(* Title: HOL/Lifting.thy
Author: Brian Huffman and Ondrej Kuncar
Author: Cezary Kaliszyk and Christian Urban
*)
header {* Lifting package *}
theory Lifting
imports Plain Equiv_Relations Transfer
keywords
"print_quotmaps" "print_quotients" :: diag and
"lift_definition" :: thy_goal and
"setup_lifting" :: thy_decl
uses
("Tools/Lifting/lifting_info.ML")
("Tools/Lifting/lifting_term.ML")
("Tools/Lifting/lifting_def.ML")
("Tools/Lifting/lifting_setup.ML")
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
lemma Quotient_abs_rep:
assumes a: "Quotient R Abs Rep T"
shows "Abs (Rep a) = a"
using a
unfolding Quotient_def
by simp
lemma Quotient_rep_reflp:
assumes a: "Quotient R Abs Rep T"
shows "R (Rep a) (Rep a)"
using a
unfolding Quotient_def
by blast
lemma Quotient_rel:
assumes a: "Quotient R Abs Rep T"
shows "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:
assumes a: "Quotient R Abs Rep T"
shows "T = (\<lambda>x y. R x x \<and> Abs x = y)"
using a
unfolding Quotient_def
by blast
lemma Quotient_refl1:
assumes a: "Quotient R Abs Rep T"
shows "R r s \<Longrightarrow> R r r"
using a unfolding Quotient_def
by fast
lemma Quotient_refl2:
assumes a: "Quotient R Abs Rep T"
shows "R r s \<Longrightarrow> R s s"
using a unfolding Quotient_def
by fast
lemma Quotient_rel_rep:
assumes a: "Quotient R Abs Rep T"
shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
using a
unfolding Quotient_def
by metis
lemma Quotient_rep_abs:
assumes a: "Quotient R Abs Rep T"
shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
using a unfolding Quotient_def
by blast
lemma Quotient_rel_abs:
assumes a: "Quotient R Abs Rep T"
shows "R r s \<Longrightarrow> Abs r = Abs s"
using a unfolding Quotient_def
by blast
lemma Quotient_symp:
assumes a: "Quotient R Abs Rep T"
shows "symp R"
using a unfolding Quotient_def using sympI by (metis (full_types))
lemma Quotient_transp:
assumes a: "Quotient R Abs Rep T"
shows "transp R"
using a unfolding Quotient_def using transpI by (metis (full_types))
lemma Quotient_part_equivp:
assumes a: "Quotient R Abs Rep T"
shows "part_equivp R"
by (metis Quotient_rep_reflp Quotient_symp Quotient_transp a part_equivpI)
lemma identity_quotient: "Quotient (op =) id id (op =)"
unfolding Quotient_def by simp
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 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)"
proof -
from 1 have Abs1: "\<And>a b. T1 a b \<Longrightarrow> Abs1 a = b"
unfolding Quotient_alt_def by simp
from 1 have Rep1: "\<And>b. T1 (Rep1 b) b"
unfolding Quotient_alt_def by simp
from 2 have Abs2: "\<And>a b. T2 a b \<Longrightarrow> Abs2 a = b"
unfolding Quotient_alt_def by simp
from 2 have Rep2: "\<And>b. T2 (Rep2 b) b"
unfolding Quotient_alt_def by simp
from 2 have R2:
"\<And>x y. R2 x y \<longleftrightarrow> T2 x (Abs2 x) \<and> T2 y (Abs2 y) \<and> Abs2 x = Abs2 y"
unfolding Quotient_alt_def by simp
show ?thesis
unfolding Quotient_alt_def
apply simp
apply safe
apply (drule Abs1, simp)
apply (erule Abs2)
apply (rule pred_compI)
apply (rule Rep1)
apply (rule Rep2)
apply (rule pred_compI, assumption)
apply (drule Abs1, simp)
apply (clarsimp simp add: R2)
apply (rule pred_compI, assumption)
apply (drule Abs1, simp)+
apply (clarsimp simp add: R2)
apply (drule Abs1, simp)+
apply (clarsimp simp add: R2)
apply (rule pred_compI, assumption)
apply (rule pred_compI [rotated])
apply (erule conversepI)
apply (drule Abs1, simp)+
apply (simp add: R2)
done
qed
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 copy_type_to_Quotient:
assumes "type_definition Rep Abs UNIV"
and T_def: "T \<equiv> (\<lambda>x y. Abs x = y)"
shows "Quotient (op =) Abs Rep T"
proof -
interpret type_definition Rep Abs UNIV by fact
from Abs_inject Rep_inverse T_def show ?thesis by (auto intro!: QuotientI)
qed
lemma copy_type_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 invariant_type_to_Quotient:
assumes "type_definition Rep Abs {x. P x}"
and T_def: "T \<equiv> (\<lambda>x y. (invariant P) x x \<and> Abs x = y)"
shows "Quotient (invariant P) Abs Rep T"
proof -
interpret type_definition Rep Abs "{x. P x}" by fact
from Rep Abs_inject Rep_inverse T_def show ?thesis by (auto intro!: QuotientI simp: invariant_def)
qed
lemma invariant_type_to_part_equivp:
assumes "type_definition Rep Abs {x. P x}"
shows "part_equivp (invariant P)"
proof (intro part_equivpI)
interpret type_definition Rep Abs "{x. P x}" by fact
show "\<exists>x. invariant P x x" using Rep by (auto simp: invariant_def)
next
show "symp (invariant P)" by (auto intro: sympI simp: invariant_def)
next
show "transp (invariant P)" by (auto intro: transpI simp: invariant_def)
qed
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)
subsection {* ML setup *}
text {* Auxiliary data for the lifting package *}
use "Tools/Lifting/lifting_info.ML"
setup Lifting_Info.setup
declare [[map "fun" = (fun_rel, fun_quotient)]]
use "Tools/Lifting/lifting_term.ML"
use "Tools/Lifting/lifting_def.ML"
use "Tools/Lifting/lifting_setup.ML"
hide_const (open) invariant
end