# Theory Lifting

theory Lifting
imports Equiv_Relations
```(*  Title:      HOL/Lifting.thy
Author:     Brian Huffman and Ondrej Kuncar
Author:     Cezary Kaliszyk and Christian Urban
*)

section ‹Lifting package›

theory Lifting
imports Equiv_Relations Transfer
keywords
"parametric" and
"print_quot_maps" "print_quotients" :: diag and
"lift_definition" :: thy_goal and
"setup_lifting" "lifting_forget" "lifting_update" :: thy_decl
begin

subsection ‹Function map›

context includes lifting_syntax
begin

lemma map_fun_id:
"(id ---> id) = id"

subsection ‹Quotient Predicate›

definition
"Quotient R Abs Rep T ⟷
(∀a. Abs (Rep a) = a) ∧
(∀a. R (Rep a) (Rep a)) ∧
(∀r s. R r s ⟷ R r r ∧ R s s ∧ Abs r = Abs s) ∧
T = (λx y. R x x ∧ Abs x = y)"

lemma QuotientI:
assumes "⋀a. Abs (Rep a) = a"
and "⋀a. R (Rep a) (Rep a)"
and "⋀r s. R r s ⟷ R r r ∧ R s s ∧ Abs r = Abs s"
and "T = (λx y. R x x ∧ 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 ∧ R s s ∧ Abs r = Abs s ⟷ R r s" ― ‹orientation does not loop on rewriting›
using a unfolding Quotient_def
by blast

lemma Quotient_cr_rel: "T = (λx y. R x x ∧ Abs x = y)"
using a unfolding Quotient_def
by blast

lemma Quotient_refl1: "R r s ⟹ R r r"
using a unfolding Quotient_def
by fast

lemma Quotient_refl2: "R r s ⟹ R s s"
using a unfolding Quotient_def
by fast

lemma Quotient_rel_rep: "R (Rep a) (Rep b) ⟷ a = b"
using a unfolding Quotient_def
by metis

lemma Quotient_rep_abs: "R r r ⟹ R (Rep (Abs r)) r"
using a unfolding Quotient_def
by blast

lemma Quotient_rep_abs_eq: "R t t ⟹ R ≤ op= ⟹ Rep (Abs t) = t"
using a unfolding Quotient_def
by blast

lemma Quotient_rep_abs_fold_unmap:
assumes "x' ≡ Abs x" and "R x x" and "Rep x' ≡ 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' ≡ Abs x"
shows "Rep x' ≡ Rep x'"
by simp

lemma Quotient_rel_abs: "R r s ⟹ 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 ⟷
(∀a b. T a b ⟶ Abs a = b) ∧
(∀b. T (Rep b) b) ∧
(∀x y. R x y ⟷ T x (Abs x) ∧ T y (Abs y) ∧ 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 ⟷
(∀a b. T a b ⟶ Abs a = b) ∧
(∀b. T (Rep b) b) ∧
(∀x y. R x y ⟷ T x (Abs y) ∧ T y (Abs x))"
unfolding Quotient_alt_def by (safe, metis+)

lemma Quotient_alt_def3:
"Quotient R Abs Rep T ⟷
(∀a b. T a b ⟶ Abs a = b) ∧ (∀b. T (Rep b) b) ∧
(∀x y. R x y ⟷ (∃z. T x z ∧ T y z))"
unfolding Quotient_alt_def2 by (safe, metis+)

lemma Quotient_alt_def4:
"Quotient R Abs Rep T ⟷
(∀a b. T a b ⟶ Abs a = b) ∧ (∀b. T (Rep b) b) ∧ R = T OO conversep T"
unfolding Quotient_alt_def3 fun_eq_iff by auto

lemma Quotient_alt_def5:
"Quotient R Abs Rep T ⟷
T ≤ BNF_Def.Grp UNIV Abs ∧ BNF_Def.Grp UNIV Rep ≤ T¯¯ ∧ R = T OO T¯¯"
unfolding Quotient_alt_def4 Grp_def by blast

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 rel_fun_def fun_eq_iff map_fun_apply
by (safe, metis+)

lemma apply_rsp:
fixes f g::"'a ⇒ '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: rel_funE)

lemma apply_rsp':
assumes a: "(R1 ===> R2) f g" "R1 x y"
shows "R2 (f x) (g y)"
using a by (auto elim: rel_funE)

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 ∘ Abs1) (Rep1 ∘ Rep2) (T1 OO T2)"
using assms unfolding Quotient_alt_def4 by fastforce

lemma equivp_reflp2:
"equivp R ⟹ reflp R"
by (erule equivpE)

subsection ‹Respects predicate›

definition Respects :: "('a ⇒ 'a ⇒ bool) ⇒ 'a set"
where "Respects R = {x. R x x}"

lemma in_respects: "x ∈ Respects R ⟷ R x x"
unfolding Respects_def by simp

lemma UNIV_typedef_to_Quotient:
assumes "type_definition Rep Abs UNIV"
and T_def: "T ≡ (λx 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 ⇒ 'b"
