(*  Title:      Nominal2_Eqvt

    Author:     Brian Huffman, 

    Author:     Christian Urban

    Equivariance, supp and freshness lemmas for various operators 

    (contains many, but not all such lemmas).

*)

theory Nominal2_Eqvt

imports Nominal2_Base 

uses ("nominal_thmdecls.ML")

     ("nominal_permeq.ML")

     ("nominal_eqvt.ML")

begin

section {* Logical Operators *}

lemma eq_eqvt:

  shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"

  unfolding permute_eq_iff permute_bool_def ..

lemma if_eqvt:

  shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"

  by (simp add: permute_fun_def permute_bool_def)

lemma True_eqvt:

  shows "p \<bullet> True = True"

  unfolding permute_bool_def ..

lemma False_eqvt:

  shows "p \<bullet> False = False"

  unfolding permute_bool_def ..

lemma imp_eqvt:

  shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"

  by (simp add: permute_bool_def)

lemma conj_eqvt:

  shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"

  by (simp add: permute_bool_def)

lemma disj_eqvt:

  shows "p \<bullet> (A \<or> B) = ((p \<bullet> A) \<or> (p \<bullet> B))"

  by (simp add: permute_bool_def)

lemma Not_eqvt:

  shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"

  by (simp add: permute_bool_def)

lemma all_eqvt:

  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"

  unfolding permute_fun_def permute_bool_def

  by (auto, drule_tac x="p \<bullet> x" in spec, simp)

lemma all_eqvt2:

  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"

  unfolding permute_fun_def permute_bool_def

  by (auto, drule_tac x="p \<bullet> x" in spec, simp)

lemma ex_eqvt:

  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"

  unfolding permute_fun_def permute_bool_def

  by (auto, rule_tac x="p \<bullet> x" in exI, simp)

lemma ex_eqvt2:

  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"

  unfolding permute_fun_def permute_bool_def

  by (auto, rule_tac x="p \<bullet> x" in exI, simp)

lemma ex1_eqvt:

  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"

  unfolding Ex1_def 

  by (simp add: ex_eqvt permute_fun_def conj_eqvt all_eqvt imp_eqvt eq_eqvt)

lemma ex1_eqvt2:

  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"

  unfolding Ex1_def ex_eqvt2 conj_eqvt all_eqvt2 imp_eqvt eq_eqvt

  by simp

lemma the_eqvt:

  assumes unique: "\<exists>!x. P x"

  shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"

  apply(rule the1_equality [symmetric])

  apply(simp add: ex1_eqvt2[symmetric])

  apply(simp add: permute_bool_def unique)

  apply(simp add: permute_bool_def)

  apply(rule theI'[OF unique])

  done

section {* Set Operations *}

lemma mem_permute_iff:

  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"

unfolding mem_def permute_fun_def permute_bool_def

by simp

lemma mem_eqvt:

  shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"

  unfolding mem_permute_iff permute_bool_def by simp

lemma not_mem_eqvt:

  shows "p \<bullet> (x \<notin> A) \<longleftrightarrow> (p \<bullet> x) \<notin> (p \<bullet> A)"

  unfolding mem_def permute_fun_def by (simp add: Not_eqvt)

lemma Collect_eqvt:

  shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"

  unfolding Collect_def permute_fun_def ..

lemma Collect_eqvt2:

  shows "p \<bullet> {x. P x} = {x. p \<bullet> (P (-p \<bullet> x))}"

  unfolding Collect_def permute_fun_def ..

lemma empty_eqvt:

  shows "p \<bullet> {} = {}"

  unfolding empty_def Collect_eqvt2 False_eqvt ..

lemma supp_set_empty:

  shows "supp {} = {}"

  by (simp add: supp_def empty_eqvt)

lemma fresh_set_empty:

  shows "a \<sharp> {}"

  by (simp add: fresh_def supp_set_empty)

lemma UNIV_eqvt:

  shows "p \<bullet> UNIV = UNIV"

  unfolding UNIV_def Collect_eqvt2 True_eqvt ..

lemma union_eqvt:

  shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"

  unfolding Un_def Collect_eqvt2 disj_eqvt mem_eqvt by simp

lemma inter_eqvt:

  shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"

  unfolding Int_def Collect_eqvt2 conj_eqvt mem_eqvt by simp

lemma Diff_eqvt:

  fixes A B :: "'a::pt set"

  shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"

  unfolding set_diff_eq Collect_eqvt2 conj_eqvt Not_eqvt mem_eqvt by simp

lemma Compl_eqvt:

  fixes A :: "'a::pt set"

  shows "p \<bullet> (- A) = - (p \<bullet> A)"

  unfolding Compl_eq_Diff_UNIV Diff_eqvt UNIV_eqvt ..

lemma insert_eqvt:

  shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"

  unfolding permute_set_eq_image image_insert ..

lemma vimage_eqvt:

  shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"

  unfolding vimage_def permute_fun_def [where f=f]

  unfolding Collect_eqvt2 mem_eqvt ..

lemma image_eqvt:

  shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"

  unfolding permute_set_eq_image

  unfolding permute_fun_def [where f=f]

  by (simp add: image_image)

lemma finite_permute_iff:

  shows "finite (p \<bullet> A) \<longleftrightarrow> finite A"

  unfolding permute_set_eq_vimage

  using bij_permute by (rule finite_vimage_iff)

lemma finite_eqvt:

  shows "p \<bullet> finite A = finite (p \<bullet> A)"

  unfolding finite_permute_iff permute_bool_def ..

