Quellcode-Bibliothek AOT_PLM.thy

theory AOT_PLM
  imports AOT_Axioms
begin
(*>*)

section
text‹

(* constrain sledgehammer to the abstraction layer *)
unbundle

subsection‹
 ‹ (case "m + n ≥

 case True
 assumes ‹ and ‹
 shows ‹ψ›
  (* NOTE: semantics needed *)
  using assms by (simp add: AOT_sem_imp)
lemmas  = "modus

subsection‹
text‹

AOT_theorem " -con-thm-thm":
 assumes ‹
 shows ‹ = ∣" using f_m by force
 using assms by simp

  "vdash-properties:1[1]":
 assumes ‹s sing bounof "- (m + n)" n]rue False by simp
 shows ‹ ase Fals
  (* NOTE: semantics needed *)
  using assms "f n n =- f (- n)"using bymp

text‹
  act_axiom_inst =
 ‹+ f n)∣- f (- m + -n) - (- f (-m) + - f(- n))∣
 (K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))›(-m)+ f-n))\<>"
 "Instantiate modally fragile axiom as modally fragile theorem."

 
 assumes ‹
 shows ‹
  (* NOTE: semantics needed *)
  using assms unfolding AOT_model_axiom_def

textjava.lang.NullPointerException
attribute_setup axiom_inst =
  <assumes 
    (fn RS{thmproperties})<close
  "Instantiate axiom as theorem."

text‹f (m + n) - (f m + f n)\<bar  D" for m n using assms by fast
  cqt_2_lambda_inst_prover =
 (fast intro: AOT_instance_of_cqt_2_intro)
  "cqt:2[lambda]" =
 (rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
  "cqt:2" =
 "cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst]
 AOT_instance_of_cqt_2_intro
  "cqt:2" = (safe intro!: "cqt:2")

  "vdash-properties:3":
 assumes ‹f (-m)∣ ∣ + B + D" if "m ≥
 shows ‹\<>\ + B + D" for z sng _bound B_onneg D_nonnbouof "-z" by (cas"z ge> 0" fastf+
 using assms by blast

  "vdash-properties:5":
 assumes ‹"ound by (loune)
 shows ‹
 using MP assms by blast

  "vdash-properties:6":
 assumes ‹ f"
 shows ‹is "?lhs⟷
 using MP assms by blast

  "vdash-properties:8":
 assumes ‹assume asm ?rhs
 shows ‹
 using assms by argo

  "vdash-properties:9":
 assumes ‹
 shows ‹
 using MP "pl:1"[axiom_inst] assms by blast

  "vdash-properties:10":
 assumes ‹ {0..} \inter 0<..})" "D > 0"
 shows ‹
 using MP assms by blast
  "→E" = "vdash-properties:10"

 ‹ (m + 1) * D ≤
 ‹

  "rule-gen": 
 assumes ‹: "D \<ge 
  ‹}›
  (* NOTE: semantics needed *)
  using assms by (metis AOT_var_of_term_inverse2 ( + D)≤f_M∣ ∈ {0<..})" using mult_left_mono[of "C + D" _ 2] D_nonneg by (metis assms(2) abs_ge_zabs_le_D1 int_set_infiniteD)
lemmas GEN = "rule"

AOT_theorem "RN[prem]":
  assumes ‹ abs_diff_le_iff minus_int_code(1,2))
  shows ‹◻ + ) (f (m * M) + f M ≥ D_nonneg add_increas minus_le_iff)
  by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
AOT_theorem RN:
  assumes ‹ f (m * M) ≥ f m
  shows ‹[where ?k=1])
  using " [prem]" assms by blast

 ‹ ?case using M_bound b fastforce
 ‹

  "df-rues-fformu[1]":
 assumes \openφ ≡_f ψ›
 shows ‹
  (* NOTE: semantics needed *)
  using ".. \le (m + 1) (C + ) + M + - (C + D)" M_boundfastforce
  by (auto simp: assms AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
AOT_axiom-rules[2]":
  assumes ‹ (f (m * M) + f M) + (f (m * M + M) - (f (m * M) + f M))" using add_left_mono[OFne_lt
   a have "  = f ((m 1) * M)" by simp add: distrib_right)
  (* NOTE: semantics needed *)
  using assms
  by (finally ?byblast
(* NOTE: for convenience also state the above as regular theorems *)
AOT_theorem "df-rules-formulas[3]":
  assumes ‹φ ≡_df ψ›
  shows ‹φ → ψ›
  using "df-rules-formulas[1]"[axiom_inst, OF assms].
AOT_theorem "df-rules-formulas[4]":
  assumes ‹φ
 shows ‹
 using"df-rules-formulas[2]"axiom_inst, OFOF assms].


  "df-rules-terms[1]":
 assumes ‹
 shows ‹ (m + 1) * D" if "m > 0" for m using *[OF that] mu_eft_mono[of D "C + D" "m + 1"] that Cnonneg Dnonneblinarrith
 (¬σ{τ\<^moreover m>0. (m + 1) * D ≤
  (* NOTE: semantics needed *)
  using assms
  by (simp add
                AOT_sem_not AOT_sem_denotes AOT_model_id_def)
AOT_axiom-rules-terms]:
  assumes ‹
 shows ‹‹Inf›‹
 by (metis "df-rules-terms[1]" case_unit_Unity assms)
(* NOTE: for convenience also state the above as regular theorems *)
AOT_theorem
  assumes ‹
 shows ‹
java.lang.NullPointerException
 using "df-rules-terms[1]"[axiom_inst, OF assms].
  "df-rules-terms[4]":
 assumes ‹
 shows ‹<noteq 
 using "df-rules-terms[2]"[axiom_inst, OF assms].

 ‹The Theory of Negations and Conditionals›
 ‹\label{PLM: 9.5}›

  "if-p-then-p": ‹φ → φ›
 by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)

  "deduction-theorem":
 assumes ‹φ \⊨ ψ›
 shows ‹φ → ψ›
  (* NOTE: semantics needed *)
  using assms by (simp add: AOT_sem_imp)
lemmas CP = "deduction-theorem"
lemmas "→I" = "deduction-theorem"

AOT_theorem "ded-thm-cor:1":
  assumes ‹Γ_-
 shows ‹
 using "→E" "→I" assms by blast
  "ded-thm-cor:2":
 assumes ‹Γ₁ \⊨ φ → (ψ → χ)› and ‹ hence "int (Inf (nat ` ({..} \>))) \<in "using wellorder_I[o "nat `({0. \<> 
 shows ‹Γ₁, Γ₂ \⊨ φ → χ›
 using "→E" "→I" assms by blast

  "ded-thm-cor:3":
 assumes ‹φ → ψ› and ‹ψ → χ›
 shows ‹φ → χ›
 using "→E" "→I" assms by blast
  "ded-thm-cor:3"[trans]
  "ded-thm-cor:4":
 assumes ‹φ → (ψ → χ)› and ‹ψ›
 shows ‹φ → χ›
 using "→E" "→I" assms by blast

  "Hypothetical Syllogism" = "ded-thm-cor:3"

  "useful-tautologies:1": ‹¬¬φ → φ›
 by (metis "pl:3"[axiom_inst] "→I" "Hypothetical Syllogism")
 "useful-tautologies:2": ‹
 by (metis "pl:3"[axiom_inst] "→I" "ded-thm-cor:4")
  "useful-tautologies:3": ‹¬φ → (φ → ψ)›
 by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "→I")
  "useful-tautologies:4": ‹(¬ψ → ¬φ) → (φ → ψ)›
 (memeson"pl:3[axiom_i] Hypo Syllogism" "→
  "useful-tautologies:5": ‹(φ → ψ) → (¬ψ → ¬φ
 by (metis "useful-tautologies:4" "Hypothetical Syllogism" "→I")

  "useful-tautologies:6": ‹by fa
 by (metis "→I" MP "useful-tautologies:4")

  "useful-tautologies:7": ‹(¬φ → ψ) → (¬ψ → φ)›
 by (metis "→I" MP "useful-tautologies:3" "useful-tautologies:5")

  "useful-tautologies:8": ‹
 by (metis "→I" MP "useful-tautologies:5")

  "useful-tautologies:9": ‹(φ → ψ) → ((¬φ → ψ) → ψ)›
 (metis "\rightarrowI" MP "useful-tautolog:6"

  "useful-tautologies:10": ‹(φ → ¬ψ) → ((φ → ψ) → ¬φ)›
 by (metis "→I" MP "pl:3"[axiom_inst])

  "dn-i-e:1":
 assumes ‹φ›
 shows ‹¬¬φ›
 using MP "useful-tautologies:2" assms by blast
  "¬¬I" = "dn-i-e:1"
  "dn-i-e:2":
 assumes ‹¬¬φ›
 shows ‹φ›
 usingMP "use-tau:1" ass byblast
  "¬¬E" = "dn-i-e:2"

  "modus-tollens:1":
 assumes ‹φ → ψ› and ‹¬ψ›
 shows ‹¬φ›
 using MP "useful-tautologies:5" assms by blast
  "modus-tollens:2":
 assumes ‹φ → ¬ψ› and ‹ψ›
 shows ‹¬φ›
 using "¬¬I" "modus-tollens:1" assms by blast
  MT = "modus-tollens:1" "modus-tollens:2"

  "contraposition:1[1]":
 assumes ‹φ → ψ›
 shows ‹¬} \<inter 
 using "→I" MT(1) assms by blast
  "contraposition:1[2]":
 assumes ‹¬ψ → ¬φ›: int) \>y ==>b (imp add infor)
 shows ‹φ → ψ›
 using "→I" "¬¬E" MT(2) assms by blast

  "contraposition:2":
 assumes ‹φ → ¬ψ›In ({b..} \inter A) = inf(Inf {0....} 🚫
 shows ‹ψ → ¬φ›
 using "→I" MT(2) assms by blast

  "reductio-aa:1":
 assumes ‹¬φ \⊨ ¬ψ› and ‹ ({b..} ∩] True partitio by simp
 shows ‹φ›
 using "→I" "¬¬E" MT(2) assms by blast
  "reductio-aa:2":
 assumes ‹φ \⊨ ¬ψ› and ‹φ \⊨ ψ›
  ‹
 using "reductio-aa:1" assms by blast
  "RAA" = "reductio-aa:1" "reductio-aa:2"

  "exc-mid": ‹φ ∨ ¬φ›
 using "df-rules-formulas[4]" "if-p-then-p" MP
 "conventions:2" by blast

  "non-contradiction": ‹¬(φ & ¬φ)›Inf({b..<0  A)}" by (metis inf.commute insertI1insertI2 nle_le)
 using "df-rules-formulas[3]" MT(2) "useful-tautologies:2"
 "conventions:1" by blast

  "con-dis-taut:1": ‹(φ & ψ) → φ›
 by (meson "→I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
  "con-dis-taut:2": ‹(φ & ψ) → ψ›
 by (metis "→I" "df-rules-formulas[3]" MT(2) RAA(2)
 "¬¬E" "conventions:1")
  "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"

  "con-dis-taut:3": ‹φ → (φ ∨ ψ)›
 by (meson "contraposition:1[2]" "df-rules-formulas[4]"
 MP "→I" "conventions:2")
  "con-dis-taut:4": ‹ψ → (φ ∨ ψ)›
 using "Hypothetical Syllogism" "df-rules-formulas[4]"
 "pl:1"[axiom_inst] "conventions:2" by blast
  "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"

  "con-dis-taut:5": ‹φ → (ψ → (φ & ψ))›
 by (metis "contraposition:2" "Hypothetical Syllogism" "→I"
 "df-rules-formulas[4]" "conventions:1")
  Adjunction = "con-dis-taut:5"

  "con-dis-taut:6": ‹(φ & φ) ≡ φ›
 by (metis Adjunction "→I" "df-rules-formulas[4]" MP
 "Conjunction Simplification"(1) "conventions:3")
  "I"Idemp of &" = "con-dis-taut:6"

  "con-dis-taut:7": ‹(φ
 False
 {
 AOT_assume ‹..} \inter A) \< (
 AOT_hence ‹¬φ → φ›
 using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
 AOT_hence ‹φ› using "if-p-then-p" RAA(1) MP by blast
 }
 moreover {
 AOT_assume ‹φ›
 AOT_hence ‹φ ∨ φ› using "Disjunction Addition"(1) MP by blast
 }
 ultimately AOT_show ‹(φ ∨ φ) ≡ φ›
 using "conventions:3"[THEN "df-rules-formulas[4]"] MP
 by (metis Adjunction "→I")
 
  "Idempotence of \<or"


  "con-dis-i-e:1":
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 shows ‹φ & ψ›
 using Adjunction MP assms by blabla
  "&I" = "con-dis-i-e:1"

  "con-dis-i-e:2:a":
 assumes ‹
 shows ‹"int set"
 using "Conjunction Simplification"(1) MP assms by blast
  "con-dis-i-e:2:b":
 assumes ‹
 shows ‹ψ›
 using "Conjunction Simplification"(2) MP assms by blast
  "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"

  "con-dis-i-e:3:a":
 assumes ‹ S))" unfold Infint_def image_c b s
 shows ‹φ ∨ ψ›
 using "Disjunction Addition"(1) MP assms by bmo h "bdd_ (uminus )" si u bdd_be bdd_above_def by (metis i ne
  "con-dis-i-e:3:b":
 assumes ‹ψ›
 shows ‹ using int_Inf_mem by
 using "Disjunction Addition"(2) MP assms by blast
  "con-dis-i-e:3:c":
 assumes ‹φ ∨ ψ› and ‹φ →ultimately show ?thesisby
 shows ‹
 by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2)
 "df-rules-formulas[3]" MT(1) RAA(1)
 "conventions:2" assms)
  "∨I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"

  "con-dis-i-e:4:a":
 assumes ‹
 shows ‹χ›
 by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
  "con-dis-i-e:4:b":
 assumes ‹φ ∨ ψ› and ‹¬φ›
 shows ‹ψ›
 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
  "con-dis-i-e:4:c":
 assumes ‹φ ∨ ψ› and ‹¬ψ›
 shows ‹φ›
 using "con-dis-i-e:4:a" RAA(1) "→I" assms by blast
  "∨E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"

  "raa-cor:1":
 assumes ‹¬φ \⊨ ψ & ¬ψ›
 shows ‹φ›
 using "&E" "∨E"(3) "∨I"(2) RAA(2) assms by blast
  "raa-cor:2":
 assumes ‹φ \⊨ ψ & ¬ψ›
 shows ‹¬φ›
 using "raa-cor:1" assms by blast
  "raa-cor:3":
 assumes ‹φ› and ‹¬ψ \⊨ ¬φ›
 shows ‹ψ›
 using RAA assms by blast
  "raa-cor:4":
 assumes ‹¬φ› and ‹¬ψ \⊨ φ›
 shows ‹ψ›
 using RAA assms by blast
  "raa-cor:5":
 assumes ‹φ› and ‹ψ \⊨ ¬φ›
 shows ‹¬ψ›
 using RAA assms by blast
  "raa-cor:6":
 assumes ‹¬φ› and ‹ψ \⊨ φ›
 shows ‹¬ψ›
 using RAA assms by blast

