Quellcode-Bibliothek Laws_Quantum.thy Sprache: Isabelle

(*
* This is an autogenerated file. Do not edit.
* The original is Laws.thy. It was converted using instantiate_laws.py.
*)

section ‹

Laws_Quantum
imports Axioms_Quantum

‹This notation is only used inside this file›
cblinfun_compose (infixl "*_u" 55)
tensor_op (infixr "⊗
register_pair ("'(_;_')")

‹ no_writes_ncagd:aues\open∀ l<k. writes(path σ shows ‹k

id_preregister[simp]
cblinfun_compose_id_left[simp]
cblinfun_compose_id_right[simp]
register_preregister[simp]
register_comp[simp]
register_of_id[simp]
register_tensor_left[simp]
register_tensor_right[simp]
preregister_mult_right[simp]
preregister_mult_left[simp]
register_id[simp]

‹Preregisters›

preregister_tensor_left[simp]: ‹preregister (λb::'b::type update. tensor_op a b)›
for a :: ‹'a::type update›
-
have ‹preregister ((λb1::('a×'b) update. (a ⊗rt dests. kD (invalidate rt dests) = kD rt"
by (rule comp_preregister; simp)
then show ?thesis
by (simp add: o_def comp_tensor_op)

preregister_tensor_right[simp]: ‹preregister (λa::'a::type update. tensor_op a b)›
for b :: ‹'b::type update›
-
have ‹preregister ((λa1::('a×open>k›)
by (rule comp_preregister, simp_all)
then show ?thesiunfold invalidate_defkD_def
by (simp add: o_def comp_tensor_op)

‹Registers›

id_update_tensor_register[simp]:
assumes ‹register F›
shows ‹register (λa::'a::type update. id_cblinfun ⊗_simp split: optio.spli)
using assms apply (rule register_comp[unfolded o_def])
by simp

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

‹

register_tensor (infixr "⊗_r" 70) where
"register_tensor F G = register_pair (λa. tensor_op (F a) id_cblinfun) (λb. tensor_op id_cblinfun (G b))"

register_tensor_is_register:
fixes F :: "'a::type update → 'b::type update" and G :: "'c::type update → 'd::type update"
shows "register F ==> register G ==> register (F ⊗
unfolding register_tensor_def
apply (rule register_pair_is_register)
by (simp_all add: comp_tensor_op)

register_tensor_apply[simp]:
fixes F :: "'a::type update → 'b::type update" and G :: "'c::type update → 'd::type update"
assumes ‹register F› and ‹register G›
java.lang.NullPointerException
unfolding register_tensor_def
apply (subst register_pair_apply)
unfolding register_tensor_def
by (simp_all add: assms comp_tensor_op)

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

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

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

register_eqI: ‹separating TYPE('b::type) A ==> preregister F \<  shows
unfolding separating_def by auto

separating_tensor:
fixes A :: ‹'a::type update set› and B :: ‹'b::type update set›
assumes [simp]: ‹separating TYPE('c::type) A›
assumes [simp]: ‹∧ the (dhops (inval rt show<open? using writes by metis
shows ‹separating TYPE('c) {a ⊗_u b | a b. a∈A ∧ b∈B}›
(unfold separating_def, intro allI impI)
fix F G :: ‹ invalidate_d b (clarsi split: : opt.spl)
assume [simp]: ‹preregister F› ‹preregister G›
have [simp]: ‹preregister (λx. F (a ⊗_u x))› for a
using _ ‹preregister F›qed
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]: ‹
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 \<  "
then have EQ: ‹F (a ⊗cla split: option..split)
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 ⊗_have ‹v ∈ writes (π ?l)› by (metis (no_types, lifting) GreatestI_nat assms(3) assms(4) less_or_eq_imp_le)
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

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)

‹The following is easier to apply using the @{method rule}-method than @{thm [source] separating_tensor}›
separating_tensor':
fixes A :: ‹'a::type update set\<close v ∈ reads (π k) ∩ ?l)›
assumes ‹separating TYPE('c::type) 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_extensionality3:
fixes F G :: ‹('a::type×'b::type×'c::type) update → 'd::type update›
assumes [simp]: ‹register F›
assumes "∧
shows "F = G"
(rule register_eqI[where A=‹{a⊗_ub⊗_uc| a b c. True}›])
have ‹separating TYPE('d) {b ⊗:
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⊗ des. di\in>>v(inva r de) ==>and> dests dip = None no is_d
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

tensor_extensionality3':
fixes F G :: ‹
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.StringIndexOutOfBoundsException: Range [25, 22) out of bounds for length 154
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› ‹ultimately
show ‹x ∈ {(a ⊗_u b) ⊗_u c |a b c. True} ==> F x = G x› for 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::type update → 'c::type update)
→ ('b::type update → 'c update) → bool› where
‹compatible F G ⟷ register F ∧