and Rep :: "'b ⇒ 'a"
assumes "type_definition Rep Abs (UNIV::'a set)"
shows "equivp (op=::'a⇒'a⇒bool)"
by (rule identity_equivp)

lemma typedef_to_Quotient:
assumes "type_definition Rep Abs S"
and T_def: "T ≡ (λx y. x = Rep y)"
shows "Quotient (eq_onp (λx. x ∈ 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: eq_onp_def fun_eq_iff)
qed

lemma typedef_to_part_equivp:
assumes "type_definition Rep Abs S"
shows "part_equivp (eq_onp (λx. x ∈ S))"
proof (intro part_equivpI)
interpret type_definition Rep Abs S by fact
show "∃x. eq_onp (λx. x ∈ S) x x" using Rep by (auto simp: eq_onp_def)
next
show "symp (eq_onp (λx. x ∈ S))" by (auto intro: sympI simp: eq_onp_def)
next
show "transp (eq_onp (λx. x ∈ S))" by (auto intro: transpI simp: eq_onp_def)
qed

lemma open_typedef_to_Quotient:
assumes "type_definition Rep Abs {x. P x}"
and T_def: "T ≡ (λx y. x = Rep y)"
shows "Quotient (eq_onp 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 (eq_onp P)"
using typedef_to_part_equivp [OF assms] by simp

lemma type_definition_Quotient_not_empty: "Quotient (eq_onp P) Abs Rep T ⟹ ∃x. P x"
unfolding eq_onp_def by (drule Quotient_rep_reflp) blast

lemma type_definition_Quotient_not_empty_witness: "Quotient (eq_onp P) Abs Rep T ⟹ P (Rep undefined)"
unfolding eq_onp_def by (drule Quotient_rep_reflp) blast

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 rel_fun_def by simp

lemma Quotient_abs_induct:
assumes "⋀y. R y y ⟹ P (Abs y)" shows "P x"
using 1 assms unfolding Quotient_def by metis

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_left_total: "left_total T"
using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto

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) (λx. x) Abs"
using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp

lemma Quotient_total_abs_induct: "(⋀y. P (Abs y)) ⟹ P x"
using 1 2 unfolding Quotient_alt_def reflp_def by metis

lemma Quotient_total_abs_eq_iff: "Abs x = Abs y ⟷ R x y"
using Quotient_rel [OF 1] 2 unfolding reflp_def by simp

end

text ‹Generating transfer rules for a type defined with ‹typedef›.›

context
fixes Rep Abs A T
assumes type: "type_definition Rep Abs A"
assumes T_def: "T ≡ (λ(x::'a) (y::'b). x = Rep y)"
begin

lemma typedef_left_unique: "left_unique T"
unfolding left_unique_def T_def
by (simp add: type_definition.Rep_inject [OF type])

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 =) (λx. x) Rep"
unfolding rel_fun_def T_def by simp

end

text ‹Generating the correspondence rule for a constant defined with
‹lift_definition›.›

lemma Quotient_to_transfer:
assumes "Quotient R Abs Rep T" and "R c c" and "c' ≡ Abs c"
shows "T c c'"
using assms by (auto dest: Quotient_cr_rel)

text ‹Proving reflexivity›

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 Quotient_composition_ge_eq:
assumes "left_total T"
assumes "R ≥ op="
shows "(T OO R OO T¯¯) ≥ op="
using assms unfolding left_total_def by fast

lemma Quotient_composition_le_eq:
assumes "left_unique T"
assumes "R ≤ op="
shows "(T OO R OO T¯¯) ≤ op="
using assms unfolding left_unique_def by blast

lemma eq_onp_le_eq:
"eq_onp P ≤ op=" unfolding eq_onp_def by blast

lemma reflp_ge_eq:
"reflp R ⟹ R ≥ op=" unfolding reflp_def by blast

text ‹Proving a parametrized correspondence relation›

definition POS :: "('a ⇒ 'b ⇒ bool) ⇒ ('a ⇒ 'b ⇒ bool) ⇒ bool" where
"POS A B ≡ A ≤ B"

definition  NEG :: "('a ⇒ 'b ⇒ bool) ⇒ ('a ⇒ 'b ⇒ bool) ⇒ bool" where
"NEG A B ≡ B ≤ A"

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 ≡ NEG B A"
unfolding POS_def NEG_def by auto

