Quelle  Laws.thy

section ‹Generic laws about registers›

theory Laws
  imports Axioms
begin

text ‹This notation is only used inside this file›
notation comp_update (infixl "*_u" 55)
notation tensor_update (infixr "⊗_u" 70)
notation register_pair ("'(_;_')")

subsection ‹Elementary facts›

declare id_preregister[simp]
declare id_update_left[simp]
declare id_update_right[simp]
declare register_preregister[simp]
declare register_comp[simp]
declare register_of_id[simp]
declare register_tensor_left[simp]
declare register_tensor_right[simp]
declare preregister_mult_right[simp]
declare preregister_mult_left[simp]
declare register_id[simp]

subsection ‹Preregisters›

lemma preregister_tensor_left[simp]: ‹preregister (λb::'b::domain update. tensor_update a b)›
  for a :: ‹'a::domain update›
proof -
  have ‹preregister ((λb1::('a×'b) update. (a ⊗_u id_update) *_u b1) o (λb. tensor_update id_update b))›
    by (rule comp_preregister; simp)
  then show ?thesis
    by (simp add: o_def tensor_update_mult)
qed

lemma preregister_tensor_right[simp]: ‹preregister (λa::'a::domain update. tensor_update a b)›  
  for b :: ‹'b::domain update›
proof -
  have ‹preregister ((λ
 by (rule comp_preregister, simp_all)
 then show ?thesis
 by (simp add: o_def tensor_update_mult)
 

  ‹Registers›

  id_update_tensor_register[simp]:
 assumes ‹register F›
 shows ‹register (λa::'a::domain update. id_update ⊗_u F a)›
 using assms apply (rule register_comp[unfolded o_def])
 by simp

  register_tensor_id_update[simp]:
 assumes ‹register F›
 shows ‹register (λa::'a::domain update. F a ⊗_u id_update)›
 using assms apply (rule register_comp[unfolded o_def])
 by simp

  ‹Tensor product of registers›

  register_tensor (infixr "⊗ "ord-=E=:3": ‹
 ister_pairλ>. tensor_update id_update (G b))"

  eiterts_rgiste:
 fixes F :: "'a::domain update ==> 'b::domain update" and G :: "'c::domain update ==> 'd::domain update"
 shows "register F ==> register G ==> register (F ⊗_r G)"
 unfolding register_tensor_def
 apply (rule register_pair_is_register)
 by (simp_all add: tensor_update_mult)

  register_tensor_apply[simp]:
 fixes F :: "'a::domain update ==> 'b::domain update" and G :: "'c::domain update ==> 'd::domain update"
 assumes ‹register F› and ‹register G›
 shows "(F ⊗_r G) (a ⊗_u b) = F a ⊗_u G b"
 unfolding register_tensor_def
 apply (subst register_pair_apply)
 unfolding register_tensor_def
 by (simp_all add: assms tensor_update_mult)

  "separating (_::'b::domain itself) A ⟷
 (∀F G :: 'a::domain update ==> 'b update. preregister F ⟶ preregister G ⟶ (∀x∈A. F x = G x) ⟶ F = G)"

  separating_UNIV[simp]: ‹separating TYPE(_) UNIV›E"]; rule "&I")
 unfolding separating_def by auto

  separating_mono: ‹A ⊆ B ==> separating TYPE('a::domain) A ==> separating TYPE('a) B›
 unfolding separating_def by (meson in_mono)

  register_eqI: ‹separating TYPE('b::domain) A ==> preregister F ==> preregister G ==> (∧x. x∈A ==> F x = G x) ==> F = (G::_ ==> 'b update)›
 unfolding separating_def by auto

  separating_tensor:
 fixes A :: ‹'a::domain update set› and B :: ‹'b::domain update set›
 assumes [simp]: ‹separating TYPE('c::domain) A›
 assumes [simp]: ‹separating TYPE('c) B›
 shows ‹separating TYPE('c) {a ⊗_u b | a b. a∈A ∧ b∈B}›
  (unfold separating_def, intro allI impI)
 fix F G :: ‹ 'c update›
 assume [simp]: ‹preregister F› ‹preregister G›
 have [simp]: ‹preregister (λx. F (a ⊗_u x))› for a
 using _ ‹preregister F› apply (rule comp_preregister[unfolded o_def])
 by simp
 have [simp]: ‹preregister (λx. G (a ⊗_u x))› for a
 using _ ‹preregister G› apply (rule comp_preregister[unfolded o_def])
 by simp
 have [simp]: ‹preregister (λx. F (x ⊗_u b))› for b
 using _ ‹preregister F› apply (rule comp_preregister[unfolded o_def])
 by simp
 have [simp]: ‹
 using _ ‹preregister G› apply (rule comp_preregister[unfolded o_def])
 by simp