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

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› "dip∉
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 ‹compatible F G›
shows ‹dests)dip sqnr d"
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: ‹compatible F G ==> register F›
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›
java.lang.NullPointerException
apply (rule tensor_extensional
using assms by (simp_all add: register_tensor_is_register register_pair_apply comp_preregister)

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

compatible_tensor_id_update_right[simp]:
fixes F :: "'a::type update → 'c::type update" and G :: "'b::type update → invalidate_changes:
assumes "compatible F G"
shows "compatible (λa. F a ⊗_u id_cblinfun) (λa. G a ⊗_u id_cblinfun)"
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_cblinfun) (λa. id_cblinfun ⊗_u G a)"
apply (rule compatibleI)
using assms by (auto simp: comp_tensor_op)

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

register_comp_pair:
assumes [simp]: ‹register F› and [simp]: ‹compatible G H›
shows "(F o G; F o H) = F o (G; H)"
(rule tensor_extensionality)
java.lang.StringIndexOutOfBoundsException: Range [8, 6) out of bounds for length 108
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 ∘ (induction ‹ arbitrary: ‹)
by (auto simp: register_pair_apply register_mult comp_tensor_op)

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 *_u S b = c *_u S b *_u R a"
by (metis assms cblinfun_compose_assoc compatible_def)

compatible_ac_rules = swap_registers cblinfun_compose_assoc[symmetric] swap_registers_right

‹Fst and Snd›

(* TODO: specify types *)
definition Fst where ‹case 0 thus ‹
Snd where ‹Snd a = id_cblinfun ⊗_u a›

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

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

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

compatible_Snd_Fst[simp] = compatible_Fst_Snd[THEN compatible_sym]

‹swap = (Snd; Fst)›

swap_apply[simp]: "swap (a ⊗_u b) = (b ⊗_next
unfolding swap_def
by (simp add: Axioms_Quantum.register_pair_apply Fst_def Snd_def comp_tensor_op)

swap_o_Fst: "swap o Fst = Snd"
by ( shows " = (ase dests dip of None → π₂(the (rt dip)) | Some rsn → rsn)
swap_o_Snd: "swap o Snd = Fst"
by (auto simp add: Fst_def Snd_def)

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

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

swap_o_swap[simp]: ‹swap o swap = id›
by (metis swap_def compatible_Snd_Fst pair_Fst_Snd register_comp_pair registe\<and cas (uc k )

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

inv_swap[simp]: ‹inv swap = swap›
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 simp: Fst_def register_pair_apply compatible_register2)

register_pair_Snd:
assumes 🚫
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)
using assms by (auto simp: register_pair_apply Fst_def Snd_def register_mult comp_tensor_op)

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

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]: ‹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
java.lang.NullPointerException
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
java.lang.NullPointerException
using assms compatible_def by metis
have ‹(F; G) (f ⊗_?case› using writes by metis
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 cblinfun_compose_assoc)
also have ‹… = H h *_u (F; G) (f ⊗_u g)›
using ‹compatible F G› by (subst register_pair_apply, auto simp: cblinfun_compose_assoc)
finally show ?thesis
by -
qed
java.lang.NullPointerException
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 "compatible F H"
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 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) ∘🚫
(* 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::type update → 'e::type update› and G :: ‹'b::type update → 'f::type update›
    and F' :: ‹ 'e update›'d::type updatejava.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 133
  assumes [simp]: ‹compatible F F'›
  assumes [simp]: ‹compatible G G'›
  shows ‹compatible (F ⊗(σ (σ using contradicts.intros(2)[OF csk[symmetrc]b u
-
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› ‹
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_
have [simp]: ‹reorder ((a ⊗_u b) ⊗_u (c ⊗_u d)) = ((a ⊗ \ \right> p" (<\ σb>pat \si>' ')< conv: \openk<\
apply (simp add: reorder_def register_pair_apply)
by (simp add: Fst_def Snd_def comp_tensor_op)
define Φ where ‹Φ c d = ((F;F') ⊗lambda>ddip. if dests dip = None then store dip
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' ⊗have cs: ‹?π 🚫 \<>( (\< l. is_kth_obs π k1 l)› using none1(1) π obs_none_no_kth_obs by auto
unfolding Φ_def by (auto simp: register_pair_apply comp_tensor_op)
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
using assms
unfolding Φ_def compatible_def by (auto simp: register_pair_apply comp_tensor_op)
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

‹Associativity of the tensor product›

assoc :: ‹(('a::type×'b::type)×'c::type) update → ('a×('b×'c)) update› where
‹assoc = ((Fst; Snd o Fst); Snd o Snd)›

assoc_is_hom[simp]: ‹preregister assoc›
by (auto simp: assoc_def)

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 comp_tensor_op)

assoc' :: ‹('a×('b×'c)) update → (('a::type×'b::type)×'c::type) update› where
‹assoc' = (Fst o Fst; (Fst o Snd; Snd))›

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

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 comp_tensor_op)