lemma NEG_POS:
"NEG A B ≡ 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 ≥ C"
assumes "B ≤ D"
shows   "(A ===> B) ≤ (C ===> D)"
using assms unfolding rel_fun_def by blast

lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) ≤ ((R OO R') ===> (S OO S'))"
unfolding OO_def rel_fun_def by blast

lemma functional_relation: "right_unique R ⟹ left_total R ⟹ ∀x. ∃!y. R x y"
unfolding right_unique_def left_total_def by blast

lemma functional_converse_relation: "left_unique R ⟹ right_total R ⟹ ∀y. ∃!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') ≤ ((R ===> S) OO (R' ===> S')) "
using functional_relation[OF 2] functional_converse_relation[OF 1]
unfolding rel_fun_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') ≤ ((R ===> S) OO (R' ===> S'))"
using functional_converse_relation[OF 2] functional_relation[OF 1]
unfolding rel_fun_def OO_def
apply clarify
apply (subst all_comm)
apply (subst all_conj_distrib[symmetric])
apply (intro choice)
by metis

subsection ‹Domains›

lemma composed_equiv_rel_eq_onp:
assumes "left_unique R"
assumes "(R ===> op=) P P'"
assumes "Domainp R = P''"
shows "(R OO eq_onp P' OO R¯¯) = eq_onp (inf P'' P)"
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def
fun_eq_iff by blast

lemma composed_equiv_rel_eq_eq_onp:
assumes "left_unique R"
assumes "Domainp R = P"
shows "(R OO op= OO R¯¯) = eq_onp P"
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def
fun_eq_iff is_equality_def by metis

lemma pcr_Domainp_par_left_total:
assumes "Domainp B = P"
assumes "left_total A"
assumes "(A ===> op=) P' P"
shows "Domainp (A OO B) = P'"
using assms
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def
by (fast intro: fun_eq_iff)

lemma pcr_Domainp_par:
assumes "Domainp B = P2"
assumes "Domainp A = P1"
assumes "(A ===> op=) P2' P2"
shows "Domainp (A OO B) = (inf P1 P2')"
using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def
by (fast intro: fun_eq_iff)

definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool"
where "rel_pred_comp R P ≡ λx. ∃y. R x y ∧ P y"

lemma pcr_Domainp:
assumes "Domainp B = P"
shows "Domainp (A OO B) = (λx. ∃y. A x y ∧ P y)"
using assms by blast

lemma pcr_Domainp_total:
assumes "left_total B"
assumes "Domainp A = P"
shows "Domainp (A OO B) = P"
using assms unfolding left_total_def
by fast

lemma Quotient_to_Domainp:
assumes "Quotient R Abs Rep T"
shows "Domainp T = (λx. R x x)"
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms])

lemma eq_onp_to_Domainp:
assumes "Quotient (eq_onp P) Abs Rep T"
shows "Domainp T = P"
by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms])

end

(* needed for lifting_def_code_dt.ML (moved from Lifting_Set) *)
lemma right_total_UNIV_transfer:
assumes "right_total A"
shows "(rel_set A) (Collect (Domainp A)) UNIV"
using assms unfolding right_total_def rel_set_def Domainp_iff by blast

subsection ‹ML setup›

ML_file "Tools/Lifting/lifting_util.ML"

named_theorems relator_eq_onp
"theorems that a relator of an eq_onp is an eq_onp of the corresponding predicate"
ML_file "Tools/Lifting/lifting_info.ML"

(* 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_bnf.ML"
ML_file "Tools/Lifting/lifting_term.ML"
ML_file "Tools/Lifting/lifting_def.ML"
ML_file "Tools/Lifting/lifting_setup.ML"
ML_file "Tools/Lifting/lifting_def_code_dt.ML"

lemma pred_prod_beta: "pred_prod P Q xy ⟷ P (fst xy) ∧ Q (snd xy)"
by(cases xy) simp

lemma pred_prod_split: "P (pred_prod Q R xy) ⟷ (∀x y. xy = (x, y) ⟶ P (Q x ∧ R y))"
by(cases xy) simp

hide_const (open) POS NEG

end
```