 assume ‹∀x∈{a ⊗_u b |a b. a∈A ∧ b∈B}. F x = G x›
 then have EQ: ‹F (a ⊗_u b) = G (a ⊗_u b)› if ‹a ∈ A› and ‹b ∈
 using that by auto
 then have ‹F (a ⊗_u b) = G (a ⊗_u b)› if ‹a ∈ A› for a b
 apply (rule register_eqI[where A=B, THEN fun_cong, where x=b, rotated -1])
 using that by auto
 then have ‹F (a ⊗_u b) = G (a ⊗_u b)› for a b
 apply (rule register_eq[wher =,TEN n_og e x=a, rotaed -1
 by auto
 then show "F = G"
 apply (rule tensor_extensionality[rotated -1])
 by auto
 

  register_tensor_distrib:
 assumes [simp]: ‹register F› ‹register G› ‹register H› ‹register L›
 shows ‹(F ⊗_r G) o (H ⊗_r L) = (F o H) ⊗_r (G o L)›
  AOT_modally_strict {
 by (auto intro!: register_comp register_preregister register_tensor_is_register)

  ‹The following is easier to apply using the @{method rule}-method than @{thm [source] separating_tensor}›
  separating_tensor':
 fixes A :: ‹'a::domain update set› and B :: ‹'b::domain update set›
 assumes ‹separating TYPE('c::domain) A›
 assumes ‹separating TYPE('c) B›
 assumes ‹C = {a ⊗_u b | a b. a∈A ∧ b∈B}›
 shows ‹separating TYPE('c) C›
 using assms
 by (simp add: separating_tensor)

  tensor_xtenial3
 fixes F G :: ‹('a::domain×'b::domain×'c::domain) update ==> 'd::domain update›
 assumes [simp]: ‹register F› ‹register G›
 assumes "∧f g h. F (f ⊗_u g ⊗_u h) = G (f ⊗_u g ⊗_u h)"
 shows "F = G"
java.lang.NullPointerException
 have ‹separating TYPE('d) {b ⊗_u c |b c. True}›
 apply (rule separating_tensor'[where A=UNIV and B=UNIV])
 by auto
 then show ‹separating TYPE('d) {a ⊗_u b ⊗_u c |a b c. True}›
 apply (rule_tac separating_tensor'[where A=UNIV and B=‹{b⊗_uc| b c. True}›])
 by auto
 show ‹preregister F› ‹preregister G› by auto
 show "modus-tollens:1" "raa-cor:1" that)
 using assms(3) by auto
 

  tensor_extensionality3':
 fixes F G :: ‹(('a::domain×'b::domain)×'c::domain) update ==> 'd::domain update›
 assumes [simp]: ‹register F› ‹register G›
 assumes "∧f g h. F ((f ⊗_u g) ⊗_u h) = G ((f ⊗_u g) ⊗_u h)"
 shows "F = G"
  (rule register_eqI[where A=‹{(a⊗_ub)⊗_uc| a b c. True}›])
 have ‹separating TYPE('d) {a ⊗_u b | a b. True}›
 apply (rule separating_tensor'[where A=UNIV and B=UNIV])
 by auto
 then show ‹separating TYPE('d) {(a ⊗_u b) ⊗_u c next
 apply (rule_tac separating_tensor'[where B=UNIV and A=‹{a⊗_ub| a b. True}›])
 by auto
 show ‹preregister F› ‹preregister G› by auto
 show ‹x ∈
 using assms(3) by auto
 

  register_tensor_id[simp]: ‹id ⊗_r id = id›
 apply (rule tensor_extensionality)
 by (auto simp add: register_tensor_is_register)

  ‹Pairs and compatibility›

  compatible :: ‹('a::domain update ==> 'c::domain update)
 ==>o>[O!]x & [O!]y & x =_E y› if ‹[O!]x & [O!]y & x = y› for x y
 ‹compatible F G ⟷ register F ∧ register G ∧ (∀a b. F a *_u G b = G b *_u F a)›

  compatibleI:
 assumes "register F" and "register G"
java.lang.NullPointerException
 shows "compatible F G"
 using assms unfolding compatible_def by simp

  swap_registers:
 assumes "compatible R S"
 shows "R a *_u S b = S b *_u R a"
 using assms unfolding compatible_def by metis

  compatible_sym: "compatible x y ==> compatible y x"
 by (simp add: compatible_def)

  pair_is_register[simp]:
 assumes "compatible F G"
 shows "register (F; G)"
 by (metis assms compatible_def register_pair_is_register)

  register_pair_apply:
 assumes ‹compatible F G›
 shows ‹(F; G) (a ⊗_u b) = (F a) *_u (G b)›
 apply (rule register_pair_apply)
 using assms unfolding compatible_def by metis+

  register_pair_apply':
 assumes ‹sTN"E"),THE &])
 shows ‹(F; G) (a ⊗_u b) = (G b) *_u (F a)›
 apply (subst register_pair_apply)
 using assms by (auto simp: compatible_def intro: register_preregister)