  "oth-class-taut:1:a": ‹(φ → ψ) ≡ ¬(φ & ¬ψ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&E" "&I" "raa-cor:3" "→I" MP)
  "oth-class-taut:1:b": ‹¬(φ → ψ) ≡ (φ & ¬ψ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&E" "&I" "raa-cor:3" "→I" MP)
  "oth-class-taut:1:c": ‹(φ → ψ) ≡ (¬φ ∨ ψ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&I" "∨I"(1, 2) "∨E"(3) "→I" MP "raa-cor:1")

  "oth-class-taut:2:a": ‹(φ & ψ) ≡ (ψ & φ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (meson "&I" "&E" "→I")
  "Commutativity of &" = "oth-class-taut:2:a"
  "oth-class-taut:2:b": ‹(φ & (ψ & χ)) ≡ ((φ & ψ) & χ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&I" "&E" "→I")
  "Associativity of &" = "oth-class-taut:2:b"
  "oth-class-taut:2:c": ‹(φ ∨ ψ) ≡ (ψ ∨ φ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
  "Commutativity of ∨" = "oth-class-taut:2:c"
  "oth-class-taut:2:d": ‹(φ ∨ (ψ ∨ χ)) ≡ ((φ ∨ ψ) ∨ χ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"])
 (metis "&I" "∨I"(1, 2) "∨E"(1) "→I")
  "Associativity of ∨" = "oth-class-taut:2:d"
  "oth-class-taut:2:e": ‹(φ ≡ ψ) ≡ (ψ ≡ φ)›
 by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]; rule "&I";
 metis "&I" "df-rules-formulas[4]" "conventions:3" "&E"
 "Hypothetical Syllogism" "→I" "df-rules-formulas[3]")
  "Commutativity of ≡" = "oth-class-taut:2:e"
  "oth-class-taut:2:f": ‹(φ ≡ (ψ ≡ χ)) ≡ ((φ ≡ ψ) ≡ χ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I"
 by metis
  "Associativity of ≡" = "oth-class-taut:2:f"

  "oth-class-taut:3:a": ‹φ ≡ φ›
 using "&I" "vdash-properties:6" "if-p-then-p"
 "df-rules-formulas[4]" "conventions:3" by blast
  "oth-class-taut:3:b": ‹φ ≡ ¬¬φ›
 using "&I" "useful-tautologies:1" "useful-tautologies:2" "→E"
 "df-rules-formulas[4]" "conventions:3" by blast
  "oth-class-taut:3:c": ‹¬(φ ≡ ¬φ)›
 by (metis "&E" "→E" RAA "df-rules-formulas[3]" "conventions:3")

  "oth-class-taut:4:a": ‹(φ → ψ) → ((ψ → χ) → (φ → χ))›
 by (metis "→E" "→I")
  "oth-class-taut:4:b": ‹(φ ≡ ψ) ≡ (¬φ ≡ ¬ψ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" RAA by metis
  "oth-class-taut:4:c": ‹(φ ≡ ψ) → ((φ → χ) ≡ (ψ → χ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" by metis
  "oth-class-taut:4:d": ‹(φ ≡ ψ) → ((χ → φ) ≡ (χ → ψ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" by metis
  "oth-class-taut:4:e": ‹(φ ≡ ψ) → ((φ & χ) ≡ (ψ & χ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" by metis
  "oth-class-taut:4:f": ‹(φ ≡ ψ) → ((χ & φ) ≡ (χ & ψ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" by metis
  "oth-class-taut:4:g": ‹(φ ≡ ψ) ≡ ((φ & ψ) ∨ (¬φ & ¬ψ))›
 (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]
 "&I" "→I"
 dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "→E"])
 AOT_show ‹φ & ψ ∨ (¬φ & ¬ψ)› if ‹(φ → ψ) & (ψ → φ)›
 using "&E" "∨I" "→E" "&I" "raa-cor:1" "→I" "∨E" that by metis
 
 AOT_show ‹ψ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹φ›
 using that "∨E" "&E" "raa-cor:3" by blast
 
 AOT_show ‹φ› if ‹φ & ψ ∨ (¬φ & ¬ψ)› and ‹ψ›
 using that "∨E" "&E" "raa-cor:3" by blast
 
  "oth-class-taut:4:h": ‹¬(φ ≡ ψ) ≡ ((φ & ¬ψ) ∨ (¬φ & ψ))›
  (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "→E"]
 "&I" "→I")
 AOT_show ‹φ & ¬ψ ∨ (¬φ & ψ)› if ‹¬(φ ≡ ψ)›
 by (metis that "&I" "∨I"(1, 2) "→I" MT(1) "df-rules-formulas[4]"
 "raa-cor:3" "conventions:3")
 
 AOT_show ‹¬(φ ≡ ψ)› if ‹φ & ¬ψ ∨ (¬φ & ψ)›
 by (metis that "&E" "∨E"(2) "→E" "df-rules-formulas[3]"
 "raa-cor:3" "conventions:3")
 
  "oth-class-taut:5:a": ‹(φ & ψ) ≡ ¬(¬φ ∨ ¬ψ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
  "oth-class-taut:5:b": ‹(φ ∨ ψ) ≡ ¬(¬φ & ¬ψ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
  "oth-class-taut:5:c": ‹¬(φ & ψ) ≡ (¬φ ∨ ¬ψ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
  "oth-class-taut:5:d": ‹¬(φ ∨ ψ) ≡ (¬φ & ¬ψ)›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis

  DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"

  "oth-class-taut:6:a":
 ‹(φ & (ψ ∨ χ)) ≡ ((φ & ψ) ∨ (φ & χ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis
  "oth-class-taut:6:b":
 ‹(φ ∨ (ψ & χ)) ≡ ((φ ∨ ψ) & (φ ∨ χ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E" RAA by metis

  "oth-class-taut:7:a": ‹((φ & ψ) → χ) → (φ → (ψ → χ))›
 by (metis "&I" "→E" "→I")
  Exportation = "oth-class-taut:7:a"
  "oth-class-taut:7:b": ‹(φ → (ψ →χ)) → ((φ & ψ) → χ)›
 by (metis "&E" "→E" "→I")
  Importation = "oth-class-taut:7:b"

  "oth-class-taut:8:a":
 ‹(φ → (ψ → χ)) ≡ (ψ → (φ → χ))›
 using "conventions:3"[THEN "df-rules-formulas[4]"] "→I" "→E" "&E" "&I"
 by metis
  Permutation = "oth-class-taut:8:a"
  "oth-class-taut:8:b":
 ‹(φ → ψ) → ((φ → χ) → (φ → (ψ & χ)))›
 by (metis "&I" "→E" "→I")
  Composition = "oth-class-taut:8:b"
  "oth-class-taut:8:c":
 ‹(φ → χ) → ((ψ → χ) → ((φ ∨ ψ) → χ))›
 by (metis "∨E"(2) "→E" "→I" RAA(1))
  "oth-class-taut:8:d":
 ‹((φ → ψ) & (χ → Θ)) → ((φ & χ) → (ψ & Θ))›
 by (metis "&E" "&I" "→E" "→I")
  "Double Composition" = "oth-class-taut:8:d"
  "oth-class-taut:8:e":
 ‹((φ & ψ) ≡ (φ & χ)) ≡ (φ → (ψ ≡ χ))›
 by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I")
  "oth-class-taut:8:f":
 ‹((φ & ψ) ≡ (χ & ψ)) ≡ (ψ → (φ ≡ χ))›
 by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I")
  "oth-class-taut:8:g":
 ‹(ψ ≡ χ) → ((φ ∨ ψ) ≡ (φ ∨ χ))›
 by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
  "oth-class-taut:8:h":
 ‹(ψ ≡ χ) → ((ψ ∨ φ) ≡ (χ ∨ φ))›
 by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I" "∨I" "∨E"(1))
  "oth-class-taut:8:i":
 ‹(φ ≡ (ψ & χ)) → (ψ → (φ ≡ χ))›
 by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
 "conventions:3"[THEN "df-rules-formulas[3]"]
 "→I" "→E" "&E" "&I")

  "intro-elim:1":
 assumes ‹φ ∨ ψ› and ‹φ ≡ χ› and ‹ψ ≡ Θ›
 shows ‹χ ∨ Θ›
 by (metis assms "∨I"(1, 2) "∨E"(1) "→I" "→E" "&E"(1)
 "conventions:3"[THEN "df-rules-formulas[3]"])

  "intro-elim:2":
 assumes ‹φ → ψ› and ‹ψ → φ›
 shows ‹φ ≡ ψ›
 by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
  "≡I" = "intro-elim:2"

  "intro-elim:3:a":
 assumes ‹φ ≡ ψ› and ‹φ›
 shows ‹ψ›
 by (metis "∨I"(1) "→I" "∨E"(1) "intro-elim:1" assms)
  "intro-elim:3:b":
 assumes ‹φ ≡ ψ› and ‹ψ›
 shows ‹φ›
 using "intro-elim:3:a" "Commutativity of ≡" assms by blast
  "intro-elim:3:c":
 assumes ‹φ ≡ ψ› and ‹¬φ›
 shows ‹¬ψ›
 using "intro-elim:3:b" "raa-cor:3" assms by blast
  "intro-elim:3:d":
 assumes ‹φ ≡ ψ› and ‹¬ψ›
 shows ‹¬φ›
 using "intro-elim:3:a" "raa-cor:3" assms by blast
  "intro-elim:3:e":
 assumes ‹φ ≡ ψ› and ‹ψ ≡ χ›
 shows ‹φ ≡ χ›
 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
  "intro-elim:3:e"[trans]
  "intro-elim:3:f":
 assumes ‹φ ≡ ψ› and ‹φ ≡ χ›
 shows ‹χ ≡ ψ›
 by (metis "≡I" "→I" "intro-elim:3:a" "intro-elim:3:b" assms)
  "≡E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c"
 "intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"

  "Commutativity of ≡"[THEN "≡E"(1), sym]

  "rule-eq-df:1":
 assumes ‹φ ≡_df ψ›
 shows ‹φ ≡ ψ›
 by (simp add: "≡I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
  "≡Df" = "rule-eq-df:1"
  "rule-eq-df:2":
 assumes ‹φ ≡_df ψ› and ‹φ›
 shows ‹ψ›
 using "≡Df" "≡E"(1) assms by blast
  "≡_dfE" = "rule-eq-df:2"
  "rule-eq-df:3":
 assumes ‹φ ≡_df ψ› and ‹ψ›
 shows ‹φ›
 using "≡Df" "≡E"(2) assms by blast
  "≡_dfI" = "rule-eq-df:3"

  "df-simplify:1":
 assumes ‹φ ≡ (ψ & χ)› and ‹ψ›
 shows ‹φ ≡ χ›
 by (metis "&E"(2) "&I" "≡E"(1, 2) "≡I" "→I" assms)
(* Note: this is a slight variation from PLM *)
AOT_theorem  "df-simplify:2":
  assumes ‹φ ≡ (ψ & χ)› and ‹χ›
  shows ‹φ ≡ ψ›
  by (metis "&E"(1) "&I" "≡E"(1, 2) "≡I" "→I" assms)
lemmas "≡S" = "df-simplify:1"  "df-simplify:2"

subsection‹The Theory of Quantification›
text‹\label{PLM: 9.6}›

AOT_theorem "rule-ui:1":
  assumes ‹∀α φ{α}› and ‹τ↓›
  shows ‹φ{τ}›
  using "→E" "cqt:1"[axiom_inst] assms by blast
AOT_theorem "rule-ui:2[const_var]":
  assumes ‹∀α φ{α}›
  shows ‹φ{β}›
  by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
AOT_theorem "rule-ui:2[lambda]":
  assumes ‹∀F φ{F}› and ‹INSTANCE_OF_CQT_2(ψ)›
  shows ‹φ{[λν₁...ν_n ψ{ν₁...ν_n}]}›
  by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
AOT_theorem "rule-ui:3":
  assumes ‹∀α φ{α}›
  shows ‹φ{α}›
  by (simp add: "rule-ui:2[const_var]" assms)
lemmas "∀E" = "rule-ui:1" "rule-ui:2[const_var]"
              "rule-ui:2[lambda]" "rule-ui:3"

AOT_theorem "cqt-orig:1[const_var]": ‹∀α φ{α} → φ{β}›
  by (simp add: "∀E"(2) "→I")
AOT_theorem "cqt-orig:1[lambda]":
  assumes ‹INSTANCE_OF_CQT_2(ψ)›
  shows ‹∀F φ{F} → φ{[λν₁...ν_n ψ{ν₁...ν_n}]}›
  by (simp add: "∀E"(3) "→I" assms)
AOT_theorem "cqt-orig:2": ‹∀α (φ → ψ{α}) → (φ → ∀α ψ{α})›
  by (metis "→I" GEN "vdash-properties:6" "∀E"(4))
AOT_theorem "cqt-orig:3": ‹∀α φ{α} → φ{α}›
  using "cqt-orig:1[const_var]".

AOT_theorem universal:
  assumes ‹for arbitrary β: φ{β}›
  shows ‹∀α φ{α}›
  using GEN assms .
lemmas "∀I" = universal

(* Generalized mechanism for \<forall>I followed by \<forall>E *)
ML‹
  get_instantiated_allI ctxt varname thm = let
  trm = Thm.concl_of thm
  trm =
 case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
 | _ => raise Term.TERM ("Expected simple theorem.", [trm])
  extractVars (Const (🍋‹AOT_term_of_var›, _) $ Var v) =
 (if fst (fst v) = fst varname then [Var v] else [])
 | extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
 | extractVars (Abs (_, _, t)) = extractVars t
 | extractVars _ = []
  vars = extractVars trm
  vars = fold Term.add_vars vars []
  var = hd vars
  trmty =
 case (snd var) of (Type (🍋‹AOT_var›, [t])) => (t)
 | _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
  trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
 Const (🍋‹AOT_term_of_var›, Type ("fun", [snd var, trmty]))
 $ Var var, trm))
  trm = Thm.cterm_of (Context.proof_of ctxt) trm
  ty = hd (Term.add_tvars (Thm.prop_of @{thm "∀I"}) [])
  typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
  allthm = Drule.instantiate_normalize (TVars.make [(ty, typ)], Vars.empty) @{thm "∀I"}
  phi = hd (Term.add_vars (Thm.prop_of allthm) [])
  allthm = Drule.instantiate_normalize (TVars.empty, Vars.make [(phi,trm)]) allthm
 
 
 
 ›

attribute_setup "∀I" =
  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
 (fn ctxt => fn thm => fold (fn arg => fn thm =>
 thm RS get_instantiated_allI ctxt arg thm) args thm))›
  "Quantify over a variable in a theorem using GEN."

attribute_setup "unvarify" =
  ‹Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
 (fn ctxt => fn thm =>
 let
 fun get_inst_allI arg thm = thm RS get_instantiated_allI ctxt arg thm
 val thm = fold get_inst_allI args thm
 val thm = fold (K (fn thm => thm RS @{thm "∀E"(1)})) args thm
 in
 thm
 end))›
  "Generalize a statement about variables to a statement about denoting terms."