section {* List Operations *}

lemma append_eqvt:

  shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"

  by (induct xs) auto

lemma supp_append:

  shows "supp (xs @ ys) = supp xs \<union> supp ys"

  by (induct xs) (auto simp add: supp_Nil supp_Cons)

lemma fresh_append:

  shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"

  by (induct xs) (simp_all add: fresh_Nil fresh_Cons)

lemma rev_eqvt:

  shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"

  by (induct xs) (simp_all add: append_eqvt)

lemma supp_rev:

  shows "supp (rev xs) = supp xs"

  by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)

lemma fresh_rev:

  shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"

  by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)

lemma set_eqvt:

  shows "p \<bullet> (set xs) = set (p \<bullet> xs)"

  by (induct xs) (simp_all add: empty_eqvt insert_eqvt)

(* needs finite support premise

lemma supp_set:

  fixes x :: "'a::pt"

  shows "supp (set xs) = supp xs"

*)

lemma map_eqvt: 

  shows "p \<bullet> (map f xs) = map (p \<bullet> f) (p \<bullet> xs)"

  by (induct xs) (simp_all, simp only: permute_fun_app_eq)

section {* Product Operations *}

lemma fst_eqvt:

  "p \<bullet> (fst x) = fst (p \<bullet> x)"

 by (cases x) simp

lemma snd_eqvt:

  "p \<bullet> (snd x) = snd (p \<bullet> x)"

 by (cases x) simp

section {* Units *}

lemma supp_unit:

  shows "supp () = {}"

  by (simp add: supp_def)

lemma fresh_unit:

  shows "a \<sharp> ()"

  by (simp add: fresh_def supp_unit)

lemma permute_eqvt_raw:

  shows "p \<bullet> permute = permute"

apply(simp add: expand_fun_eq permute_fun_def)

apply(subst permute_eqvt)

apply(simp)

done

section {* Equivariance automation *}

text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw} *}

use "nominal_thmdecls.ML"

setup "Nominal_ThmDecls.setup"

lemmas [eqvt] = 

  (* connectives *)

  eq_eqvt if_eqvt imp_eqvt disj_eqvt conj_eqvt Not_eqvt 

  True_eqvt False_eqvt ex_eqvt all_eqvt ex1_eqvt

  imp_eqvt [folded induct_implies_def]

  (* nominal *)

  (* datatypes *)

  permute_prod.simps append_eqvt rev_eqvt set_eqvt

  map_eqvt fst_eqvt snd_eqvt Pair_eqvt permute_list.simps

  (* sets *)

  empty_eqvt UNIV_eqvt union_eqvt inter_eqvt mem_eqvt

  Diff_eqvt Compl_eqvt insert_eqvt Collect_eqvt image_eqvt

lemmas [eqvt_raw] =

  permute_eqvt_raw[THEN eq_reflection] (* the normal version of this lemma loops *)

text {* helper lemmas for the eqvt_tac *}

definition

  "unpermute p = permute (- p)"

lemma eqvt_apply:

  fixes f :: "'a::pt \<Rightarrow> 'b::pt" 

  and x :: "'a::pt"

  shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"

  unfolding permute_fun_def by simp

lemma eqvt_lambda:

  fixes f :: "'a::pt \<Rightarrow> 'b::pt"

  shows "p \<bullet> (\<lambda>x. f x) \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"

  unfolding permute_fun_def unpermute_def by simp

lemma eqvt_bound:

  shows "p \<bullet> unpermute p x \<equiv> x"

  unfolding unpermute_def by simp

(* provides perm_simp methods *)

use "nominal_permeq.ML"

setup Nominal_Permeq.setup

ML {*

  (Scan.repeat (Args.const true))) []

*}

method_setup perm_simp =

 {* pushes permutations inside *}

method_setup perm_strict_simp =

 {* pushes permutations inside, raises an error if it cannot solve all permutations *}

declare [[trace_eqvt = true]]

lemma 

  fixes B::"'a::pt"

  shows "p \<bullet> (B = C)"

apply(perm_simp)

oops

lemma 

  fixes B::"bool"

  shows "p \<bullet> (B = C)"

apply(perm_simp)

oops

lemma 

  fixes B::"bool"

  shows "p \<bullet> (A \<longrightarrow> B = C)"

apply (perm_simp) 

oops

lemma 

  shows "p \<bullet> (\<lambda>(x::'a::pt). A \<longrightarrow> (B::'a \<Rightarrow> bool) x = C) = foo"

apply(perm_simp)

oops

lemma 

  shows "p \<bullet> (\<lambda>B::bool. A \<longrightarrow> (B = C)) = foo"

apply (perm_simp)

oops

lemma 

  shows "p \<bullet> (\<lambda>x y. \<exists>z. x = z \<and> x = y \<longrightarrow> z \<noteq> x) = foo"

apply (perm_simp)

oops

lemma 

  shows "p \<bullet> (\<lambda>f x. f (g (f x))) = foo"

apply (perm_simp)

oops

lemma 

  fixes p q::"perm"

  and   x::"'a::pt"

  shows "p \<bullet> (q \<bullet> x) = foo"

apply(perm_simp)

oops

lemma 

  fixes p q r::"perm"

  and   x::"'a::pt"

  shows "p \<bullet> (q \<bullet> r \<bullet> x) = foo"

apply(perm_simp)

oops

lemma 

  fixes p r::"perm"

  shows "p \<bullet> (\<lambda>q::perm. q \<bullet> (r \<bullet> x)) = foo"

apply (perm_simp)

oops

lemma 

  fixes C D::"bool"

  shows "B (p \<bullet> (C = D))"

apply(perm_simp)

oops

declare [[trace_eqvt = false]]

text {* Problem: there is no raw eqvt-rule for The *}

lemma "p \<bullet> (THE x. P x) = foo"

apply(perm_strict_simp exclude: The)

apply(perm_simp exclude: The)

oops

(* automatic equivariance procedure for 

   inductive definitions *)

use "nominal_eqvt.ML"

end