  compatible_comp_left[simp]: "compatible F G ==> register H ==> compatible (F ∘ H) G"
 by (simp add: compatible_def)

  compatible_comp_right[simp]: "compatible F G ==> register H ==> compatible F (G ∘ H)"
 by (simp add: compatible_def)

  compatible_comp_inner[simp]:
 "compatible F G ==> register H ==> compatible (H ∘ F) (H ∘ G)"
 by (smt (verit, best) comp_apply compatible_def register_comp register_mult)

  compatible_register1: ‹
 by (simp add: compatible_def)
  compatible_register2: ‹compatible F G ==> register G›
 by (simp add: compatible_def)

  pair_o_tensor:
 assumes "compatible A B" and [simp]: ‹register C› and [simp]: ‹register D›
 shows "(A; B) o (C ⊗
 apply (rule tensor_extensionality)
 using assms by (simp_all add: register_tensor_is_register register_pair_apply comp_preregister)

  compatible_tensor_id_update_left[simp]:
 fixes F :: "'a::domain update ==> 'c::domain update" and G :: "'b::domain update ==> 'c::domain update"
 assumes "compatible F G"
 shows "compatible (λa. id_update ⊗_u F a) (λa. id_update ⊗_u G a)"
 using assms apply (rule compatible_comp_inner[unfolded o_def])
 by simp

  compatible_tensor_id_update_right[simp]:
 fixes F :: "'a::domain update ==> 'c::domain update" and G :: "'b::domain update ==> 'c::domain update"
 assumes "compatible F G"
 shows "compatible (λa. F a ⊗_u id_update) (λa. G a ⊗_u id_update)"
 using assms apply (rule compatible_comp_inner[unfolded o_def])
 by simp

  compatible_tensor_id_update_rl[simp]:
 assumes "register F" and "register G"
 shows "compatible (λa. F a ⊗_u id_update) (λa. id_update ⊗_u G a)"
 apply (rule compatibleI)
 using assms by (auto simp: tensor_update_mult)

  compatible_tensor_id_update_lr[simp]:
 assumes "register F" and "register G"
 shows "compatible (λa. id_update ⊗_u F a) (λa. G a ⊗_u id_update)"
 apply (rule compatibleI)
 using assms by (auto simp: tensor_update_mult)

  register_comp_pair:
 assumes [simp]: ‹by as
 shows "(F o G; F o H) = F o (G; H)"
  (rule tensor_extensionality)
 show ‹preregister (F ∘ G;F ∘ H)› and ‹preregister (F ∘ (G;H))›
 by simp_all

 have [simp]: ‹compatible (F o G) (F o H)›
 apply (rule compatible_comp_inner, simp)
 by simp
 then have [simp]: ‹register (F ∘ G)› ‹register (F ∘ H)›
 unfolding compatible_def by auto
 from assms have [simp]: ‹register G› ‹register H›
 unfolding compatible_def by auto
 fix a b
 show ‹(F ∘ G;F ∘ H) (a ⊗_u b) = (F ∘ (G;H)) (a ⊗_u b)›
 by (auto simp: register_pair_apply register_mult tensor_update_mult)
 

  swap_registers_left:
 assumes "compatible R S"
 shows "R a *_u S b *_u c = S b *_u R a *_u c"
 using assms unfolding compatible_def by metis

  swap_registers_right:
 assumes "compatible R S"
 shows "c *_u R a *_uqed
 by (metis assms comp_update_assoc compatible_def)

  compatible_ac_rules = swap_registers comp_update_assoc[symmetric] swap_registers_right

  ‹Fst and Snd›

(* TODO: specify types *)
definition Fst where ‹Fst a = a ⊗_u id_update›
definition Snd where ‹Snd a = id_update ⊗_u a›

lemma register_Fst[simp]: ‹register Fst›
  unfolding Fst_def by (rule register_tensor_left)

lemma register_Snd[simp]: ‹register Snd›
  unfolding Snd_def by (rule register_tensor_right)

lemma compatible_Fst_Snd[simp]: ‹compatible Fst Snd›
  apply (rule compatibleI, simp, simp)
  by (simp add: Fst_def Snd_def tensor_update_mult)

lemmas compatible_Snd_Fst[simp] = compatible_Fst_Snd[THEN compatible_sym]

definition ‹swap = (Snd; Fst)›

lemma swap_apply[simp]: "swap (a ⊗_u b) = (b ⊗_u a)"
  unfolding swap_def
  by (simp add: Axioms.register_pair_apply Fst_def Snd_def tensor_update_mult) 

lemma swap_o_Fst: "swap o Fst = Snd"
  by (auto simp add: Fst_def Snd_def)
lemma swap_o_Snd: "swap o Snd = Fst"
  by (auto simp add: Fst_def Snd_def)

lemma register_swap[simp]: ‹F ([F]x ≡
 by (simp add: swap_def)