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

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

pair_o_assoc[simp]:
assumes [simp]: ‹compatible F G› ‹compatible G H› ‹compatible F H›
shows ‹(F; (G; H)) ∘ assoc = ((F; G); H)›
(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 cblinfun_compose_assoc)

pair_o_assoc'[simp]:
assumes [simp]: ‹compatible F G› ‹compatible G H› ‹compatible F H›
shows ‹((F; G); H) ∘ assoc' = (F; (G; H))›
(rule tensor_extensionality3)
show ‹register (((F; G); H) ∘ assoc')›
by simp
show ‹register (F; (G; H))›
by simp
show ‹(((F; G); H) ∘ assoc') (f ⊗^sub>u g ⊗_u h) = (F; (G; H)) (f ⊗_u g ⊗_u h)› for f g h
by (simp add: register_pair_apply assoc'_apply cblinfun_compose_assoc)

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

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

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

java.lang.StringIndexOutOfBoundsException: Range [19, 17) out of bounds for length 61
by (simp add: pointfree_idE)

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

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

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

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

‹Iso-registers›

‹iso_register F ⟷ register F ∧ (∃G. register G ∧ F o G = id ∧ G o F = id)›
for F :: ‹_::type update → _::type update›

iso_registerI:
assumes ‹register F› ‹?thesis› by blast
shows ‹iso_register F›
using assms(1) assms(2) assms(3) assms(4) iso_register_def by blast

iso_register_inv: ‹iso_register F ==> iso_register (inv F)›
by (metis inv_unique_comp iso_register_def)

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

iso_register_inv_comp2: ‹
using inv_unique_comp iso_register_def by blast

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

iso_register_is_register: ‹iso_register F ==> register F›
using iso_register_def by blast

iso_register_comp[simp]:
assumes [simp]: ‹iso_register F› ‹iso_register G›
shows ‹iso_register (F o G)›
-
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_registerde)
have 1: ‹F ∘ G ∘ (G' ∘ F') = id›
by (metis ‹F ∘ F' = id› ‹G ∘ G' = id› fcomp_assoc fcomp_comp id_fcomp)
have 2: ‹G' ∘ F' ∘ (F ∘ G) = id›
by (metis (no_types, lifting) ‹ F' = id›F' ∘ ‹ G = id›

show ?thesis
apply (rule iso_registerI[where G=‹G' o F'›])
using 1 2 by (auto simp: register_tensor_is_register iso_register_is_register register_tensor_distrib)

iso_register_tensor_is_iso_register[simp]:
assumes [simp]: ‹iso_register F› ‹iso_register G›
shows ‹iso_register (F ⊗_r G)›🚫
-
from assms obtain F' G' where [simp]: ‹register F'› ‹register G'› ‹F o F' = id qedqed
‹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)

ilqed
using iso_register_def o_bij by auto

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_ thus hesis🚫
iso_register_inv bij_is_surj iso_register_bij surj_f_inv_f)
by (metis eq_id_iff register_tensor_id)

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

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

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

‹equivalent_registers F G ⟷ (register F ∧ (∃I. iso_register I ∧then obtain k i wer iki ‹ and notin\pi': ‹ (∃ππ'\<esup by auto
for F G :: ‹_::type update → _::type update›

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

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

java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 101
using that by (auto intro!: exI[of _ id] simp: equivalent_registers_def)

equivalent_registers_register_left: ‹equivalent_registers F G ==> register F›
using equivalent_registers_def by auto

equivalent_registers_register_right: ‹registershow \op>thesis› by blast
by (metis equivalent_registers_def iso_register_def register_comp that)

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)

equivalent_registers_trans[trans]:
assumes ‹equivalent_registers F G› and ‹equivalent_registers G H›
shows ‹equivalent_registers F H›
-
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)

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

equivalent_registers_pair_right:
assumes [simp]: ‹compatible F G›
assumes eq: ‹equivalent_registers G H›
shows ‹equivalent_registers (F;G) (F;H)›
-
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)

equivalent_registers_pair_left:
assumes [simp]: ‹compatible F G›
assumes eq: ‹equivalent_registers F H›
shows ‹equivalent_registers (F;G) (H;G)›
-
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)

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)

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)

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.›
"compatibility_warn" ("compatible x y" | "register x") = ‹
val thy_string = Markup.markup (Theory.get_markup 🚫

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
›

register_attribute_rule_immediate
register_attribute_rule

[register_attribute_rule] = conjunct1 conjunct2 iso_register_is_register iso_regilemma obs_neq_some_contradict' fie σ' defines ππ ≡<igma
[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
java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 145
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‹?thesis›
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 =>tt OEt >ft tt
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 Simplifier.map_ss (fn ctxt => ctxt addsimps thms) context end
in
Attrib.setup 🚫 j') java.lang.StringIndexOutOfBoundsException: Range [163, 162) out of bounds for length 327
(Scan.succeed (Thmtribute
  "Add register-related rules to the simplifier"
end
›

subsection ‹Notation›

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

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

end

Messung
C	H	Z	G
87	96	100	94

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