(* Note: rereplace-lem does not apply to the embedding *)

AOT_theorem "cqt-basic:1": ‹∀α∀β φ{α,β} ≡ ∀β∀α φ{α,β}›
  by (metis "≡I" "∀E"(2) "∀I" "→I")

AOT_theorem "cqt-basic:2":
  ‹∀α(φ{α} ≡ ψ{α}) ≡ (∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α}))›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
  AOT_hence ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
    using "≡E"(1,2) "→I" by blast+
  AOT_thus ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
    by (auto intro: "&I" "∀I")
next
  AOT_assume ‹∀α(φ{α} → ψ{α}) & ∀α(ψ{α} → φ{α})›
  AOT_hence ‹φ{α} → ψ{α}› and ‹ψ{α} → φ{α}› for α
    using "∀E"(2) "&E" by blast+
  AOT_hence ‹φ{α} ≡ ψ{α}› for α
    using "≡I" by blast
  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› by (auto intro: "∀I")
qed

AOT_theorem "cqt-basic:3": ‹∀α(φ{α} ≡ ψ{α}) → (∀α φ{α} ≡ ∀α ψ{α})›
proof(rule "→I")
  AOT_assume ‹∀α(φ{α} ≡ ψ{α})›
  AOT_hence 1: ‹φ{α} ≡ ψ{α}› for α using "∀E"(2) by blast
  {
    AOT_assume ‹∀α φ{α}›
    AOT_hence ‹∀α ψ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
  }
  moreover {
    AOT_assume ‹∀α ψ{α}›
    AOT_hence ‹∀α φ{α}› using 1 "∀I" "∀E"(4) "≡E" by metis
  }
  ultimately AOT_show ‹∀α φ{α} ≡ ∀α ψ{α}›
    using "≡I" "→I" by auto
qed

AOT_theorem "cqt-basic:4": ‹∀α(φ{α} & ψ{α}) → (∀α φ{α} & ∀α ψ{α})›
proof(rule "→I")
  AOT_assume 0: ‹∀α(φ{α} & ψ{α})›
  AOT_have ‹φ{α}› and ‹ψ{α}› for α using "∀E"(2) 0 "&E" by blast+
  AOT_thus ‹∀α φ{α} & ∀α ψ{α}›
    by (auto intro: "∀I" "&I")
qed

AOT_theorem "cqt-basic:5": ‹(∀α₁...∀α_n(φ{α₁...α_n})) → φ{α₁...α_n}›
  using "cqt-orig:3" by blast

AOT_theorem "cqt-basic:6": ‹∀α∀α φ{α} ≡ ∀α φ{α}›
  by (meson "≡I" "→I" GEN "cqt-orig:1[const_var]")

AOT_theorem "cqt-basic:7": ‹(φ → ∀α ψ{α}) ≡ ∀α(φ → ψ{α})›
  by (metis "→I" "vdash-properties:6" "rule-ui:3" "≡I" GEN)

AOT_theorem "cqt-basic:8": ‹(∀α φ{α} ∨ ∀α ψ{α}) → ∀α (φ{α} ∨ ψ{α})›
  by (simp add: "∨I"(3) "→I" GEN "cqt-orig:1[const_var]")

AOT_theorem "cqt-basic:9":
  ‹(∀α (φ{α} → ψ{α}) & ∀α (ψ{α} → χ{α})) → ∀α(φ{α} → χ{α})›
proof -
  {
    AOT_assume ‹∀α (φ{α} → ψ{α})›
    moreover AOT_assume ‹∀α (ψ{α} → χ{α})›
    ultimately AOT_have ‹φ{α} → ψ{α}› and ‹ψ{α} → χ{α}› for α
      using "∀E" by blast+
    AOT_hence ‹φ{α} → χ{α}› for α by (metis "→E" "→I")
    AOT_hence ‹∀α(φ{α} → χ{α})› using "∀I" by fast
  }
  thus ?thesis using "&I" "→I" "&E" by meson
qed

AOT_theorem "cqt-basic:10":
  ‹(∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})) → ∀α (φ{α} ≡ χ{α})›
proof(rule "→I"; rule "∀I")
  fix β
  AOT_assume ‹∀α(φ{α} ≡ ψ{α}) & ∀α(ψ{α} ≡ χ{α})›
  AOT_hence ‹φ{β} ≡ ψ{β}› and ‹ψ{β} ≡ χ{β}› using "&E" "∀E" by blast+
  AOT_thus ‹φ{β} ≡ χ{β}› using "≡I" "≡E" by blast
qed

AOT_theorem "cqt-basic:11": ‹∀α(φ{α} ≡ ψ{α}) ≡ ∀α (ψ{α} ≡ φ{α})›
proof (rule "≡I"; rule "→I")
  AOT_assume 0: ‹∀α(φ{α} ≡ ψ{α})›
  {
    fix α
    AOT_have ‹φ{α} ≡ ψ{α}› using 0 "∀E" by blast
    AOT_hence ‹ψ{α} ≡ φ{α}› using "≡I" "≡E" "→I" "→E" by metis
  }
  AOT_thus ‹∀α(ψ{α} ≡ φ{α})› using "∀I" by fast
next
  AOT_assume 0: ‹∀α(ψ{α} ≡ φ{α})›
  {
    fix α
    AOT_have ‹ψ{α} ≡ φ{α}› using 0 "∀E" by blast
    AOT_hence ‹φ{α} ≡ ψ{α}› using "≡I" "≡E" "→I" "→E" by metis
  }
  AOT_thus ‹∀α(φ{α} ≡ ψ{α})› using "∀I" by fast
qed

AOT_theorem "cqt-basic:12": ‹∀α φ{α} → ∀α (ψ{α} → φ{α})›
  by (simp add: "∀E"(2) "→I" GEN)

AOT_theorem "cqt-basic:13": ‹∀α φ{α} ≡ ∀β φ{β}›
  using "≡I" "→I" by blast

AOT_theorem "cqt-basic:14":
  ‹(∀α₁...∀α_n (φ{α₁...α_n} → ψ{α₁...α_n})) →
 ((∀α₁...∀α_n φ{α₁...α_n}) → (∀α₁...∀α_n ψ{α₁...α_n}))›
  using "cqt:3"[axiom_inst] by auto

AOT_theorem "cqt-basic:15":
  ‹(∀α₁...∀α_n (φ → ψ{α₁...α_n})) → (φ → (∀α₁...∀α_n ψ{α₁...α_n}))›
  using "cqt-orig:2" by auto

AOT_theorem "universal-cor":
  assumes ‹for arbitrary β: φ{β}›
  shows ‹∀α φ{α}›
  using GEN assms .

AOT_theorem "existential:1":
  assumes ‹φ{τ}› and ‹τ↓›
  shows ‹∃α φ{α}›
proof(rule "raa-cor:1")
  AOT_assume ‹¬∃α φ{α}›
  AOT_hence ‹∀α ¬φ{α}›
    using "≡_dfI" "conventions:4" RAA "&I" by blast
  AOT_hence ‹¬φ{τ}› using assms(2) "∀E"(1) "→E" by blast
  AOT_thus ‹φ{τ} & ¬φ{τ}› using assms(1) "&I" by blast
qed

AOT_theorem "existential:2[const_var]":
  assumes ‹φ{β}›
  shows ‹∃α φ{α}›
  using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast

AOT_theorem "existential:2[lambda]":
  assumes ‹φ{[λν₁...ν_n ψ{ν₁...ν_n}]}› and ‹INSTANCE_OF_CQT_2(ψ)›
  shows ‹∃α φ{α}›
  using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
lemmas "∃I" = "existential:1" "existential:2[const_var]"
              "existential:2[lambda]" 

AOT_theorem "instantiation":
  assumes ‹for arbitrary β: φ{β} \⊨ ψ› and ‹∃α φ{α}›
  shows ‹ψ›
  by (metis (no_types, lifting) "≡_dfE" GEN "raa-cor:3" "conventions:4" assms)
lemmas "∃E" = "instantiation"

AOT_theorem "cqt-further:1": ‹∀α φ{α} → ∃α φ{α}›
  using "∀E"(4) "∃I"(2) "→I" by metis

AOT_theorem "cqt-further:2": ‹¬∀α φ{α} → ∃α ¬φ{α}›
  using "∀I" "∃I"(2) "→I" RAA by metis

AOT_theorem "cqt-further:3": ‹∀α φ{α} ≡ ¬∃α ¬φ{α}›
  using "∀E"(4) "∃E" "→I" RAA
  by (metis "cqt-further:2" "≡I" "modus-tollens:1")

AOT_theorem "cqt-further:4": ‹¬∃α φ{α} → ∀α ¬φ{α}›
  using "∀I" "∃I"(2)"→I" RAA by metis

AOT_theorem "cqt-further:5": ‹∃α (φ{α} & ψ{α}) → (∃α φ{α} & ∃α ψ{α})›
  by (metis (no_types, lifting) "&E" "&I" "∃E" "∃I"(2) "→I")

AOT_theorem "cqt-further:6": ‹∃α (φ{α} ∨ ψ{α}) → (∃α φ{α} ∨ ∃α ψ{α})›
  by (metis (mono_tags, lifting) "∃E" "∃I"(2) "∨E"(3) "∨I"(1, 2) "→I" RAA(2))

(* NOTE: vacuous in the embedding *)
AOT_theorem "cqt-further:7": ‹∃α φ{α} ≡ ∃β φ{β}›
  by (simp add: "oth-class-taut:3:a")

AOT_theorem "cqt-further:8":
  ‹(∀α φ{α} & ∀α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
  by (metis (mono_tags, lifting) "&E" "≡I" "∀E"(2) "→I" GEN)

AOT_theorem "cqt-further:9":
  ‹(¬∃α φ{α} & ¬∃α ψ{α}) → ∀α (φ{α} ≡ ψ{α})›
  by (metis (mono_tags, lifting) "&E" "≡I" "∃I"(2) "→I" GEN "raa-cor:4")

AOT_theorem "cqt-further:10":
  ‹(∃α φ{α} & ¬∃α ψ{α}) → ¬∀α (φ{α} ≡ ψ{α})›
proof(rule "→I"; rule "raa-cor:2")
  AOT_assume 0: ‹∃α φ{α} & ¬∃α ψ{α}›
  then AOT_obtain α where ‹φ{α}› using "∃E" "&E"(1) by metis
  moreover AOT_assume ‹∀α (φ{α} ≡ ψ{α})›
  ultimately AOT_have ‹ψ{α}› using "∀E"(4) "≡E"(1) by blast
  AOT_hence ‹∃α ψ{α}› using "∃I" by blast
  AOT_thus ‹∃α ψ{α} & ¬∃α ψ{α}› using 0 "&E"(2) "&I" by blast
qed

AOT_theorem "cqt-further:11": ‹∃α∃β φ{α,β} ≡ ∃β∃α φ{α,β}›
  using "≡I" "→I" "∃I"(2) "∃E" by metis

subsection‹Logical Existence, Identity, and Truth›
text‹\label{PLM: 9.7}›

AOT_theorem "log-prop-prop:1": ‹[λ φ]↓›
  using "cqt:2[lambda0]"[axiom_inst] by auto

AOT_theorem "log-prop-prop:2": ‹φ↓›
  by (rule "≡_dfI"[OF "existence:3"]) "cqt:2[lambda]"

AOT_theorem "exist-nec": ‹τ↓ → ◻τ↓›
proof -
  AOT_have ‹∀β ◻β↓›
    by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
  AOT_thus ‹τ↓ → ◻τ↓›
    using "cqt:1"[axiom_inst] "→E" by blast
qed

(* TODO: replace this mechanism by a "proof by types" command *)
class AOT_Term_id = AOT_Term +
  assumes "t=t-proper:1"[AOT]: ‹[v ⊨ τ = τ' → τ↓]›
      and "t=t-proper:2"[AOT]: ‹[v ⊨ τ = τ' → τ'↓]›

instance κ :: AOT_Term_id
proof
  AOT_modally_strict {
    AOT_show ‹κ = κ' → κ↓› for κ κ'
    proof(rule "→I")
      AOT_assume ‹κ = κ'›
      AOT_hence ‹O!κ ∨ A!κ›
        by (rule "∨I"(3)[OF "≡_dfE"[OF "identity:1"]])
           (meson "→I" "∨I"(1) "&E"(1))+
      AOT_thus ‹κ↓›
        by (rule "∨E"(1))
           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
    qed
  }
next
  AOT_modally_strict {
    AOT_show ‹κ = κ' → κ'↓› for κ κ'
    proof(rule "→I")
      AOT_assume ‹κ = κ'›
      AOT_hence ‹O!κ' ∨ A!κ'›
        by (rule "∨I"(3)[OF "≡_dfE"[OF "identity:1"]])
           (meson "→I" "∨I" "&E")+
      AOT_thus ‹κ'↓›
        by (rule "∨E"(1))
           (metis "cqt:5:a"[axiom_inst] "→I" "→E" "&E"(2))+
    qed
  }
qed

instance rel :: (AOT_κs) AOT_Term_id
proof
  AOT_modally_strict {
    AOT_show ‹Π = Π' → Π↓› for Π Π' :: ‹<'a>›
    proof(rule "→I")
      AOT_assume ‹Π = Π'›
      AOT_thus ‹Π↓› using "≡_dfE"[OF "identity:3"[of Π Π']] "&E" by blast
    qed
  }
next
  AOT_modally_strict {
    AOT_show ‹Π = Π' → Π'↓› for Π Π' :: ‹<'a>›
    proof(rule "→I")
      AOT_assume ‹Π = Π'›
      AOT_thus ‹Π'↓› using "≡_dfE"[OF "identity:3"[of Π Π']] "&E" by blast
    qed
  }
qed

instance o :: AOT_Term_id
proof
  AOT_modally_strict {
    fix φ ψ
    AOT_show ‹φ = ψ → φ↓›
    proof(rule "→I")
      AOT_assume ‹φ = ψ›
      AOT_thus ‹φ↓› using "≡_dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
    qed
  }
next
  AOT_modally_strict {
    fix φ ψ
    AOT_show ‹φ = ψ → ψ↓›
    proof(rule "→I")
      AOT_assume ‹φ = ψ›
      AOT_thus ‹ψ↓› using "≡_dfE"[OF "identity:4"[of φ ψ]] "&E" by blast
    qed
  }
qed

instance prod :: (AOT_Term_id, AOT_Term_id) AOT_Term_id
proof
  AOT_modally_strict {
    fix τ τ' :: ‹'a×'b›
    AOT_show ‹τ = τ' → τ↓›
    proof (induct τ; induct τ'; rule "→I")
      fix τ₁ τ₁' :: 'a and τ₂  τ₂' :: 'b
      AOT_assume ‹«(τ₁, τ₂)¬ = «(τ₁', τ₂')¬›
      AOT_hence ‹(τ₁ = τ₁') & (τ₂ = τ₂')› by (metis "≡_dfE" tuple_identity_1)
      AOT_hence ‹τ₁↓› and ‹τ₂↓›
        using "t=t-proper:1" "&E" "vdash-properties:10" by blast+
      AOT_thus ‹«(τ₁, τ₂)¬↓› by (metis "≡_dfI" "&I" tuple_denotes)
    qed
  }
next
  AOT_modally_strict {
    fix τ τ' :: ‹'a×'b›
    AOT_show ‹τ = τ' → τ'↓›
    proof (induct τ; induct τ'; rule "→I")
      fix τ₁ τ₁' :: 'a and τ₂  τ₂' :: 'b
      AOT_assume ‹«(τ₁, τ₂)¬ = «(τ₁', τ₂')¬›
      AOT_hence ‹(τ₁ = τ₁') & (τ₂ = τ₂')› by (metis "≡_dfE" tuple_identity_1)
      AOT_hence ‹τ₁'↓› and ‹τ₂'↓›
        using "t=t-proper:2" "&E" "vdash-properties:10" by blast+
      AOT_thus ‹«(τ₁', τ₂')¬↓› by (metis "≡_dfI" "&I" tuple_denotes)
    qed
  }
qed