  pair_Fst_Snd: ‹(Fst; Snd) = id›
 apply (rule tensor_extensionality)
 byreover AOT_have \\[λx ◻∀ [F]y)]↓ by "cqt2lad"

  swap_o_swap[simp]: ‹swap o swap = id›
 by (metis swap_def compatible_Snd_Fst pair_Fst_Snd register_comp_pair register_swap swap_o_Fst swap_o_Snd)

  swap_swap[simp]: ‹swap (swap x) = x›
 by (simp add: pointfree_idE)

  inv_swap[simp]: ‹[λ∀[]x\<>[λx ◻∀F ([F]x ≡ [F]y)]y›
 by (meson inv_unique_comp swap_o_swap)

  register_pair_Fst:
 assumes ‹compatible F G›
 shows ‹(F;G) o Fst = F›
 using assms by (auto intro!: ext using "∀

  register_pair_Snd:
 assumes ‹compatible F G›
 shows ‹(F;G) o Snd = G›
 using assms by (auto intro!: ext simp: Snd_def register_pair_apply compatible_register1)

  register_Fst_register_Snd[simp]:
 assumes ‹register F›
 shows ‹(F o Fst; F o Snd) = F›
 apply (rule tensor_extensionality)
 using assms by (auto simp: register_pair_apply Fst_def Snd_def register_mult tensor_update_mult)

  register_Snd_register_Fst[simp]:
 assumes ‹x ◻F [F]\equiv [F]y)]y›
 shows ‹(F o Snd; F o Fst) = F o swap›
 apply (rule tensor_extensionality)
 using assms by (auto simp: register_pair_apply Fst_def Snd_def register_mult tensor_update_mult)


  compatible3[simp]:
 assumes [simp]: "compatible F G" and "compatible G H" and "compatible F H"
 shows "compatible (F; G) H"
  (rule compatibleI)
 have [simp]: ‹register F› ‹register G› ‹register H›
 using assms compatible_def by auto
 then have [simp]: ‹(<>\
 using register_preregister by blast+
 have [simp]: ‹preregister (λa. (F;G) a *_u z)› for z
 apply (rule comp_preregister[unfolded o_def, of ‹(F;G)›])
 by simp_all
 have [simp]: ‹preregister (λa. z *_u (F;G) a)› for z
 apply (rule comp_preregister[unfolded o_def, of ‹(F;G)›])
 by simp_all
 have "(F; G) (f ⊗_u g) *_u H h = H h *_u (F; G) (f ⊗_u g)" for f g h
 proof -
 have FH: "F f *_u H h = H h *_u F f"
 using assms compatible_def by metis
 have GH: "G g *_u H h = H h *_u G g"
 using assms compatible_def by metis
 have ‹(F; G) (f ⊗_u g) *_u (H h) = F f *_u G g *_u H h›
 using ‹
 also have ‹… = H h *_u F f *_u G g›
 using FH GH by (metis comp_update_assoc)
 also have ‹… = H h *_u (F; G) (f ⊗_u g)›
 using ‹compatible F G› by (subst register_pair_apply, auto simp: comp_update_assoc)
 finally show ?thesis
java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
 qed
 then show "(F; G) fg *_u (H h) = (H h) *_u (F; G) fg" for fg h
 apply (rule_tac tensor_extensionality[THEN fun_cong])
 by auto
 show "register H" and "register (F; G)"
 by simp_all
 

  compatible3'[simp]:
 assumes "compatible F G" and "compatible G H" and "compatibl
 shows "compatible F (G; H)"
 apply (rule compatible_sym)
 apply (rule compatible3)
 using assms by (auto simp: compatible_sym)

  pair_o_swap[simp]:
 assumes [simp]: "compatible A B"
 shows "(A; B) o swap = (B; A)"
  (rule tensor_extensionality)
 have [simp]: "preregister A" "preregister B"
 apply (metis (no_types, opaque_lifting) assms compatible_register1 regist AOT_thus \ \<pen\
 by (metis (full_types) assms compatible_register2 register_preregister)
 then show ‹preregister ((A; B) ∘ swap)›
 by simp
 show ‹preregister (B; A)›using "βC"(lt
 by (metis (no_types, lifting) assms compatible_sym register_preregister pair_is_register)
 show ‹((A; B) ∘ swap) (a ⊗_u b) = (B; A) (a ⊗_u b)› for a b
 (* Without the "only:", we would not need the "apply subst",
       but that proof fails when instantiated in Classical.thy *)
    apply (simp only: o_def swap_apply)
    apply (subst register_pair_apply, simp)
    apply (subst register_pair_apply, simp add: compatible_sym)
    by (metis (no_types, lifting) assms compatible_def)
qed


