Theory Laws

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 ((λa1::('a×'b) update. (id_update ⊗⇩u b) *⇩u a1) o (λa. tensor_update a id_update))›
    by (rule comp_preregister, simp_all)
  then show ?thesis
    by (simp add: o_def tensor_update_mult)
qed

subsection ‹Registers›

lemma 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

lemma 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

subsection ‹Tensor product of registers›

definition register_tensor  (infixr "⊗⇩r" 70) where
  "register_tensor F G = register_pair (λa. tensor_update (F a) id_update) (λb. tensor_update id_update (G b))"

lemma register_tensor_is_register: 
  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)

lemma 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)

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

lemma separating_UNIV[simp]: ‹separating TYPE(_) UNIV›
  unfolding separating_def by auto

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

lemma 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

lemma 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}›
proof (unfold separating_def, intro allI impI)
  fix F G :: ‹('a×'b) update ⇒ '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]: ‹preregister (λx. G (x ⊗⇩u b))› for b
    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 ∈ B› for a 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_eqI[where A=A, THEN fun_cong, where x=a, rotated -1])
    by auto
  then show "F = G"
    apply (rule tensor_extensionality[rotated -1])
    by auto
qed

lemma 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)›
  apply (rule tensor_extensionality)
  by (auto intro!: register_comp register_preregister register_tensor_is_register)

text ‹The following is easier to apply using the @{method rule}-method than @{thm [source] separating_tensor}›
lemma 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)

lemma 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"
proof (rule register_eqI[where A=‹{a⊗⇩ub⊗⇩uc| a b c. True}›])
  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 ‹x ∈ {a ⊗⇩u b ⊗⇩u c |a b c. True} ⟹ F x = G x› for x
    using assms(3) by auto
qed

lemma 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"
proof (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 |a b c. True}›
    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 ∈ {(a ⊗⇩u b) ⊗⇩u c |a b c. True} ⟹ F x = G x› for x
    using assms(3) by auto
qed

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

subsection ‹Pairs and compatibility›

definition compatible :: ‹('a::domain update ⇒ 'c::domain update)
                       ⇒ ('b::domain update ⇒ 'c update) ⇒ bool› where
  ‹compatible F G ⟷ register F ∧ register G ∧ (∀a b. F a *⇩u G b = G b *⇩u F a)›

lemma compatibleI:
  assumes "register F" and "register G"
  assumes ‹⋀a b. (F a) *⇩u (G b) = (G b) *⇩u (F a)›
  shows "compatible F G"
  using assms unfolding compatible_def by simp

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

lemma compatible_sym: "compatible x y ⟹ compatible y x"
  by (simp add: compatible_def)

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

lemma 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+

lemma register_pair_apply':
  assumes ‹compatible F G›
  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)



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

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

lemma 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)

lemma compatible_register1: ‹compatible F G ⟹ register F›
  by (simp add: compatible_def)
lemma compatible_register2: ‹compatible F G ⟹ register G›
  by (simp add: compatible_def)

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

lemma 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

lemma 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

lemma 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)

lemma 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)

lemma register_comp_pair:
  assumes [simp]: ‹register F› and [simp]: ‹compatible G H›
  shows "(F o G; F o H) = F o (G; H)"
proof (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)
qed

lemma 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

lemma swap_registers_right:
  assumes "compatible R S"
  shows "c *⇩u R a *⇩u S b = c *⇩u S b *⇩u R a"
  by (metis assms comp_update_assoc compatible_def)

lemmas compatible_ac_rules = swap_registers comp_update_assoc[symmetric] swap_registers_right

subsection ‹Fst and Snd›

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]: ‹register swap›
  by (simp add: swap_def)

lemma pair_Fst_Snd: ‹(Fst; Snd) = id›
  apply (rule tensor_extensionality)
  by (simp_all add: register_pair_apply Fst_def Snd_def tensor_update_mult)

lemma 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)

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

lemma inv_swap[simp]: ‹inv swap = swap›
  by (meson inv_unique_comp swap_o_swap)

lemma register_pair_Fst:
  assumes ‹compatible F G›
  shows ‹(F;G) o Fst = F›
  using assms by (auto intro!: ext simp: Fst_def register_pair_apply compatible_register2)

lemma 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)

lemma 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)

lemma register_Snd_register_Fst[simp]: 
  assumes ‹register F›
  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)


lemma compatible3[simp]:
  assumes [simp]: "compatible F G" and "compatible G H" and "compatible F H"
  shows "compatible (F; G) H"
proof (rule compatibleI)
  have [simp]: ‹register F› ‹register G› ‹register H›
    using assms compatible_def by auto
  then have [simp]: ‹preregister F› ‹preregister G› ‹preregister H›
    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 ‹compatible F G› by (subst register_pair_apply, auto)
    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
      by -
  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
qed