(* This is the end of the "proof by types" and
   makes the results available on new theorems *)
AOT_register_type_constraints
  Term: ‹_::AOT_Term_id› ‹_::AOT_Term_id›
AOT_register_type_constraints
  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id}›
AOT_register_type_constraints
  Relation: ‹<_::{AOT_κs, AOT_Term_id}>›

AOT_theorem "id-rel-nec-equiv:1":
  ‹Π = Π' → ◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π']x₁...x_n)›
proof(rule "→I")
  AOT_assume assumption: ‹Π = Π'›
  AOT_hence ‹Π↓› and ‹Π'↓›
    using "t=t-proper:1" "t=t-proper:2" MP by blast+
  moreover AOT_have ‹∀F∀G (F = G → ((◻∀x₁...∀x_n ([F]x₁...x_n ≡ [F]x₁...x_n)) →
 ◻∀x₁...∀x_n ([F]x₁...x_n ≡ [G]x₁...x_n)))›
    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
  ultimately AOT_have ‹Π = Π' → ((◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π]x₁...x_n)) →
 ◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π']x₁...x_n))›
    using "∀E"(1) by blast
  AOT_hence ‹(◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π]x₁...x_n)) →
 ◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π']x₁...x_n)›
    using assumption "→E" by blast
  moreover AOT_have ‹◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π]x₁...x_n)›
    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
  ultimately AOT_show ‹◻∀x₁...∀x_n ([Π]x₁...x_n ≡ [Π']x₁...x_n)›
    using "→E" by blast
qed

AOT_theorem "id-rel-nec-equiv:2": ‹φ = ψ → ◻(φ ≡ ψ)›
proof(rule "→I")
  AOT_assume assumption: ‹φ = ψ›
  AOT_hence ‹φ↓› and ‹ψ↓›
    using "t=t-proper:1" "t=t-proper:2" MP by blast+
  moreover AOT_have ‹∀p∀q (p = q → ((◻(p ≡ p) → ◻(p ≡ q))))›
    apply (rule GEN)+ using "l-identity"[axiom_inst] by force
  ultimately AOT_have ‹φ = ψ → (◻(φ ≡ φ) → ◻(φ ≡ ψ))›
    using "∀E"(1) by blast
  AOT_hence ‹◻(φ ≡ φ) → ◻(φ ≡ ψ)›
    using assumption "→E" by blast
  moreover AOT_have ‹◻(φ ≡ φ)›
    by (simp add: RN "oth-class-taut:3:a" "universal-cor")
  ultimately AOT_show ‹◻(φ ≡ ψ)›
    using "→E" by blast
qed

AOT_theorem "rule=E":
  assumes ‹φ{τ}› and ‹τ = σ›
  shows ‹φ{σ}›
proof -
  AOT_have ‹τ↓› and ‹σ↓›
    using assms(2) "t=t-proper:1" "t=t-proper:2" "→E" by blast+
  moreover AOT_have ‹∀α∀β(α = β → (φ{α} → φ{β}))›
    apply (rule GEN)+ using "l-identity"[axiom_inst] by blast
  ultimately AOT_have ‹τ = σ → (φ{τ} → φ{σ})›
    using "∀E"(1) by blast
  AOT_thus ‹φ{σ}› using assms "→E" by blast
qed

AOT_theorem "propositions-lemma:1": ‹[λ φ] = φ›
proof -
  AOT_have ‹φ↓› by (simp add: "log-prop-prop:2")
  moreover AOT_have ‹∀p [λ p] = p›
    using "lambda-predicates:3[zero]"[axiom_inst] "∀I" by fast
  ultimately AOT_show ‹[λ φ] = φ›
    using "∀E" by blast
qed

AOT_theorem "propositions-lemma:2": ‹[λ φ] ≡ φ›
proof -
  AOT_have ‹[λ φ] ≡ [λ φ]› by (simp add: "oth-class-taut:3:a")
  AOT_thus ‹[λ φ] ≡ φ› using "propositions-lemma:1" "rule=E" by blast
qed

text‹propositions-lemma:3 through propositions-lemma:5 hold implicitly›

AOT_theorem "propositions-lemma:6": ‹(φ ≡ ψ) ≡ ([λ φ] ≡ [λ ψ])›
  by (metis "≡E"(1) "≡E"(5) "Associativity of ≡" "propositions-lemma:2")

text‹dr-alphabetic-rules holds implicitly›

AOT_theorem "oa-exist:1": ‹O!↓›
proof -
  AOT_have ‹[λx ♢[E!]x]↓› by "cqt:2[lambda]"
  AOT_hence 1: ‹O! = [λx ♢[E!]x]›
    using "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1)] "→E" by blast
  AOT_show ‹O!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
qed

AOT_theorem "oa-exist:2": ‹A!↓›
proof -
  AOT_have ‹[λx ¬♢[E!]x]↓› by "cqt:2[lambda]"
  AOT_hence 1: ‹A! = [λx ¬♢[E!]x]›
    using "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1)] "→E" by blast
  AOT_show ‹A!↓› using "t=t-proper:1"[THEN "→E", OF 1] by simp
qed

AOT_theorem "oa-exist:3": ‹O!x ∨ A!x›
proof(rule "raa-cor:1")
  AOT_assume ‹¬(O!x ∨ A!x)›
  AOT_hence A: ‹¬O!x› and B: ‹¬A!x›
    using "Disjunction Addition"(1) "modus-tollens:1"
          "∨I"(2) "raa-cor:5" by blast+
  AOT_have C: ‹O! = [λx ♢[E!]x]›
    by (rule "df-rules-terms[4]"[OF "oa:1", THEN "&E"(1), THEN "→E"]) "cqt:2"
  AOT_have D: ‹A! = [λx ¬♢[E!]x]›
    by (rule "df-rules-terms[4]"[OF "oa:2", THEN "&E"(1), THEN "→E"]) "cqt:2"
  AOT_have E: ‹¬[λx ♢[E!]x]x›
    using A C "rule=E" by fast
  AOT_have F: ‹¬[λx ¬♢[E!]x]x›
    using B D "rule=E" by fast
  AOT_have G: ‹[λx ♢[E!]x]x ≡ ♢[E!]x›
    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2"
  AOT_have H: ‹[λx ¬♢[E!]x]x ≡ ¬♢[E!]x›
    by (rule "lambda-predicates:2"[axiom_inst, THEN "→E"]) "cqt:2"
  AOT_show ‹¬♢[E!]x & ¬¬♢[E!]x› using G E "≡E" H F "≡E" "&I"> by metis
qed

AOT_theorem "p-identity-thm2:1": ‹F = G ≡ ◻∀x(x[F] ≡ x[G])›
proof -
  AOT_have ‹F = G ≡ F↓ & G↓ & ◻∀x(x[F] ≡ x[G])›
    using "identity:2" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹F↓› and ‹G↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately AOT_show ‹F = G ≡ ◻∀x(x[F] ≡ x[G])›
    using "≡S"(1) "&I" by blast
qed

AOT_theorem "p-identity-thm2:2[2]":
  ‹F = G ≡ ∀y₁([λx [F]xy₁] = [λx [G]xy₁] & [λx [F]y₁x] = [λx [G]y₁x])›
proof -
  AOT_have ‹F = G ≡ F↓ & G↓ &
 ∀y₁([λx [F]xy₁] = [λx [G]xy₁] & [λx [F]y₁x] = [λx [G]y₁x])›
    using "identity:3[2]" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹F↓› and ‹G↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately show ?thesis
    using "≡S"(1) "&I" by blast
qed
    
AOT_theorem "p-identity-thm2:2[3]":
  ‹F = G ≡ ∀y₁∀y₂([λx [F]xy₁y₂] = [λx [G]xy₁y₂] &
 [λx [F]y₁xy₂] = [λx [G]y₁xy₂] &
 [λx [F]y₁y₂x] = [λx [G]y₁y₂x])›
proof -
  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y₁∀y₂([λx [F]xy₁y₂] = [λx [G]xy₁y₂] &
 [λx [F]y₁xy₂] = [λx [G]y₁xy₂] &
 [λx [F]y₁y₂x] = [λx [G]y₁y₂x])›
    using "identity:3[3]" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹F↓› and ‹G↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately show ?thesis
    using "≡S"(1) "&I" by blast
qed

AOT_theorem "p-identity-thm2:2[4]":
  ‹F = G ≡ ∀y₁∀y₂∀y₃([λx [F]xy₁y₂y₃] = [λx [G]xy₁y₂y₃] &
 [λx [F]y₁xy₂y₃] = [λx [G]y₁xy₂y₃] &
 [λx [F]y₁y₂xy₃] = [λx [G]y₁y₂xy₃] &
 [λx [F]y₁y₂y₃x] = [λx [G]y₁y₂y₃x])›
proof -
  AOT_have ‹F = G ≡ F↓ & G↓ & ∀y₁∀y₂∀y₃([λx [F]xy₁y₂y₃] = [λx [G]xy₁y₂y₃] &
 [λx [F]y₁xy₂y₃] = [λx [G]y₁xy₂y₃] &
 [λx [F]y₁y₂xy₃] = [λx [G]y₁y₂xy₃] &
 [λx [F]y₁y₂y₃x] = [λx [G]y₁y₂y₃x])›
    using "identity:3[4]" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹F↓› and ‹G↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately show ?thesis
    using "≡S"(1) "&I" by blast
qed

AOT_theorem "p-identity-thm2:2":
  ‹F = G ≡ ∀x₁...∀x_n «AOT_sem_proj_id x₁x_n (λ τ . «[F]τ¬) (λ τ . «[G]τ¬)¬›
proof -
  AOT_have ‹F = G ≡ F↓ & G↓ &
 ∀x₁...∀x_n «AOT_sem_proj_id x₁x_n (λ τ . «[F]τ¬) (λ τ . «[G]τ¬)¬›
    using "identity:3" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹F↓› and ‹G↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately show ?thesis
    using "≡S"(1) "&I" by blast
qed

AOT_theorem "p-identity-thm2:3":
  ‹p = q ≡ [λx p] = [λx q]›
proof -
  AOT_have ‹p = q ≡ p↓ & q↓ & [λx p] = [λx q]›
    using "identity:4" "df-rules-formulas[3]" "df-rules-formulas[4]"
          "→E" "&E" "≡I" "→I" by blast
  moreover AOT_have ‹p↓› and ‹q↓›
    by (auto simp: "cqt:2[const_var]"[axiom_inst])
  ultimately show ?thesis
    using "≡S"(1) "&I" by blast
qed

class AOT_Term_id_2 = AOT_Term_id + assumes "id-eq:1": ‹[v ⊨ α = α]›

instance κ :: AOT_Term_id_2
proof
  AOT_modally_strict {
    fix x
    {
      AOT_assume ‹O!x›
      moreover AOT_have ‹◻∀F([F]x ≡ [F]x)›
        using RN GEN "oth-class-taut:3:a" by fast
      ultimately AOT_have ‹O!x & O!x & ◻∀F([F]x ≡ [F]x)› using "&I" by simp
    }
    moreover {
      AOT_assume ‹A!x›
      moreover AOT_have ‹◻∀F(x[F] ≡ x[F])›
        using RN GEN "oth-class-taut:3:a" by fast
      ultimately AOT_have ‹A!x & A!x & ◻∀F(x[F] ≡ x[F])› using "&I" by simp
    }
    ultimately AOT_have ‹(O!x & O!x & ◻∀F([F]x ≡ [F]x)) ∨
 (A!x & A!x & ◻∀F(x[F] ≡ x[F]))›
      using "oa-exist:3" "∨I"(1) "∨I"(2) "∨E"(3) "raa-cor:1" by blast
    AOT_thus ‹x = x›
      using "identity:1"[THEN "df-rules-formulas[4]"] "→E" by blast
  }
qed

instance rel :: ("{AOT_κs,AOT_Term_id_2}") AOT_Term_id_2
proof
  AOT_modally_strict {
    fix F :: "<'a> AOT_var"
    AOT_have 0: ‹[λx₁...x_n [F]x₁...x_n] = F›
      by (simp add: "lambda-predicates:3"[axiom_inst])
    AOT_have ‹[λx₁...x_n [F]x₁...x_n]↓›
      by "cqt:2[lambda]"
    AOT_hence ‹[λx₁...x_n [F]x₁...x_n] = [λx₁...x_n [F]x₁...x_n]›
      using "lambda-predicates:1"[axiom_inst] "→E" by blast
    AOT_show ‹F = F› using "rule=E" 0 by force 
  }
qed

instance o :: AOT_Term_id_2
proof
  AOT_modally_strict {
    fix p
    AOT_have 0: ‹[λ p] = p›
      by (simp add: "lambda-predicates:3[zero]"[axiom_inst])
    AOT_have ‹[λ p]↓›
      by (rule "cqt:2[lambda0]"[axiom_inst])
    AOT_hence ‹[λ p] = [λ p]›
      using "lambda-predicates:1[zero]"[axiom_inst] "→E" by blast
    AOT_show ‹p = p› using "rule=E" 0 by force
  }
qed

instance prod :: (AOT_Term_id_2, AOT_Term_id_2) AOT_Term_id_2
proof
  AOT_modally_strict {
    fix α :: ‹('a×'b) AOT_var›
    AOT_show ‹α = α›
    proof (induct)
      AOT_show ‹τ = τ› if ‹τ↓› for τ :: ‹'a×'b›
        using that
      proof (induct τ)
        fix τ₁ :: 'a and τ₂ :: 'b
        AOT_assume ‹«(τ₁,τ₂)¬↓›
        AOT_hence ‹τ₁↓› and ‹τ₂↓›
          using "≡_dfE" "&E" tuple_denotes by blast+
        AOT_hence ‹τ₁ = τ₁› and ‹τ₂ = τ₂›
          using "id-eq:1"[unvarify α] by blast+
        AOT_thus ‹«(τ₁, τ₂)¬ = «(τ₁, τ₂)¬›
          by (metis "≡_dfI" "&I" tuple_identity_1)
      qed
    qed
  }
qed

AOT_register_type_constraints
  Term: ‹_::AOT_Term_id_2› ‹_::AOT_Term_id_2›
AOT_register_type_constraints
  Individual: ‹κ› ‹_::{AOT_κs, AOT_Term_id_2}›
AOT_register_type_constraints
  Relation: ‹<_::{AOT_κs, AOT_Term_id_2}>›

AOT_theorem "id-eq:2": ‹α = β → β = α›
  by (meson "rule=E" "deduction-theorem")

AOT_theorem "id-eq:3": ‹α = β & β = γ → α = γ›
  using "rule=E" "→I" "&E" by blast

AOT_theorem "id-eq:4": ‹α = β ≡ ∀γ (α = γ ≡ β = γ)›
proof (rule "≡I"; rule "→I")
  AOT_assume 0: ‹α = β›
  AOT_hence 1: ‹β = α› using "id-eq:2" "→E" by blast
  AOT_show ‹∀γ (α = γ ≡ β = γ)›
    by (rule GEN) (metis "≡I" "→I" 0 "1" "rule=E")
next
  AOT_assume ‹∀γ (α = γ ≡ β = γ)›
  AOT_hence ‹α = α ≡ β = α› using "∀E"(2) by blast
  AOT_hence ‹α = α → β = α› using "≡E"(1) "→I" by blast
  AOT_hence ‹β = α› using "id-eq:1" "→E" by blast
  AOT_thus ‹α = β› using "id-eq:2" "→E" by blast
qed

AOT_theorem "rule=I:1":
  assumes ‹τ↓›
  shows ‹τ = τ›
proof -
  AOT_have ‹∀α (α = α)›
    by (rule GEN) (metis "id-eq:1")
  AOT_thus ‹τ = τ› using assms "∀E" by blast
qed

AOT_theorem "rule=I:2[const_var]": "α = α"
  using "id-eq:1".