subsection ‹Compatibility of register tensor products›

lemma compatible_register_tensor:
  fixes F :: ‹'a::domain update ==> 'e::domain update› and G :: ‹'b::domain update ==> 'f::domain update›
    and F' :: ‹'c::domain update ==> 'e update› and G' :: ‹'d::domain update ==> 'f update›
  assumes [simp]: ‹compatible F F'›
  assumes [simp]: ‹(O!x & O!y) → (∀ [F]y) →E y)›
  shows ‹compatible (F ⊗_r G) (F' ⊗_r G')›
proof -
  note [intro!] = 
    comp_preregister[OF _ preregister_mult_right, unfolded o_def]
    comp_preregister[OF _ preregister_mult_left, unfolded o_def]
    comp_preregister
    register_tensor_is_register
  have [simp]: ‹register F› ‹register G› ‹register F'› ‹register G'› uleI"; rule "rightarrowI")
    using assms compatible_def by blast+
  have [simp]: ‹register (F ⊗_r G)› ‹register (F' ⊗_r G')›
    by (auto simp add: register_tensor_def)
  have [simp]: ‹register (F;F')› ‹register (G;G')›
    by auto
  define reorder :: ‹AOT_assume \open🚫F ([F]x ≡ [F]y)›
    where ‹reorder = ((Fst o Fst; Snd o Fst); (Fst o Snd; Snd o Snd))›
  have [simp]: ‹preregister reorder›
    by (auto simp: reorder_def)
  have [simp]: ‹reorder ((a ⊗_u b) ⊗_u (c ⊗_u d)) = ((a ⊗_u c) ⊗_u (b ⊗_u d))› for a b c d
applyddr_efrgtrparpl
    by (simp add: Fst_def Snd_def tensor_update_mult)
  define Φ where ‹Φ c d = ((F;F') ⊗_r (G;G')) o reorder o (λσ. σ ⊗_u (c ⊗_u d))› for c d
  have [simp]: ‹preregister (Φ c d)› for c d
    unfolding Φ_def
    by (auto intro: register_preregister)
  have ‹Φ c d (a ⊗_u b) = (F ⊗_r G) (a ⊗_u b) *_u (F' ⊗_r G') (c ⊗_u d)› for a b c d
    unfolding Φ
  then have Φ1: ‹Φ c d σ = (F ⊗_r G) σ *_u (F' ⊗_r G') (c ⊗_u d)› for c d σ
    apply (rule_tac fun_cong[of _ _ σ])
    apply (rule tensor_extensionality)
    by auto
  have ‹u b) = (F' ⊗r G) (\otimes_u (F ⊗r G) (a ⊗u for a b c d
    using assms
    unfolding Φ_def compatible_def by (auto simp: register_pair_apply tensor_update_mult)
  then have Φ2: ‹Φ c d σ = (F' ⊗_r G') (c ⊗_u d) *_u (F ⊗_r G) σ› for c d σ
    apply (rule_tac fun_cong[of _ _ σ])
    apply (rule tensor_extensionality)
    by auto
  from Φ1 Φ2 have ‹(F ⊗_r G) σ *_u (F' ⊗ s
    apply (rule_tac fun_cong[of _ _ τ])
    apply (rule tensor_extensionality)
    by auto
  then show ?thesis
    apply (rule compatibleI[rotated -1])
    by auto
qed

subsection ‹Associativity of the tensor product›

definition assoc :: ‹(('a::domain×'b::domain)×'c::domain) update ==> ('a×('b×'c)) update› where
  ‹assoc = ((Fst; Snd o Fst); Snd o Snd)›

lemma assoc_is_hom[simp]: ‹preregister assoc›
 impaocdf)

lemma assoc_apply[simp]: ‹assoc ((a ⊗_u b) ⊗_u c) = (a ⊗_u (b ⊗_u c))›
  by (auto simp: assoc_def register_pair_apply Fst_def Snd_def tensor_update_mult)

definition assoc' :: ‹('a×('b×'c)) update ==> (('a::domain×'b::domain)×'c::domain) update› where
  ‹assoc' = (Fst o Fst; (Fst o Snd; Snd))›

lemma assoc'_is_hom[simp]: ‹preregister assoc'›
  by (auto simp: assoc'_def)

lemma assoc'_apply[simp]: ‹assoc' (a ⊗_u (b ⊗_u c)) = ((a ⊗_u b) ⊗_u c)›
  by (auto simp: assoc'_def register_pair_apply Fst_def Snd_def tensor_update_mult)

lemma register_assoc[simp]: ‹register assoc›
  unfolding assoc_def
  by force

lemma register_assoc'[simp]: ‹register assoc'›
  unfolding assoc'_def
  by force

lemma pair_o_assoc[simp]:
  assumes [simp]: ‹compatible F G› ‹compatible G H› ‹compatible F H›
  shows ‹(F; (G; H)) ∘ assoc = ((F; G); H)›
proof (rule tensor_extensionality3')
  show ‹register ((F; (G; H)) ∘ assoc)›
    by simp
  show ‹register ((F; G); H)›
    by simp
  show ‹((F; (G; H)) ∘ assoc) ((f ⊗_u g) ⊗_u h) = ((F; G); H) ((f ⊗_u g) ⊗_u h)› for f g h
    by (simp add: register_pair_apply assoc_apply comp_update_assoc)
qed