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

lemma pair_o_swap[simp]:
  assumes [simp]: "compatible A B"
  shows "(A; B) o swap = (B; A)"
proof (rule tensor_extensionality)
  have [simp]: "preregister A" "preregister B"
     apply (metis (no_types, opaque_lifting) assms compatible_register1 register_preregister)
    by (metis (full_types) assms compatible_register2 register_preregister)
  then show ‹preregister ((A; B) ∘ swap)›
    by simp
  show ‹preregister (B; A)›
    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]: ‹compatible G G'›
  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'›
    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 :: ‹(('a×'b) × ('c×'d)) update ⇒ (('a×'c) × ('b×'d)) update›
    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
    apply (simp add: reorder_def register_pair_apply)
    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_def by (auto simp: register_pair_apply tensor_update_mult)
  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 ‹Φ c d (a ⊗⇩u b) = (F' ⊗⇩r G') (c ⊗⇩u d) *⇩u (F ⊗⇩r G) (a ⊗⇩u b)› for a b c d
    unfolding Φ_def apply (auto simp: register_pair_apply)
    by (metis assms(1) assms(2) compatible_def 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' ⊗⇩r G') τ = (F' ⊗⇩r G') τ *⇩u (F ⊗⇩r G) σ› for τ σ
    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›
  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)

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 ‹(((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 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]: ‹inv assoc' = assoc›
  by (simp add: inv_equality)

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

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

subsection ‹Iso-registers›

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: ‹iso_register F ⟹ iso_register (inv F)›
  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

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)
  show ?thesis
    apply (rule iso_registerI[where G=‹G' o F'›])
       apply (auto simp: register_tensor_is_register iso_register_is_register register_tensor_distrib)
     apply (metis ‹F ∘ F' = id› ‹G ∘ G' = id› fcomp_assoc fcomp_comp id_fcomp)
    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)
qed


lemma iso_register_tensor_is_iso_register[simp]:
  assumes [simp]: ‹iso_register F› ‹iso_register G›
  shows ‹iso_register (F ⊗⇩r 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)
  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 ‹inv (F ⊗⇩r G) = inv F ⊗⇩r inv G›
  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'›
  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_register_left: ‹equivalent_registers F G ⟹ register F›
  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=‹id ⊗⇩r I›])
    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 equivalent_registers_comp:
  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)

subsection ‹Compatibility simplification›

text ‹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›.›
simproc_setup "compatibility_warn" ("compatible x y" | "register x") = ‹
let val thy_string = Markup.markup (Theory.get_markup \<^theory>) (Context.theory_base_name \<^theory>)
in
fn m => fn ctxt => fn ct => let
  val (x,y) = case Thm.term_of ct of
                 Const(\<^const_name>‹compatible›,_ ) $ x $ y => (x, SOME y)
               | Const(\<^const_name>‹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 " ^ Lazy.force str ^ " contains a bound variable.\n" ^
                      "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
end›


named_theorems register_attribute_rule_immediate
named_theorems register_attribute_rule

lemmas [register_attribute_rule] = conjunct1 conjunct2 iso_register_is_register iso_register_is_register[OF iso_register_inv]
lemmas [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

text ‹The following declares an attribute ‹[register]›. When the attribute is applied to a fact
  of the form \<^term>‹register F›, \<^term>‹iso_register F›, \<^term>‹compatible F G› or a conjunction of these,
  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 ‹Laws_Complement›, support for \<^term>‹is_unit_register F› and \<^term>‹complements F G› is
  added to this attribute.›

setup ‹
let
fun add thm results = 
  Net.insert_term (K true) (Thm.concl_of thm, thm) results
  handle Net.INSERT => results
fun try_rule f thm rule state = case SOME (rule OF [thm]) handle THM _ => NONE  of
  NONE => state | SOME th => f th state
fun collect (rules,rules_immediate) thm results =
  results |> fold (try_rule add thm) rules_immediate |> fold (try_rule (collect (rules,rules_immediate)) thm) rules
fun 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 Simplifier.map_ss (fn ctxt => ctxt addsimps thms) context end
in
Attrib.setup \<^binding>‹register›
 (Scan.succeed (Thm.declaration_attribute declare))
  "Add register-related rules to the simplifier"
end
›

subsection ‹Notation›

no_notation comp_update (infixl "*⇩u" 55)
no_notation tensor_update (infixr "⊗⇩u" 70)

bundle register_notation begin
notation register_tensor (infixr "⊗⇩r" 70)
notation register_pair ("'(_;_')")
end

bundle no_register_notation begin
no_notation register_tensor (infixr "⊗⇩r" 70)
no_notation register_pair ("'(_;_')")
end

end