AOT_theorem "rule=I:2[lambda]":
  assumes ‹INSTANCE_OF_CQT_2(φ)›
  shows "[λν₁...ν_n φ{ν₁...ν_n}] = [λν₁...ν_n φ{ν₁...ν_n}]"
proof -
  AOT_have ‹∀α (α = α)›
    by (rule GEN) (metis "id-eq:1")
  moreover AOT_have ‹[λν₁...ν_n φ{ν₁...ν_n}]↓›
    using assms by (rule "cqt:2[lambda]"[axiom_inst])
  ultimately AOT_show ‹[λν₁...ν_n φ{ν₁...ν_n}] = [λν₁...ν_n φ{ν₁...ν_n}]›
    using assms "∀E" by blast
qed

lemmas "=I" = "rule=I:1" "rule=I:2[const_var]" "rule=I:2[lambda]"

AOT_theorem "rule-id-df:1":
  assumes ‹τ{α₁...α_n} =_df σ{α₁...α_n}› and ‹σ{τ₁...τ_n}↓›
  shows ‹τ{τ₁...τ_n} = σ{τ₁...τ_n}›
proof -
  AOT_have ‹σ{τ₁...τ_n}↓ → τ{τ₁...τ_n} = σ{τ₁...τ_n}›
    using "df-rules-terms[3]" assms(1) "&E" by blast
  AOT_thus ‹τ{τ₁...τ_n} = σ{τ₁...τ_n}›
    using assms(2) "→E" by blast
qed

AOT_theorem "rule-id-df:1[zero]":
  assumes ‹τ =_df σ› and ‹σ↓›
  shows ‹τ = σ›
proof -
  AOT_have ‹σ↓ → τ = σ›
    using "df-rules-terms[4]" assms(1) "&E" by blast
  AOT_thus ‹τ = σ›
    using assms(2) "→E" by blast
qed

AOT_theorem "rule-id-df:2:a":
  assumes ‹τ{α₁...α_n} =_df σ{α₁...α_n}› and ‹σ{τ₁...τ_n}↓› and ‹φ{τ{τ₁...τ_n}}›
  shows ‹φ{σ{τ₁...τ_n}}›
proof -
  AOT_have ‹τ{τ₁...τ_n} = σ{τ₁...τ_n}› using "rule-id-df:1" assms(1,2) by blast
  AOT_thus ‹φ{σ{τ₁...τ_n}}› using assms(3) "rule=E" by blast
qed

AOT_theorem "rule-id-df:2:a[2]":
  assumes ‹τ{«(α₁,α₂)¬} =_df σ{«(α₁,α₂)¬}›
    and ‹σ{«(τ₁,τ₂)¬}↓›
      and ‹φ{τ{«(τ₁,τ₂)¬}}›
  shows ‹φ{σ{«(τ₁::'a::AOT_Term_id_2,τ₂::'b::AOT_Term_id_2)¬}}›
proof -
  AOT_have ‹τ{«(τ₁,τ₂)¬} = σ{«(τ₁,τ₂)¬}›
    using "rule-id-df:1" assms(1,2) by auto
  AOT_thus ‹φ{σ{«(τ₁,τ₂)¬}}› using assms(3) "rule=E" by blast
qed

AOT_theorem "rule-id-df:2:a[zero]":
  assumes ‹τ =_df σ› and ‹σ↓› and ‹φ{τ}›
  shows ‹φ{σ}›
proof -
  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
  AOT_thus ‹φ{σ}› using assms(3) "rule=E" by blast
qed

lemmas "=_dfE" = "rule-id-df:2:a" "rule-id-df:2:a[zero]"

AOT_theorem "rule-id-df:2:b":
  assumes ‹τ{α₁...α_n} =_df σ{α₁...α_n}› and ‹σ{τ₁...τ_n}↓› and ‹φ{σ{τ₁...τ_n}}›
  shows ‹φ{τ{τ₁...τ_n}}›
proof -
  AOT_have ‹τ{τ₁...τ_n} = σ{τ₁...τ_n}›
    using "rule-id-df:1" assms(1,2) by blast
  AOT_hence ‹σ{τ₁...τ_n} = τ{τ₁...τ_n}›
    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
  AOT_thus ‹φ{τ{τ₁...τ_n}}› using assms(3) "rule=E" by blast
qed

AOT_theorem "rule-id-df:2:b[2]":
  assumes ‹τ{«(α₁,α₂)¬} =_df σ{«(α₁,α₂)¬}›
      and ‹σ{«(τ₁,τ₂)¬}↓›
      and ‹φ{σ{«(τ₁,τ₂)¬}}›
  shows ‹φ{τ{«(τ₁::'a::AOT_Term_id_2,τ₂::'b::AOT_Term_id_2)¬}}›
proof -
  AOT_have ‹τ{«(τ₁,τ₂)¬} = σ{«(τ₁,τ₂)¬}›
    using "=I"(1) "rule-id-df:2:a[2]" RAA(1) assms(1,2) "→I" by metis
  AOT_hence ‹σ{«(τ₁,τ₂)¬} = τ{«(τ₁,τ₂)¬}›
    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
  AOT_thus ‹φ{τ{«(τ₁,τ₂)¬}}› using assms(3) "rule=E" by blast
qed

AOT_theorem "rule-id-df:2:b[zero]":
  assumes ‹τ =_df σ› and ‹σ↓› and ‹φ{σ}›
  shows ‹φ{τ}›
proof -
  AOT_have ‹τ = σ› using "rule-id-df:1[zero]" assms(1,2) by blast
  AOT_hence ‹σ = τ›
    using "rule=E" "=I"(1) "t=t-proper:1" "→E" by fast
  AOT_thus ‹φ{τ}› using assms(3) "rule=E" by blast
qed

lemmas "=_dfI" = "rule-id-df:2:b" "rule-id-df:2:b[zero]"

AOT_theorem "free-thms:1": ‹τ↓ ≡ ∃β (β = τ)›
  by (metis "∃E" "rule=I:1" "t=t-proper:2" "→I" "∃I"(1) "≡I" "→E")

AOT_theorem "free-thms:2": ‹∀α φ{α} → (∃β (β = τ) → φ{τ})›
  by (metis "∃E" "rule=E" "cqt:2[const_var]"[axiom_inst] "→I" "∀E"(1))

AOT_theorem "free-thms:3[const_var]": ‹∃β (β = α)›
  by (meson "∃I"(2) "id-eq:1")

AOT_theorem "free-thms:3[lambda]":
  assumes ‹INSTANCE_OF_CQT_2(φ)›
  shows ‹∃β (β = [λν₁...ν_n φ{ν₁...ν_n}])›
  by (meson "=I"(3) assms "cqt:2[lambda]"[axiom_inst] "existential:1")

AOT_theorem "free-thms:4[rel]":
  ‹([Π]κ₁...κ_n ∨ κ₁...κ_n[Π]) → ∃β (β = Π)›
  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst]
            "cqt:5:b"[axiom_inst] "→I" "∃I"(1))

AOT_theorem "free-thms:4[vars]":
  ‹([Π]κ₁...κ_n ∨ κ₁...κ_n[Π]) → ∃β₁...∃β_n (β₁...β_n = κ₁...κ_n)›
  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst]
            "cqt:5:b"[axiom_inst] "→I" "∃I"(1))

AOT_theorem "free-thms:4[1,rel]":
  ‹([Π]κ ∨ κ[Π]) → ∃β (β = Π)›
  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a"[axiom_inst]
            "cqt:5:b"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[1,1]":
  ‹([Π]κ ∨ κ[Π]) → ∃β (β = κ)›
  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a"[axiom_inst]
            "cqt:5:b"[axiom_inst] "→I" "∃I"(1))

AOT_theorem "free-thms:4[2,rel]":
  ‹([Π]κ₁κ₂ ∨ κ₁κ₂[Π]) → ∃β (β = Π)›
  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[2]"[axiom_inst]
            "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[2,1]":
  ‹([Π]κ₁κ₂ ∨ κ₁κ₂[Π]) → ∃β (β = κ₁)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[2]"[axiom_inst]
            "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[2,2]":
  ‹([Π]κ₁κ₂ ∨ κ₁κ₂[Π]) → ∃β (β = κ₂)›
  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[2]"[axiom_inst]
            "cqt:5:b[2]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[3,rel]":
  ‹([Π]κ₁κ₂κ₃ ∨ κ₁κ₂κ₃[Π]) → ∃β (β = Π)›
  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[3]"[axiom_inst]
            "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[3,1]":
  ‹([Π]κ₁κ₂κ₃ ∨ κ₁κ₂κ₃[Π]) → ∃β (β = κ₁)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst]
            "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[3,2]":
  ‹([Π]κ₁κ₂κ₃ ∨ κ₁κ₂κ₃[Π]) → ∃β (β = κ₂)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[3]"[axiom_inst]
            "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[3,3]":
  ‹([Π]κ₁κ₂κ₃ ∨ κ₁κ₂κ₃[Π]) → ∃β (β = κ₃)›
  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[3]"[axiom_inst]
            "cqt:5:b[3]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[4,rel]":
  ‹([Π]κ₁κ₂κ₃κ₄ ∨ κ₁κ₂κ₃κ₄[Π]) → ∃β (β = Π)›
  by (metis "rule=I:1" "&E"(1) "∨E"(1) "cqt:5:a[4]"[axiom_inst]
            "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[4,1]":
  ‹([Π]κ₁κ₂κ₃κ₄ ∨ κ₁κ₂κ₃κ₄[Π]) → ∃β (β = κ₁)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst]
            "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[4,2]":
  ‹([Π]κ₁κ₂κ₃κ₄ ∨ κ₁κ₂κ₃κ₄[Π]) → ∃β (β = κ₂)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst]
            "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[4,3]":
  ‹([Π]κ₁κ₂κ₃κ₄ ∨ κ₁κ₂κ₃κ₄[Π]) → ∃β (β = κ₃)›
  by (metis "rule=I:1" "&E" "∨E"(1) "cqt:5:a[4]"[axiom_inst]
            "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))
AOT_theorem "free-thms:4[4,4]":
  ‹([Π]κ₁κ₂κ₃κ₄ ∨ κ₁κ₂κ₃κ₄[Π]) → ∃β (β = κ₄)›
  by (metis "rule=I:1" "&E"(2) "∨E"(1) "cqt:5:a[4]"[axiom_inst]
            "cqt:5:b[4]"[axiom_inst] "→I" "∃I"(1))

AOT_theorem "ex:1:a": ‹∀α α↓›
  by (rule GEN) (fact "cqt:2[const_var]"[axiom_inst])
AOT_theorem "ex:1:b": ‹∀α∃β(β = α)›
  by (rule GEN) (fact "free-thms:3[const_var]")

AOT_theorem "ex:2:a": ‹◻α↓›
  by (rule RN) (fact "cqt:2[const_var]"[axiom_inst])
AOT_theorem "ex:2:b": ‹◻∃β(β = α)›
  by (rule RN) (fact "free-thms:3[const_var]")

AOT_theorem "ex:3:a": ‹◻∀α α↓›
  by (rule RN) (fact "ex:1:a")
AOT_theorem "ex:3:b": ‹◻∀α∃β(β = α)›
  by (rule RN) (fact "ex:1:b")

AOT_theorem "ex:4:a": ‹∀α ◻α↓›
  by (rule GEN; rule RN) (fact "cqt:2[const_var]"[axiom_inst])
AOT_theorem "ex:4:b": ‹∀α◻∃β(β = α)›
  by (rule GEN; rule RN) (fact "free-thms:3[const_var]")

AOT_theorem "ex:5:a": ‹◻∀α ◻α↓›
  by (rule RN) (simp add: "ex:4:a")
AOT_theorem "ex:5:b": ‹◻∀α◻∃β(β = α)›
  by (rule RN) (simp add: "ex:4:b")

AOT_theorem "all-self=:1": ‹◻∀α(α = α)›
  by (rule RN; rule GEN) (fact "id-eq:1")
AOT_theorem "all-self=:2": ‹∀α◻(α = α)›
  by (rule GEN; rule RN) (fact "id-eq:1")

AOT_theorem "id-nec:1": ‹α = β → ◻(α = β)›
proof(rule "→I")
  AOT_assume ‹α = β›
  moreover AOT_have ‹◻(α = α)›
    by (rule RN) (fact "id-eq:1")
  ultimately AOT_show ‹◻(α = β)› using "rule=E" by fast
qed

AOT_theorem "id-nec:2": ‹τ = σ → ◻(τ = σ)›
proof(rule "→I")
  AOT_assume asm: ‹τ = σ›
  moreover AOT_have ‹τ↓›
    using calculation "t=t-proper:1" "→E" by blast
  moreover AOT_have ‹◻(τ = τ)›
    using calculation "all-self=:2" "∀E"(1) by blast
  ultimately AOT_show ‹◻(τ = σ)› using "rule=E" by fast
qed

AOT_theorem "term-out:1": ‹φ{α} ≡ ∃β (β = α & φ{β})›
proof (rule "≡I"; rule "→I")
  AOT_assume asm: ‹φ{α}›
  AOT_show ‹∃β (β = α & φ{β})›
    by (rule "∃I"(2)[where β=α]; rule "&I")
       (auto simp: "id-eq:1" asm)
next
  AOT_assume 0: ‹∃β (β = α & φ{β})›
  AOT_obtain β where ‹β = α & φ{β}›
    using "∃E"[rotated, OF 0] by blast
  AOT_thus ‹φ{α}› using "&E" "rule=E" by blast
qed

AOT_theorem "term-out:2": ‹τ↓ → (φ{τ} ≡ ∃α(α = τ & φ{α}))›
proof(rule "→I")
  AOT_assume ‹τ↓›
  moreover AOT_have ‹∀α (φ{α} ≡ ∃β (β = α & φ{β}))›
    by (rule GEN) (fact "term-out:1")
  ultimately AOT_show ‹φ{τ} ≡ ∃α(α = τ & φ{α})›
    using "∀E" by blast
qed

AOT_theorem "term-out:3":
  ‹(φ{α} & ∀β(φ{β} → β = α)) ≡ ∀β(φ{β} ≡ β = α)›
  apply (rule "≡I"; rule "→I")
   apply (frule "&E"(1))
   apply (drule "&E"(2))
   apply (rule GEN; rule "≡I"; rule "→I")
  using "rule-ui:2[const_var]" "vdash-properties:5"
    apply blast
   apply (meson "rule=E" "id-eq:1")
  apply (rule "&I")
  using "id-eq:1" "≡E"(2) "rule-ui:3"
   apply blast
  apply (rule GEN; rule "→I")
  using "≡E"(1) "rule-ui:2[const_var]"
  by blast

(* Note: generalized alphabetic variant of the last theorem. *)
AOT_theorem "term-out:4":
  ‹(φ{β} & ∀α(φ{α} → α = β)) ≡ ∀α(φ{α} ≡ α = β)›
  using "term-out:3" .