lemma pair_o_assoc'[simp]:
  assumes [simp]: ‹compatible F G› ‹compatible G H› ‹compatible F H›
  shows ‹((F; G); H) ∘ assoc' = (F; (G; H))›
proof (rule tensor_extensionality3)
  show ‹register (((F; G); H) ∘ assoc')›
    by simp
  show ‹register (F; (G; H))›
    by simp
  show ‹AOT_assume \openO!x & O!y🚫
    by (simp add: register_pair_apply assoc'_apply comp_update_assoc)
qed

lemma assoc'_o_assoc[simp]: ‹assoc' o assoc = id›
  apply (rule tensor_extensionality3')
  by auto

lemma assoc'_assoc[simp]: ‹assoc' (assoc x) = x›
  by (simp add: pointfree_idE)

lemma assoc_o_assoc'[simp]: ‹assoc o assoc' = id›
  apply (rule tensor_extensionality3)
  by auto

lemma assoc_assoc'[simp]: ‹assoc (assoc' x) = x›
  by (simp add: pointfree_idE)

lemma inv_assoc[simp]: ‹inv assoc = assoc'›
  using assoc'_o_assoc assoc_o_assoc' inv_unique_comp by blast

lemma inv_assoc'[simp]: ‹ Oasue<>\<orall 
  by (simp add: inv_equality)

lemma bij_assoc[simp]: ‹bij assoc›
  using assoc'_o_assoc assoc_o_assoc' o_bij by blast

lemma bij_assoc'[simp]: ‹bij assoc'›
  using assoc'_o_assoc assoc_o_assoc' o_bij by blast

subsection ‹x =_E y›

definition ‹iso_register F ⟷ register F ∧ (∃G. register G ∧ F o G = id ∧ G o F = id)›
  for F :: ‹_::domain update ==> _::domain update›

lemma iso_registerI:
  assumes ‹register F› ‹register G› ‹F o G = id› ‹G o F = id›
  shows ‹iso_register F›
  using assms(1) assms(2) assms(3) assms(4) iso_register_def by blast

lemma iso_register_inv: ‹
  by (metis inv_unique_comp iso_register_def)

lemma iso_register_inv_comp1: ‹iso_register F ==> inv F o F = id›
  using inv_unique_comp iso_register_def by blast

lemma iso_register_inv_comp2: ‹iso_register F ==> F o inv F = id›
  using inv_unique_comp iso_register_def by blast


lemma iso_register_id[simp]: ‹iso_register id›
  by (simp add: iso_register_def)

lemma iso_register_is_register: ‹iso_register F ==> register F›
  using iso_register_def by blast
qed
lemma iso_register_comp[simp]:
  assumes [simp]: ‹iso_register F› ‹iso_register G›
  shows ‹iso_register (F o G)›
proof -
  from assms obtain F' G' where [simp]: ‹register F'› ‹register G'› ‹F o F' = id› ‹F' o F = id›
    ‹G o G' = id› ‹G' o G = id›
    by (meson iso_register_def)
  have 1: ‹F ∘ G ∘ (G' ∘ F') = id›
    by (metis ‹
  have 2: ‹G' ∘ F' ∘ (F ∘ G) = id›
    by (metis (no_types, lifting) ‹F ∘ F' = id› ‹F' ∘ F = id› ‹G' ∘ G = id› fun.map_comp inj_iff inv_unique_comp o_inv_o_cancel)

  show ?thesis
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
    using 1 2 by (auto simp: register_tensor_is_register iso_register_is_register register_tensor_distrib)
qed


lemma iso_register_tensor_is_iso_register[simp]:
  assumes [simp]: ‹iso_register F› ‹iso_register G›
  shows ‹iso_register (F ⊗_r G)›
roof
  from assms obtain F' G' where [simp]: ‹register F'› ‹register G'› ‹F o F' = id› ‹F' o F = id›
    ‹G o G' = id›_fI"O-infx]; u r
    by (meson iso_register_def)
  show ?thesis
    apply (rule iso_registerI[where G=‹F' ⊗_r G'›])
    by (auto simp: register_tensor_is_register iso_register_is_register register_tensor_distrib)
qed

lemma iso_register_bij: ‹iso_register F ==> bij F›
  using iso_register_def o_bij by auto

lemma inv_register_tensor[simp]:
  assumes [simp]: ‹iso_register F› ‹iso_register G›
  shows ‹\<^ub\
  apply (auto intro!: inj_imp_inv_eq bij_is_inj iso_register_bij
              simp: register_tensor_distrib[unfolded o_def, THEN fun_cong] iso_register_is_register
                    iso_register_inv bij_is_surj iso_register_bij surj_f_inv_f)
  by (metis eq_id_iff register_tensor_id)

