)"
using assms unfolding POS_def OO_def by blast
lemma NEG_pcr_rule:
assumes "NEG (A OO B) C"
shows "NEG (A OO B OO X) (C OO X)"
using assms unfolding NEG_def OO_def by blast
lemma POS_apply:
assumes "POS R R'"
assumes "R f g"
shows "R' f g"
using assms unfolding POS_def by auto
text {* Proving a parametrized correspondence relation *}
lemma fun_mono:
assumes "A \<ge> C"
assumes "B \<le> D"
shows "(A ===> B) \<le> (C ===> D)"
using assms unfolding fun_rel_def by blast
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))"
unfolding OO_def fun_rel_def by blast
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y"
unfolding right_unique_def left_total_def by blast
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y"
unfolding left_unique_def right_total_def by blast
lemma neg_fun_distr1:
assumes 1: "left_unique R" "right_total R"
assumes 2: "right_unique R'" "left_total R'"
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) "
using functional_relation[OF 2] functional_converse_relation[OF 1]
unfolding fun_rel_def OO_def
apply clarify
apply (subst all_comm)
apply (subst all_conj_distrib[symmetric])
apply (intro choice)
by metis
lemma neg_fun_distr2:
assumes 1: "right_unique R'" "left_total R'"
assumes 2: "left_unique S'" "right_total S'"
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))"
using functional_converse_relation[OF 2] functional_relation[OF 1]
unfolding fun_rel_def OO_def
apply clarify
apply (subst all_comm)
apply (subst all_conj_distrib[symmetric])
apply (intro choice)
by metis
subsection {* ML setup *}
ML_file "Tools/Lifting/lifting_util.ML"
ML_file "Tools/Lifting/lifting_info.ML"
setup Lifting_Info.setup
lemmas [reflexivity_rule] = reflp_equality reflp_Quotient_composition
(* setup for the function type *)
declare fun_quotient[quot_map]
declare fun_mono[relator_mono]
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2
ML_file "Tools/Lifting/lifting_term.ML"
ML_file "Tools/Lifting/lifting_def.ML"
ML_file "Tools/Lifting/lifting_setup.ML"
hide_const (open) invariant POS NEG
end