src/HOL/Lifting.thy
 author kuncar Tue, 03 Apr 2012 16:26:48 +0200 changeset 47308 9caab698dbe4 child 47325 ec6187036495 permissions -rw-r--r--
new package Lifting - initial commit
```
(*  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
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 and function relation *}

notation map_fun (infixr "--->" 55)

lemma map_fun_id:
"(id ---> id) = id"
by (simp add: fun_eq_iff)

definition
fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
where
"fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"

lemma fun_relI [intro]:
assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
shows "(R1 ===> R2) f g"
using assms by (simp add: fun_rel_def)

lemma fun_relE:
assumes "(R1 ===> R2) f g" and "R1 x y"
obtains "R2 (f x) (g y)"
using assms by (simp add: fun_rel_def)

lemma fun_rel_eq:
shows "((op =) ===> (op =)) = (op =)"
by (auto simp add: fun_eq_iff elim: fun_relE)

lemma fun_rel_eq_rel:
shows "((op =) ===> R) = (\<lambda>f g. \<forall>x. R (f x) (g x))"
by (simp add: fun_rel_def)

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

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
```