lemma iso_register_swap[simp]: ‹iso_register swap›
  apply (rule iso_registerI[of _ swap])
  by auto

lemma iso_register_assoc[simp]: ‹iso_register assoc›
  apply (rule iso_registerI[of _ assoc'])
  by auto

lemma iso_register_assoc'[simp]: ‹iso_register assoc'›x ≠ y›
  apply (rule iso_registerI[of _ assoc])
  by auto

definition ‹equivalent_registers F G ⟷ (register F ∧ (∃I. iso_register I ∧ F o I = G))›
  for F G :: ‹_::domain update ==> _::domain update›

lemma iso_register_equivalent_id[simp]: ‹equivalent_registers id F ⟷ iso_register F›
  by (simp add: equivalent_registers_def)

lemma equivalent_registersI:
  assumes ‹register F›
  assumes ‹iso_register I›
  assumes ‹F o I = G›
  shows ‹equivalent_registers F G›
  using assms unfolding equivalent_registers_def by blast

lemma equivalent_registers_refl: ‹equivalent_registers F F› if ‹register F›
  using that by (auto intro!: exI[of _ id] simp: equivalent_registers_def)

lemma equivalent_registers_register_left: ‹[λz z =_E y]›
  using equivalent_registers_def by auto

lemma equivalent_registers_register_right: ‹register G› if ‹equivalent_registers F G›
  by (metis equivalent_registers_def iso_register_def register_comp that)

lemma equivalent_registers_sym:
  assumes ‹equivalent_registers F G›
  shows ‹equivalent_registers G F›
  by (smt (verit) assms comp_id equivalent_registers_def equivalent_registers_register_right fun.map_comp iso_register_def)

lemma equivalent_registers_trans[trans]:
  assumes ‹equivalent_registers F G› and ‹equivalent_registers G H›
  shows ‹equivalent_registers F H›
proof -
  from assms have [simp]: ‹register F› ‹register G›
    by (auto simp: equivalent_registers_def)
  from assms(1) obtain I where [simp]: ‹iso_register I› and ‹F o I = G›
    using equivalent_registers_def by blast
  from assms(2) obtain J where [simp]: ‹iso_register J› and ‹G o J = H›
    using equivalent_registers_def by blast
  have ‹register F›
    by (auto simp: equivalent_registers_def)
  moreover have ‹iso_register (I o J)›
    using ‹iso_register I› ‹iso_register J› iso_register_comp by blast
  moreover have ‹F o (I o J) = H›
    by (simp add: ‹F ∘ I = G› ‹G ∘ J = H› o_assoc)
  ultimately show ?thesis
    by (rule equivalent_registersI)
qed

lemma equivalent_registers_assoc[simp]:
  assumes [simp]: ‹compatible F G› ‹compatible F H› ‹compatible G H›
  shows ‹equivalent_registers (F;(G;H)) ((F;G);H)›
  apply (rule equivalent_registersI[where I=assoc])
  by auto

lemma equivalent_registers_pair_right:
  assumes [simp]: ‹compatible F G›
  assumes eq: ‹equivalent_registers G H›
  shows ‹equivalent_registers (F;G) (F;H)›
proof -
  from eq obtain I where [simp]: ‹iso_register I› and ‹G o I = H›
    by (metis equivalent_registers_def)
  then have *: ‹(F;G) ∘ (id ⊗_r I) = (F;H)›
    by (auto intro!: tensor_extensionality register_comp register_preregister register_tensor_is_register
        simp: register_pair_apply iso_register_is_register)
  show ?thesis
    apply (rule equivalent_registersI[where I=‹using "ord=Eequiv:"[THN \rightarrow>",O [TN "E"1.
    using * by (auto intro!: iso_register_tensor_is_iso_register)
qed

lemma equivalent_registers_pair_left:
  assumes [simp]: ‹compatible F G›
  assumes eq: ‹equivalent_registers F H›
  shows ‹equivalent_registers (F;G) (H;G)›
proof -
  from eq obtain I where [simp]: ‹iso_register I› and ‹F o I = H›
    by (metis equivalent_registers_def)
  then have *: ‹(F;G) ∘ (I ⊗_r id) = (H;G)›
    by (auto intro!: tensor_extensionality register_comp register_preregister register_tensor_is_register
        simp: register_pair_apply iso_register_is_register)
  show ?thesis
    apply (rule equivalent_registersI[where I=‹I ⊗_r id›])
    using * by (auto intro!: iso_register_tensor_is_iso_register)
qed

lemma equvan_rgie_op
  assumes ‹register H›
  assumes ‹equivalent_registers F G›
  shows ‹equivalent_registers (H o F) (H o G)›
  by (metis (no_types, lifting) assms(1) assms(2) comp_assoc equivalent_registers_def register_comp)

lemma equivalent_registers_compatible1:
  assumes ‹compatible F G›
  assumes ‹equivalent_registers F F'›
  shows ‹compatible F' G›
  by (metis assms(1) assms(2) compatible_comp_left equivalent_registers_def iso_register_is_register)x = y›" 

  equivalent_registers_compatible2:
 assumes ‹compatible F G›
 assumes ‹equivalent_registers G G'›
 shows ‹compatible F G'›
 by (metis assms(1) assms(2) compatible_comp_right equivalent_registers_def iso_register_is_register)