(* TODO: Provide a nicer mechanism for introducing custom binders. *)
AOT_define AOT_exists_unique :: ‹α ==> φ ==> φ› "uniqueness:1":
  ‹«AOT_exists_unique φ¬ ≡_df ∃α (φ{α} & ∀β (φ{β} → β = α))›
syntax (input) "_AOT_exists_unique" :: ‹α ==> φ ==> φ› (‹∃!_ _› [1,40])
syntax (output) "_AOT_exists_unique" :: ‹α ==> φ ==> φ› (‹∃!_'(_')› [1,40])
AOT_syntax_print_translations
  "_AOT_exists_unique τ φ" <= "CONST AOT_exists_unique (_abs τ φ)"
syntax
   "_AOT_exists_unique_ellipse" :: ‹id_position ==> id_position ==> φ ==> φ›
   (‹∃!_...∃!_ _› [1,40])
parse_ast_translation‹
 (🍋‹_AOT_exists_unique_ellipse›,
 fn ctx => fn [a,b,c] => Ast.mk_appl (Ast.Constant "AOT_exists_unique")
 [parseEllipseList "_AOT_vars" ctx [a,b],c]),
 (🍋‹_AOT_exists_unique›,
 AOT_restricted_binder
 🍋‹AOT_exists_unique›
 🍋‹AOT_conj›)]›
print_translation‹AOT_syntax_print_translations [
 AOT_preserve_binder_abs_tr'
 🍋‹AOT_exists_unique›
 🍋‹_AOT_exists_unique›
 (🍋‹_AOT_exists_unique_ellipse›, true)
 🍋‹AOT_conj›,
 AOT_binder_trans
 @{theory}
 @{binding "AOT_exists_unique_binder"}
 🍋‹_AOT_exists_unique›
 ›


context AOT_meta_syntax
begin
notation AOT_exists_unique (binder ‹\∃!› 20)
end
context AOT_no_meta_syntax
begin
no_notation AOT_exists_unique (binder ‹\∃!› 20)
end

AOT_theorem "uniqueness:2": ‹∃!α φ{α} ≡ ∃α∀β(φ{β} ≡ β = α)›
proof(rule "≡I"; rule "→I")
    AOT_assume ‹∃!α φ{α}›
    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
      using "uniqueness:1" "≡_dfE" by blast
    then AOT_obtain α where ‹φ{α} & ∀β (φ{β} → β = α)›
      using "instantiation"[rotated] by blast
    AOT_hence ‹∀β(φ{β} ≡ β = α)›
      using "term-out:3" "≡E" by blast
    AOT_thus ‹∃α∀β(φ{β} ≡ β = α)›
      using "∃I" by fast
next
    AOT_assume ‹∃α∀β(φ{β} ≡ β = α)›
    then AOT_obtain α where ‹∀β (φ{β} ≡ β = α)›
      using "instantiation"[rotated] by blast
    AOT_hence ‹φ{α} & ∀β (φ{β} → β = α)›
      using "term-out:3" "≡E" by blast
    AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
      using "∃I" by fast
    AOT_thus ‹∃!α φ{α}›
      using "uniqueness:1" "≡_dfI" by blast
qed

AOT_theorem "uni-most": ‹∃!α φ{α} → ∀β∀γ((φ{β} & φ{γ}) → β = γ)›
proof(rule "→I"; rule GEN; rule GEN; rule "→I")
  fix β γ
  AOT_assume ‹∃!α φ{α}›
  AOT_hence ‹∃α∀β(φ{β} ≡ β = α)›
    using "uniqueness:2" "≡E" by blast
  then AOT_obtain α where ‹∀β(φ{β} ≡ β = α)›
    using "instantiation"[rotated] by blast
  moreover AOT_assume ‹φ{β} & φ{γ}›
  ultimately AOT_have ‹β = α› and ‹γ = α›
    using "∀E"(2) "&E" "≡E"(1,2) by blast+
  AOT_thus ‹β = γ›
    by (metis "rule=E" "id-eq:2" "→E")
qed

AOT_theorem "nec-exist-!": ‹∀α(φ{α} → ◻φ{α}) → (∃!α φ{α} → ∃!α ◻φ{α})›
proof (rule "→I"; rule "→I")
  AOT_assume a: ‹∀α(φ{α} → ◻φ{α})›
  AOT_assume ‹∃!α φ{α}›
  AOT_hence ‹∃α (φ{α} & ∀β (φ{β} → β = α))›
    using "uniqueness:1" "≡_dfE" by blast
  then AOT_obtain α where ξ: ‹φ{α} & ∀β (φ{β} → β = α)›
    using "instantiation"[rotated] by blast
  AOT_have ‹◻φ{α}›
    using ξ a "&E" "∀E" "→E" by fast
  moreover AOT_have ‹∀β (◻φ{β} → β = α)›
    apply (rule GEN; rule "→I")
    using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "→E"]
          "qml:2"[axiom_inst, THEN "→E"] by blast
  ultimately AOT_have ‹(◻φ{α} & ∀β (◻φ{β} → β = α))›
    using "&I" by blast
  AOT_thus ‹∃!α ◻φ{α}›
    using "uniqueness:1" "≡_dfI" "∃I" by fast
qed

subsection‹The Theory of Actuality and Descriptions›
text‹\label{PLM: 9.8}›

AOT_theorem "act-cond": ‹\A(φ → ψ) → (\Aφ → \Aψ)›
  using "→I" "≡E"(1) "logic-actual-nec:2"[axiom_inst] by blast

AOT_theorem "nec-imp-act": ‹◻φ → \Aφ›
  by (metis "act-cond" "contraposition:1[2]" "≡E"(4)
            "qml:2"[THEN act_closure, axiom_inst]
            "qml-act:2"[axiom_inst] RAA(1) "→E" "→I")

AOT_theorem "act-conj-act:1": ‹\A(\Aφ → φ)›
  using "→I" "≡E"(2) "logic-actual-nec:2"[axiom_inst]
        "logic-actual-nec:4"[axiom_inst] by blast

AOT_theorem "act-conj-act:2": ‹\A(φ → \Aφ)›
  by (metis "→I" "≡E"(2, 4) "logic-actual-nec:2"[axiom_inst]
            "logic-actual-nec:4"[axiom_inst] RAA(1))

AOT_theorem "act-conj-act:3": ‹(\Aφ & \Aψ) → \A(φ & ψ)›
proof -
  AOT_have ‹◻(φ → (ψ → (φ & ψ)))›
    by (rule RN) (fact Adjunction)
  AOT_hence ‹\A(φ → (ψ → (φ & ψ)))›
    using "nec-imp-act" "→E" by blast
  AOT_hence ‹\Aφ → \A(ψ → (φ & ψ))›
    using "act-cond" "→E" by blast
  moreover AOT_have ‹\A(ψ → (φ & ψ)) → (\Aψ → \A(φ & ψ))›
    by (fact "act-cond")
  ultimately AOT_have ‹\Aφ → (\Aψ → \A(φ & ψ))›
    using "→I" "→E" by metis
  AOT_thus ‹(\Aφ & \Aψ) → \A(φ & ψ)›
    by (metis Importation "→E")
qed

AOT_theorem "act-conj-act:4": ‹\A(\Aφ ≡ φ)›
proof -
  AOT_have ‹(\A(\Aφ → φ) & \A(φ → \Aφ)) → \A((\Aφ → φ) & (φ → \Aφ))›
    by (fact "act-conj-act:3")
  moreover AOT_have ‹\A(\Aφ → φ) & \A(φ → \Aφ)›
    using "&I" "act-conj-act:1" "act-conj-act:2" by simp
  ultimately AOT_have ζ: ‹\A((\Aφ → φ) & (φ → \Aφ))›
    using "→E" by blast
  AOT_have ‹\A(((\Aφ → φ) & (φ → \Aφ)) → (\Aφ ≡ φ))›
    using "conventions:3"[THEN "df-rules-formulas[2]",
                          THEN act_closure, axiom_inst] by blast
  AOT_hence ‹\A((\Aφ → φ) & (φ → \Aφ)) → \A(\Aφ ≡ φ)›
    using "act-cond" "→E" by blast
  AOT_thus ‹\A(\Aφ ≡ φ)› using ζ "→E" by blast
qed

(* TODO: Consider introducing AOT_inductive. *)
inductive arbitrary_actualization for φ where
  ‹arbitrary_actualization φ «\Aφ¬›
| ‹arbitrary_actualization φ «\Aψ¬› if ‹arbitrary_actualization φ ψ›
declare arbitrary_actualization.cases[AOT]
        arbitrary_actualization.induct[AOT]
        arbitrary_actualization.simps[AOT]
        arbitrary_actualization.intros[AOT]
syntax arbitrary_actualization :: ‹φ' ==> φ' ==> AOT_prop›
  (‹ARBITRARY'_ACTUALIZATION'(_,_')›)

notepad
begin
  AOT_modally_strict {
    fix φ
    AOT_have ‹ARBITRARY_ACTUALIZATION(\Aφ ≡ φ, \A(\Aφ ≡ φ))›
      using AOT_PLM.arbitrary_actualization.intros by metis
    AOT_have ‹ARBITRARY_ACTUALIZATION(\Aφ ≡ φ, \A\A(\Aφ ≡ φ))›
      using AOT_PLM.arbitrary_actualization.intros by metis
    AOT_have ‹ARBITRARY_ACTUALIZATION(\Aφ ≡ φ, \A\A\A(\Aφ ≡ φ))›
      using AOT_PLM.arbitrary_actualization.intros by metis
  }
end


AOT_theorem "closure-act:1":
  assumes ‹ARBITRARY_ACTUALIZATION(\Aφ ≡ φ, ψ)›
  shows ‹ψ›
using assms proof(induct)
  case 1
  AOT_show ‹\A(\Aφ ≡ φ)›
    by (simp add: "act-conj-act:4")
next
  case (2 ψ)
  AOT_thus ‹\Aψ›
    by (metis arbitrary_actualization.simps "≡E"(1)
              "logic-actual-nec:4"[axiom_inst])
qed

AOT_theorem "closure-act:2": ‹∀α \A(\Aφ{α} ≡ φ{α})›
  by (simp add: "act-conj-act:4" "∀I")

AOT_theorem "closure-act:3": ‹\A∀α \A(\Aφ{α} ≡ φ{α})›
  by (metis (no_types, lifting) "act-conj-act:4" "≡E"(1,2) "∀I"
            "logic-actual-nec:3"[axiom_inst]
            "logic-actual-nec:4"[axiom_inst])

AOT_theorem "closure-act:4": ‹\A∀α₁...∀α_n \A(\Aφ{α₁...α_n} ≡ φ{α₁...α_n})›
  using "closure-act:3" .

AOT_act_theorem "RA[1]":
  assumes ‹\⊨ φ›
  shows ‹\⊨ \Aφ›
  ― ‹While this proof is rejected in PLM,
 we merely state it as modally-fragile rule,
 which addresses the concern in PLM.›
  using "¬¬E" assms "≡E"(3) "logic-actual"[act_axiom_inst]
        "logic-actual-nec:1"[axiom_inst] "modus-tollens:2" by blast
AOT_theorem "RA[2]":
  assumes ‹\⊨_\◻ φ›
  shows ‹\⊨_\◻ \Aφ›
  ― ‹This rule is in fact a consequence of RN and
 does not require an appeal to the semantics itself.›
  using RN assms "nec-imp-act" "vdash-properties:5" by blast
AOT_theorem "RA[3]":
  assumes ‹Γ \⊨_\◻ φ›
  shows ‹\AΓ \⊨_\◻ \Aφ›
  text‹This rule is only derivable from the semantics,
 but apparently no proof actually relies on it.
 If this turns out to be required, it is valid to derive it from the
 semantics just like RN, but we refrain from doing so, unless necessary.›
  (*  using assms by (meson AOT_sem_act imageI) *)
  oops ― ‹discard the rule›

AOT_act_theorem "ANeg:1": ‹¬\Aφ ≡ ¬φ›
  by (simp add: "RA[1]" "contraposition:1[1]" "deduction-theorem"
                "≡I" "logic-actual"[act_axiom_inst])

AOT_act_theorem "ANeg:2": ‹¬\A¬φ ≡ φ›
  using "ANeg:1" "≡I" "≡E"(5) "useful-tautologies:1"
        "useful-tautologies:2" by blast

AOT_theorem "Act-Basic:1": ‹\Aφ ∨ \A¬φ›
  by (meson "∨I"(1,2) "≡E"(2) "logic-actual-nec:1"[axiom_inst] "raa-cor:1")

AOT_theorem "Act-Basic:2": ‹\A(φ & ψ) ≡ (\Aφ & \Aψ)›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹\A(φ & ψ)›
  moreover AOT_have ‹\A((φ & ψ) → φ)›
    by (simp add: "RA[2]" "Conjunction Simplification"(1))
  moreover AOT_have ‹\A((φ & ψ) → ψ)›
    by (simp add: "RA[2]" "Conjunction Simplification"(2))
  ultimately AOT_show ‹\Aφ & \Aψ›
    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
next
  AOT_assume ‹\Aφ & \Aψ›
  AOT_thus ‹\A(φ & ψ)›
    using "act-conj-act:3" "vdash-properties:6" by blast
qed

AOT_theorem "Act-Basic:3": ‹\A(φ ≡ ψ) ≡ (\A(φ → ψ) & \A(ψ → φ))›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹\A(φ ≡ ψ)›
  moreover AOT_have ‹\A((φ ≡ ψ) → (φ → ψ))›
    by (simp add: "RA[2]" "deduction-theorem" "≡E"(1))
  moreover AOT_have ‹\A((φ ≡ ψ) → (ψ → φ))›
    by (simp add: "RA[2]" "deduction-theorem" "≡E"(2))
  ultimately AOT_show ‹\A(φ → ψ) & \A(ψ → φ)›
    using "act-cond"[THEN "→E", THEN "→E"] "&I" by metis
next
  AOT_assume ‹\A(φ → ψ) & \A(ψ → φ)›
  AOT_hence ‹\A((φ → ψ) & (ψ → φ))›
    by (metis "act-conj-act:3" "vdash-properties:10")
  moreover AOT_have ‹\A(((φ → ψ) & (ψ → φ)) → (φ ≡ ψ))›
    by (simp add: "conventions:3" "RA[2]" "df-rules-formulas[2]"
                  "vdash-properties:1[2]")
  ultimately AOT_show ‹\A(φ ≡ ψ)›
    using "act-cond"[THEN "→E", THEN "→E"] by metis
qed