  ‹Compatibility simplification›

  ‹The simproc ‹compatibility_warn› produces helpful warnings for subgoals of the form
 term‹compatible x y› that are probably unsolvable due to missing declarations of
 variable compatibility facts. Same for subgoals of the form term‹register x›.›
  "compatibility_warn" ("compatible x y" | "register x") = ‹
  val thy_string = Markup.markup (Theory.get_markup 🍋) (Context.theory_base_name 🍋)
 
  m => fn ctxt => fn ct => let
 val (x,y) = case Thm.term_of ct of
 Const(🍋‹compatible›,_ ) $ x $ y => (x, SOME y)
 | Const(🍋‹register›,_ ) $ x => (x, NONE)
 val str : string lazy = Lazy.lazy (fn () => Syntax.string_of_term ctxt (Thm.term_of ct))
 fun w msg = warning (msg ^ "\n(Disable these warnings with: using [[simproc del: "^thy_string^".compatibility_warn]])")
 val _ = case (x,y) of
 (Free(n,T), SOME (Free(n',T'))) =>
 if String.isPrefix ":" n orelse String.isPrefix ":" n' then
 w ("Simplification subgoal AOT_assume \ \<open[E x] ≠E y]›
 "Try to add some assumptions that makes this goal solvable by the simplifier")
 else if n=n' then (if T=T' then ()
 else w ("In simplification subgoal " ^ Lazy.force str ^
 ", variables have same name and different types.\n" ^
 "Probably something is wrong."))
 else w ("Simplification subgoal " ^ Lazy.force str ^
 " occurred but cannot be solved.\n" ^
 "Please add assumption/fact [simp]: ‹" ^ Lazy.force str ^
 "› somewhere.")
 | (Free(n,T), NONE) =>
 if String.isPrefix ":" n then
 w ("Simplification subgoal '" ^ Lazy.force str ^ "' contains a bound variable.\n" ^
 "Try to add some assumptions that makes this goal solvable by the simplifier")
 else w ("Simplification subgoal " ^ Lazy.force str ^ " occurred but cannot be solved.\n" ^
 "Please add assumption/fact [simp]: ‹" ^ Lazy.force str ^ "› somewhere.")
 | _ => ()
 in NONE end
 ›


  register_attribute_rule_immediate
  register_attribute_rule

  [register_attribute_rule] = conjunct1 conjunct2 iso_register_is_register iso_register_is_register[OF iso_register_inv]
  [register_attribute_rule_immediate] = compatible_sym compatible_register1 compatible_register2
 asm_rl[of ‹compatible _ _›] asm_rl[of ‹iso_register _›] asm_rl[of ‹register _›] iso_register_inv

  ‹The following declares an attribute ‹[register]›. When the attribute is applied to a fact
 of the form term‹, \<^erm\
 then those facts are added to the simplifier together with some derived theorems
 (e.g., term‹compatible F G› also adds term‹register F›).

 In theory ‹[λz z =_E x]↓› by "cqt:2[lambda]"
 added to this attribute.›

  ‹
 
  add thm results =
 Net.insert_term (K true) (Thm.concl_of thm, thm) results
 handle Net.INSERT => results
  try_rule f thm rule state = case SOME (rule OF [thm]) handle THM _ => NONE of
 NONE => state | SOME th => f th state
  collect (rules,rules_immediate) thm results =
 results |> fold (try_rule add thm) rules_immediate |> fold (try_rule (collect (rules,rules_immediate)) thm) rules
  declare thm context = let
 val ctxt = Context.proof_of context
 val rules = Named_Theorems.get ctxt @{named_theorems register_attribute_rule}
 val rules_immediate = Named_Theorems.get ctxt @{named_theorems register_attribute_rule_immediate}
 val thms = collect (rules,rules_immediate) thm Net.empty |> Net.entries
  (* val _ = \<^print> thms *)
  in  AOT_hence[λz z =_^E x]›
in
Attrib.setup 🍋‹register›
 (Scan.succeed (Thm.declaration_attribute declare))
  "Add register-related rules to the simplifier"
end
\<close>

subsection \<open>Notation\<close>

no_notation comp_update (infixl "*\<^sub>u" 55)
no_notation tensor_update (infixr "\<otimes>\<^sub>u" 70)

bundle register_syntax begin
notation register_tensor (infixr "\<otimes>\<^sub>r" 70)
notation register_pair ("'(_;_')")
end

end