AOT_theorem "Act-Basic:4": ‹(\A(φ → ψ) & \A(ψ → φ)) ≡ (\Aφ ≡ \Aψ)›
proof (rule "≡I"; rule "→I")
  AOT_assume 0: ‹\A(φ → ψ) & \A(ψ → φ)›
  AOT_show ‹\Aφ ≡ \Aψ›
    using 0 "&E" "act-cond"[THEN "→E", THEN "→E"] "≡I" "→I" by metis
next
  AOT_assume ‹\Aφ ≡ \Aψ›
  AOT_thus ‹\A(φ → ψ) & \A(ψ → φ)›
    by (metis "→I" "logic-actual-nec:2"[axiom_inst] "≡E"(1,2) "&I")
qed

AOT_theorem "Act-Basic:5": ‹\A(φ ≡ ψ) ≡ (\Aφ ≡ \Aψ)›
  using "Act-Basic:3" "Act-Basic:4" "≡E"(5) by blast

AOT_theorem "Act-Basic:6": ‹\Aφ ≡ ◻\Aφ›
  by (simp add: "≡I" "qml:2"[axiom_inst] "qml-act:1"[axiom_inst])

AOT_theorem "Act-Basic:7": ‹\A◻φ → ◻\Aφ›
  by (metis "Act-Basic:6" "→I" "→E" "≡E"(1,2) "nec-imp-act"
            "qml-act:2"[axiom_inst])

AOT_theorem "Act-Basic:8": ‹◻φ → ◻\Aφ›
  using "Hypothetical Syllogism" "nec-imp-act" "qml-act:1"[axiom_inst] by blast

AOT_theorem "Act-Basic:9": ‹\A(φ ∨ ψ) ≡ (\Aφ ∨ \Aψ)›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹\A(φ ∨ ψ)›
  AOT_thus ‹\Aφ ∨ \Aψ›
  proof (rule "raa-cor:3")
    AOT_assume ‹¬(\Aφ ∨ \Aψ)›
    AOT_hence ‹¬\Aφ & ¬\Aψ›
      by (metis "≡E"(1) "oth-class-taut:5:d")
    AOT_hence ‹\A¬φ & \A¬ψ›
      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] "&E" "&I" by metis
    AOT_hence ‹\A(¬φ & ¬ψ)›
      using "≡E" "Act-Basic:2" by metis
    moreover AOT_have ‹\A((¬φ & ¬ψ) ≡ ¬(φ ∨ ψ))›
      using "RA[2]" "≡E"(6) "oth-class-taut:3:a" "oth-class-taut:5:d" by blast
    moreover AOT_have ‹\A(¬φ & ¬ψ) ≡ \A(¬(φ ∨ ψ))›
      using calculation(2) by (metis "Act-Basic:5" "≡E"(1))
    ultimately AOT_have ‹\A(¬(φ ∨ ψ))› using "≡E" by blast
    AOT_thus ‹¬\A(φ ∨ ψ)›
      using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by auto
  qed
next
  AOT_assume ‹\Aφ ∨ \Aψ›
  AOT_thus ‹\A(φ ∨ ψ)›
    by (meson "RA[2]" "act-cond" "∨I"(1) "∨E"(1) "Disjunction Addition"(1,2))
qed

AOT_theorem "Act-Basic:10": ‹\A∃α φ{α} ≡ ∃α \Aφ{α}›
proof -
  AOT_have θ: ‹¬\A∀α ¬φ{α} ≡ ¬∀α \A¬φ{α}›
    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
       (metis "logic-actual-nec:3"[axiom_inst])
  AOT_have ξ: ‹¬∀α \A¬φ{α} ≡ ¬∀α ¬\Aφ{α}›
    by (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
       (rule "logic-actual-nec:1"[THEN universal_closure,
               axiom_inst, THEN "cqt-basic:3"[THEN "→E"]])
  AOT_have ‹\A(∃α φ{α}) ≡ \A(¬∀α ¬φ{α})›
    using "conventions:4"[THEN "df-rules-formulas[1]",
                          THEN act_closure, axiom_inst]
          "conventions:4"[THEN "df-rules-formulas[2]",
                          THEN act_closure, axiom_inst]
    "Act-Basic:4"[THEN "≡E"(1)] "&I" "Act-Basic:5"[THEN "≡E"(2)] by metis
  also AOT_have ‹… ≡ ¬\A∀α ¬φ{α}›
    by (simp add: "logic-actual-nec:1" "vdash-properties:1[2]")
  also AOT_have ‹… ≡ ¬∀α \A ¬φ{α}› using θ by blast
  also AOT_have ‹… ≡ ¬∀α ¬\A φ{α}› using ξ by blast
  also AOT_have ‹… ≡ ∃α \A φ{α}›
    using "conventions:4"[THEN "≡Df"] by (metis "≡E"(6) "oth-class-taut:3:a")
  finally AOT_show ‹\A∃α φ{α} ≡ ∃α \Aφ{α}› .
qed


AOT_theorem "Act-Basic:11":
  ‹\A∀α(φ{α} ≡ ψ{α}) ≡ ∀α(\Aφ{α} ≡ \Aψ{α})›
proof(rule "≡I"; rule "→I")
  AOT_assume ‹\A∀α(φ{α} ≡ ψ{α})›
  AOT_hence ‹∀α\A(φ{α} ≡ ψ{α})›
    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] by blast
  AOT_hence ‹\A(φ{α} ≡ ψ{α})› for α using "∀E" by blast
  AOT_hence ‹\Aφ{α} ≡ \Aψ{α}› for α by (metis "Act-Basic:5" "≡E"(1))
  AOT_thus ‹∀α(\Aφ{α} ≡ \Aψ{α})› by (rule "∀I")
next
  AOT_assume ‹∀α(\Aφ{α} ≡ \Aψ{α})›
  AOT_hence ‹\Aφ{α} ≡ \Aψ{α}› for α using "∀E" by blast
  AOT_hence ‹\A(φ{α} ≡ ψ{α})› for α by (metis "Act-Basic:5" "≡E"(2))
  AOT_hence ‹∀α \A(φ{α} ≡ ψ{α})› by (rule "∀I")
  AOT_thus ‹\A∀α(φ{α} ≡ ψ{α})›
    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
qed

AOT_act_theorem "act-quant-uniq":
  ‹∀β(\Aφ{β} ≡ β = α) ≡ ∀β(φ{β} ≡ β = α)›
proof(rule "≡I"; rule "→I")
  AOT_assume ‹∀β(\Aφ{β} ≡ β = α)›
  AOT_hence ‹\Aφ{β} ≡ β = α› for β using "∀E" by blast
  AOT_hence ‹φ{β} ≡ β = α› for β
    using "≡I" "→I" "RA[1]" "≡E"(1,2) "logic-actual"[act_axiom_inst] "→E"
    by metis
  AOT_thus ‹∀β(φ{β} ≡ β = α)› by (rule "∀I")
next
  AOT_assume ‹∀β(φ{β} ≡ β = α)›
  AOT_hence ‹φ{β} ≡ β = α› for β using "∀E" by blast
  AOT_hence ‹\Aφ{β} ≡ β = α› for β
    using "≡I" "→I" "RA[1]" "≡E"(1,2) "logic-actual"[act_axiom_inst] "→E"
    by metis
  AOT_thus ‹∀β(\Aφ{β} ≡ β = α)› by (rule "∀I")
qed

AOT_act_theorem "fund-cont-desc": ‹x = \ιx(φ{x}) ≡ ∀z(φ{z} ≡ z = x)›
  using descriptions[axiom_inst] "act-quant-uniq" "≡E"(5) by fast

AOT_act_theorem hintikka: ‹x = \ιx(φ{x}) ≡ (φ{x} & ∀z (φ{z} → z = x))›
  using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3"
        "fund-cont-desc" "≡E"(5) by blast


locale russell_axiom =
  fixes ψ
  assumes ψ_denotes_asm: "[v ⊨ ψ{κ}] ==> [v ⊨ κ↓]"
begin
AOT_act_theorem "russell-axiom":
  ‹ψ{\ιx φ{x}} ≡ ∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
proof -
  AOT_have b: ‹∀x (x = \ιx φ{x} ≡ (φ{x} & ∀z(φ{z} → z = x)))›
    using hintikka "∀I" by fast
  show ?thesis
  proof(rule "≡I"; rule "→I")
    AOT_assume c: ‹ψ{\ιx φ{x}}›
    AOT_hence d: ‹\ιx φ{x}↓›
      using ψ_denotes_asm by blast
    AOT_hence ‹∃y (y = \ιx φ{x})›
      by (metis "rule=I:1" "existential:1")
    then AOT_obtain a where a_def: ‹a = \ιx φ{x}›
      using "instantiation"[rotated] by blast
    moreover AOT_have ‹a = \ιx φ{x} ≡ (φ{a} & ∀z(φ{z} → z = a))›
      using b "∀E" by blast
    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a)›
      using "≡E" by blast
    moreover AOT_have ‹ψ{a}›
    proof - 
      AOT_have 1: ‹∀x∀y(x = y → y = x)›
        by (simp add: "id-eq:2" "universal-cor")
      AOT_have ‹a = \ιx φ{x} → \ιx φ{x} = a›
        by (rule "∀E"(1)[where τ="«\<iota>x φ{x}¬"]; rule "∀E"(2)[where β=a])
           (auto simp: 1 d "universal-cor")
      AOT_thus ‹ψ{a}›
        using a_def c "rule=E" "→E" by blast
    qed
    ultimately AOT_have ‹φ{a} & ∀z(φ{z} → z = a) & ψ{a}› by (rule "&I")
    AOT_thus ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})› by (rule "∃I")
  next
    AOT_assume ‹∃x(φ{x} & ∀z(φ{z} → z = x) & ψ{x})›
    then AOT_obtain b where g: ‹φ{b} & ∀z(φ{z} → z = b) & ψ{b}›
      using "instantiation"[rotated] by blast
    AOT_hence h: ‹b = \ιx φ{x} ≡ (φ{b} & ∀z(φ{z} → z = b))›
      using b "∀E" by blast
    AOT_have ‹φ{b} & ∀z(φ{z} → z = b)› and j: ‹ψ{b}›
      using g "&E" by blast+
    AOT_hence ‹b = \ιx φ{x}› using h "≡E" by blast
    AOT_thus ‹ψ{\ιx φ{x}}› using j "rule=E" by blast
  qed
qed
end

interpretation "russell-axiom[exe,1]": russell_axiom ‹λ κ . «[Π]꬛
  by standard (metis "cqt:5:a[1]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,2,1,1]": russell_axiom ‹λ κ . «[Π]κκ'¬›
  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[exe,2,1,2]": russell_axiom ‹λ κ . «[Π]κ'꬛
  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,2,2]": russell_axiom ‹λ κ . «[Π]κ꬛
  by standard (metis "cqt:5:a[2]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,3,1,1]": russell_axiom ‹λ κ . «[Π]κκ'κ''¬›
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[exe,3,1,2]": russell_axiom ‹λ κ . «[Π]κ'κκ''¬›
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[exe,3,1,3]": russell_axiom ‹λ κ . «[Π]κ'κ''꬛
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,3,2,1]": russell_axiom ‹λ κ . «[Π]κκκ'¬›
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[exe,3,2,2]": russell_axiom ‹λ κ . «[Π]κκ'꬛
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,3,2,3]": russell_axiom ‹λ κ . «[Π]κ'κ꬛
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[exe,3,3]": russell_axiom ‹λ κ . «[Π]κκ꬛
  by standard (metis "cqt:5:a[3]"[axiom_inst, THEN "→E"] "&E"(2))

interpretation "russell-axiom[enc,1]": russell_axiom ‹λ κ . «κ[Π]¬›
  by standard (metis "cqt:5:b[1]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,2,1]": russell_axiom ‹λ κ . «κκ'[Π]¬›
  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[enc,2,2]": russell_axiom ‹λ κ . «κ'κ[Π]¬›
  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,2,3]": russell_axiom ‹λ κ . «κκ[Π]¬›
  by standard (metis "cqt:5:b[2]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,3,1,1]": russell_axiom ‹λ κ . «κκ'κ''[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[enc,3,1,2]": russell_axiom ‹λ κ . «κ'κκ''[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[enc,3,1,3]": russell_axiom ‹λ κ . «κ'κ''κ[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,3,2,1]": russell_axiom ‹λ κ . «κκκ'[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E")
interpretation "russell-axiom[enc,3,2,2]": russell_axiom ‹λ κ . «κκ'κ[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,3,2,3]": russell_axiom ‹λ κ . «κ'κκ[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))
interpretation "russell-axiom[enc,3,3]": russell_axiom ‹λ κ . «κκκ[Π]¬›
  by standard (metis "cqt:5:b[3]"[axiom_inst, THEN "→E"] "&E"(2))

AOT_act_theorem "!-exists:1": ‹\ιx φ{x}↓ ≡ ∃!x φ{x}›
proof(rule "≡I"; rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})› by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  AOT_hence ‹φ{a} & ∀z (φ{z} → z = a)›
    using hintikka "≡E" by blast
  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
    by (rule "∃I")
  AOT_thus ‹∃!x φ{x}›
    using "uniqueness:1"[THEN "≡_dfI"] by blast
next
  AOT_assume ‹∃!x φ{x}›
  AOT_hence ‹∃x (φ{x} & ∀z (φ{z} → z = x))›
    using "uniqueness:1"[THEN "≡_dfE"] by blast
  then AOT_obtain b where ‹φ{b} & ∀z (φ{z} → z = b)›
    using "instantiation"[rotated] by blast
  AOT_hence ‹b = \ιx φ{x}›
    using hintikka "≡E" by blast
  AOT_thus ‹\ιx φ{x}↓›
    by (metis "t=t-proper:2" "vdash-properties:6")
qed

AOT_act_theorem "!-exists:2": ‹∃y(y=\ιx φ{x}) ≡ ∃!x φ{x}›
  using "!-exists:1" "free-thms:1" "≡E"(6) by blast

AOT_act_theorem "y-in:1": ‹x = \ιx φ{x} → φ{x}›
  using "&E"(1) "→I" hintikka "≡E"(1) by blast

(* Note: generalized alphabetic variant of the last theorem *)
AOT_act_theorem "y-in:2": ‹z = \ιx φ{x} → φ{z}› using "y-in:1".

AOT_act_theorem "y-in:3": ‹\ιx φ{x}↓ → φ{\ιx φ{x}}›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})›
    by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  moreover AOT_have ‹φ{a}›
    using calculation hintikka "≡E"(1) "&E" by blast
  ultimately AOT_show ‹φ{\ιx φ{x}}› using "rule=E" by blast
qed

AOT_act_theorem "y-in:4": ‹∃y (y = \ιx φ{x}) → φ{\ιx φ{x}}›
  using "y-in:3"[THEN "→E"] "free-thms:1"[THEN "≡E"(2)] "→I" by blast


AOT_theorem "act-quant-nec":
  ‹∀β (\Aφ{β} ≡ β = α) ≡ ∀β(\A\Aφ{β} ≡ β = α)›
proof(rule "≡I"; rule "→I")
  AOT_assume ‹∀β (\Aφ{β} ≡ β = α)›
  AOT_hence ‹\Aφ{β} ≡ β = α› for β using "∀E" by blast
  AOT_hence ‹\A\Aφ{β} ≡ β = α› for β 
    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
  AOT_thus ‹∀β(\A\Aφ{β} ≡ β = α)›
    by (rule "∀I")
next
  AOT_assume ‹∀β(\A\Aφ{β} ≡ β = α)›
  AOT_hence ‹\A\Aφ{β} ≡ β = α› for β using "∀E" by blast
  AOT_hence ‹\Aφ{β} ≡ β = α› for β
    by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
  AOT_thus ‹∀β (\Aφ{β} ≡ β = α)›
    by (rule "∀I")
qed

AOT_theorem "equi-desc-descA:1": ‹x = \ιx φ{x} ≡ x = \ιx(\Aφ{x})›
proof -
  AOT_have ‹x = \ιx φ{x} ≡ ∀z (\Aφ{z} ≡ z = x)›
    using descriptions[axiom_inst] by blast
  also AOT_have ‹... ≡ ∀z (\A\Aφ{z} ≡ z = x)›
  proof(rule "≡I"; rule "→I"; rule "∀I")
    AOT_assume ‹∀z (\Aφ{z} ≡ z = x)›
    AOT_hence ‹\Aφ{a} ≡ a = x› for a
      using "∀E" by blast
    AOT_thus ‹\A\Aφ{a} ≡ a = x› for a
      by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(5))
  next
    AOT_assume ‹∀z (\A\Aφ{z} ≡ z = x)›
    AOT_hence ‹\A\Aφ{a} ≡ a = x› for a
      using "∀E" by blast
    AOT_thus ‹\Aφ{a} ≡ a = x› for a
      by (metis "Act-Basic:5" "act-conj-act:4" "≡E"(1) "≡E"(6))
  qed
  also AOT_have ‹... ≡ x = \ιx(\Aφ{x})›
    using "Commutativity of ≡"[THEN "≡E"(1)] descriptions[axiom_inst] by fast
  finally show ?thesis .
qed

AOT_theorem "equi-desc-descA:2": ‹\ιx φ{x}↓ → \ιx φ{x} = \ιx(\Aφ{x})›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})›
    by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  moreover AOT_have ‹a = \ιx(\Aφ{x})›
    using calculation "equi-desc-descA:1"[THEN "≡E"(1)] by blast
  ultimately AOT_show ‹\ιx φ{x} = \ιx(\Aφ{x})›
    using "rule=E" by fast
qed

AOT_theorem "nec-hintikka-scheme":
  ‹x = \ιx φ{x} ≡ \Aφ{x} & ∀z(\Aφ{z} → z = x)›
proof -
  AOT_have ‹x = \ιx φ{x} ≡ ∀z(\Aφ{z} ≡ z = x)›
    using descriptions[axiom_inst] by blast
  also AOT_have ‹… ≡ (\Aφ{x} & ∀z(\Aφ{z} → z = x))›
    using "Commutativity of ≡"[THEN "≡E"(1)] "term-out:3" by fast
  finally show ?thesis.
qed

AOT_theorem "equiv-desc-eq:1":
  ‹\A∀x(φ{x} ≡ ψ{x}) → ∀x (x = \ιx φ{x} ≡ x = \ιx ψ{x})›
proof(rule "→I"; rule "∀I")
  fix β
  AOT_assume ‹\A∀x(φ{x} ≡ ψ{x})›
  AOT_hence ‹\A(φ{x} ≡ ψ{x})› for x
    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(1)] "∀E"(2) by blast
  AOT_hence 0: ‹\Aφ{x} ≡ \Aψ{x}› for x
    by (metis "Act-Basic:5" "≡E"(1))
  AOT_have ‹β = \ιx φ{x} ≡ \Aφ{β} & ∀z(\Aφ{z} → z = β)›
    using "nec-hintikka-scheme" by blast
  also AOT_have ‹... ≡ \Aψ{β} & ∀z(\Aψ{z} → z = β)›
  proof (rule "≡I"; rule "→I")
    AOT_assume 1: ‹\Aφ{β} & ∀z(\Aφ{z} → z = β)›
    AOT_hence ‹\Aφ{z} → z = β› for z
      using "&E" "∀E" by blast
    AOT_hence ‹\Aψ{z} → z = β› for z
      using 0 "≡E" "→I" "→E" by metis
    AOT_hence ‹∀z(\Aψ{z} → z = β)›
      using "∀I" by fast
    moreover AOT_have ‹\Aψ{β}›
      using "&E" 0[THEN "≡E"(1)] 1 by blast
    ultimately AOT_show ‹\Aψ{β} & ∀z(\Aψ{z} → z = β)›
      using "&I" by blast
  next
    AOT_assume 1: ‹\Aψ{β} & ∀z(\Aψ{z} → z = β)›
    AOT_hence ‹\Aψ{z} → z = β› for z
      using "&E" "∀E" by blast
    AOT_hence ‹\Aφ{z} → z = β› for z
      using 0 "≡E" "→I" "→E" by metis
    AOT_hence ‹∀z(\Aφ{z} → z = β)›
      using "∀I" by fast
    moreover AOT_have ‹\Aφ{β}›
      using "&E" 0[THEN "≡E"(2)] 1 by blast
    ultimately AOT_show ‹\Aφ{β} & ∀z(\Aφ{z} → z = β)›
      using "&I" by blast
  qed
  also AOT_have ‹... ≡ β = \ιx ψ{x}›
    using "Commutativity of ≡"[THEN "≡E"(1)] "nec-hintikka-scheme" by blast
  finally AOT_show ‹β = \ιx φ{x} ≡ β = \ιx ψ{x}› .
qed

AOT_theorem "equiv-desc-eq:2":
  ‹\ιx φ{x}↓ & \A∀x(φ{x} ≡ ψ{x}) → \ιx φ{x} = \ιx ψ{x}›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓ & \A∀x(φ{x} ≡ ψ{x})›
  AOT_hence 0: ‹∃y (y = \ιx φ{x})› and
            1: ‹∀x (x = \ιx φ{x} ≡ x = \ιx ψ{x})›
    using "&E" "free-thms:1"[THEN "≡E"(1)] "equiv-desc-eq:1" "→E" by blast+
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  moreover AOT_have ‹a = \ιx ψ{x}›
    using calculation 1 "∀E" "≡E"(1) by fast
  ultimately AOT_show ‹\ιx φ{x} = \ιx ψ{x}›
    using "rule=E" by fast
qed

AOT_theorem "equiv-desc-eq:3":
  ‹\ιx φ{x}↓ & ◻∀x(φ{x} ≡ ψ{x}) → \ιx φ{x} = \ιx ψ{x}›
  using "→I" "equiv-desc-eq:2"[THEN "→E", OF "&I"] "&E"
        "nec-imp-act"[THEN "→E"] by metis

(* Note: this is a special case of "exist-nec" *)
AOT_theorem "equiv-desc-eq:4": ‹\ιx φ{x}↓ → ◻\ιx φ{x}↓›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})›
    by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  AOT_thus ‹◻\ιx φ{x}↓›
    using "ex:2:a" "rule=E" by fast
qed

AOT_theorem "equiv-desc-eq:5": ‹\ιx φ{x}↓ → ∃y ◻(y = \ιx φ{x})›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})›
    by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  AOT_hence ‹◻(a = \ιx φ{x})›
    by (metis "id-nec:2" "vdash-properties:10")
  AOT_thus ‹∃y ◻(y = \ιx φ{x})›
    by (rule "∃I")
qed

AOT_act_theorem "equiv-desc-eq2:1":
  ‹∀x (φ{x} ≡ ψ{x}) → ∀x (x = \ιx φ{x} ≡ x = \ιx ψ{x})›
  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"]
        "equiv-desc-eq:1"[THEN "→E"]
        "RA[1]" "deduction-theorem" by blast

AOT_act_theorem "equiv-desc-eq2:2":
  ‹\ιx φ{x}↓ & ∀x (φ{x} ≡ ψ{x}) → \ιx φ{x} = \ιx ψ{x}›
  using "→I" "logic-actual"[act_axiom_inst, THEN "→E"]
        "equiv-desc-eq:2"[THEN "→E", OF "&I"]
        "RA[1]" "deduction-theorem" "&E" by metis

context russell_axiom
begin
AOT_theorem "nec-russell-axiom":
  ‹ψ{\ιx φ{x}} ≡ ∃x(\Aφ{x} & ∀z(\Aφ{z} → z = x) & ψ{x})›
proof -
  AOT_have b: ‹∀x (x = \ιx φ{x} ≡ (\Aφ{x} & ∀z(\Aφ{z} → z = x)))›
    using "nec-hintikka-scheme" "∀I" by fast
  show ?thesis
  proof(rule "≡I"; rule "→I")
    AOT_assume c: ‹ψ{\ιx φ{x}}›
    AOT_hence d: ‹\ιx φ{x}↓›
      using ψ_denotes_asm by blast
    AOT_hence ‹∃y (y = \ιx φ{x})›
      by (metis "rule=I:1" "existential:1")
    then AOT_obtain a where a_def: ‹a = \ιx φ{x}›
      using "instantiation"[rotated] by blast
    moreover AOT_have ‹a = \ιx φ{x} ≡ (\Aφ{a} & ∀z(\Aφ{z} → z = a))›
      using b "∀E" by blast
    ultimately AOT_have ‹\Aφ{a} & ∀z(\Aφ{z} → z = a)›
      using "≡E" by blast
    moreover AOT_have ‹ψ{a}›
    proof - 
      AOT_have 1: ‹∀x∀y(x = y → y = x)›
        by (simp add: "id-eq:2" "universal-cor")
      AOT_have ‹a = \ιx φ{x} → \ιx φ{x} = a›
        by (rule "∀E"(1)[where τ="«\<iota>x φ{x}¬"]; rule "∀E"(2)[where β=a])
           (auto simp: d "universal-cor" 1)
      AOT_thus ‹ψ{a}›
        using a_def c "rule=E" "→E" by metis
    qed
    ultimately AOT_have ‹\Aφ{a} & ∀z(\Aφ{z} → z = a) & ψ{a}›
      by (rule "&I")
    AOT_thus ‹∃x(\Aφ{x} & ∀z(\Aφ{z} → z = x) & ψ{x})›
      by (rule "∃I")
  next
    AOT_assume ‹∃x(\Aφ{x} & ∀z(\Aφ{z} → z = x) & ψ{x})›
    then AOT_obtain b where g: ‹\Aφ{b} & ∀z(\Aφ{z} → z = b) & ψ{b}›
      using "instantiation"[rotated] by blast
    AOT_hence h: ‹b = \ιx φ{x} ≡ (\Aφ{b} & ∀z(\Aφ{z} → z = b))›
      using b "∀E" by blast
    AOT_have ‹\Aφ{b} & ∀z(\Aφ{z} → z = b)› and j: ‹ψ{b}›
      using g "&E" by blast+
    AOT_hence ‹b = \ιx φ{x}›
      using h "≡E" by blast
    AOT_thus ‹ψ{\ιx φ{x}}›
      using j "rule=E" by blast
  qed
qed
end

AOT_theorem "actual-desc:1": ‹\ιx φ{x}↓ ≡ ∃!x \Aφ{x}›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})›
    by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}›
    using "instantiation"[rotated] by blast
  moreover AOT_have ‹a = \ιx φ{x} ≡ ∀z(\Aφ{z} ≡ z = a)›
    using descriptions[axiom_inst] by blast
  ultimately AOT_have ‹∀z(\Aφ{z} ≡ z = a)›
    using "≡E" by blast
  AOT_hence ‹∃x∀z(\Aφ{z} ≡ z = x)› by (rule "∃I")
  AOT_thus ‹∃!x \Aφ{x}›
    using "uniqueness:2"[THEN "≡E"(2)] by fast
next
  AOT_assume ‹∃!x \Aφ{x}›
  AOT_hence ‹∃x∀z(\Aφ{z} ≡ z = x)›
    using "uniqueness:2"[THEN "≡E"(1)] by fast
  then AOT_obtain a where ‹∀z(\Aφ{z} ≡ z = a)›
    using "instantiation"[rotated] by blast
  moreover AOT_have ‹a = \ιx φ{x} ≡ ∀z(\Aφ{z} ≡ z = a)›
    using descriptions[axiom_inst] by blast
  ultimately AOT_have ‹a = \ιx φ{x}›
    using "≡E" by blast
  AOT_thus ‹\ιx φ{x}↓›
    by (metis "t=t-proper:2" "vdash-properties:6")
qed

AOT_theorem "actual-desc:2": ‹x = \ιx φ{x} → \Aφ{x}›
  using "&E"(1) "contraposition:1[2]" "≡E"(1) "nec-hintikka-scheme"
        "reductio-aa:2" "vdash-properties:9" by blast

(* Note: generalized alphabetic variant of the last theorem *)
AOT_theorem "actual-desc:3": ‹z = \ιx φ{x} → \Aφ{z}›
  using "actual-desc:2".

AOT_theorem "actual-desc:4": ‹\ιx φ{x}↓ → \Aφ{\ιx φ{x}}›
proof(rule "→I")
  AOT_assume ‹\ιx φ{x}↓›
  AOT_hence ‹∃y (y = \ιx φ{x})› by (metis "rule=I:1" "existential:1")
  then AOT_obtain a where ‹a = \ιx φ{x}› using "instantiation"[rotated] by blast
  AOT_thus ‹\Aφ{\ιx φ{x}}›
    using "actual-desc:2" "rule=E" "→E" by fast
qed

AOT_theorem "actual-desc:5": ‹\ιx φ{x} = \ιx ψ{x} → \A∀x(φ{x} ≡ ψ{x})›
proof(rule "→I")
  AOT_assume 0: ‹\ιx φ{x} = \ιx ψ{x}›
  AOT_hence φ_down: ‹\ιx φ{x}↓› and ψ_down: ‹\ιx ψ{x}↓›
    using "t=t-proper:1" "t=t-proper:2" "vdash-properties:6" by blast+
  AOT_hence ‹∃y (y = \ιx φ{x})› and ‹∃y (y = \ιx ψ{x})›
    by (metis "rule=I:1" "existential:1")+
  then AOT_obtain a and b where a_eq: ‹a = \ιx φ{x}› and b_eq: ‹b = \ιx ψ{x}›
    using "instantiation"[rotated] by metis

  AOT_have ‹∀α∀β (α = β → β = α)›
    by (rule "∀I"; rule "∀I"; rule "id-eq:2")
  AOT_hence ‹∀β (\ιx φ{x} = β → β = \ιx φ{x})›
    using "∀E" φ_down by blast
  AOT_hence ‹\ιx φ{x} = \ιx ψ{x} → \ιx ψ{x} = \ιx φ{x}›
    using "∀E" ψ_down by blast
  AOT_hence 1: ‹\ιx ψ{x} = \ιx φ{x}› using 0
    "→E" by blast

  AOT_have ‹\Aφ{x} ≡ \Aψ{x}› for x
  proof(rule "≡I"; rule "→I")
    AOT_assume ‹\Aφ{x}›
    moreover AOT_have ‹\Aφ{x} → x = a› for x
      using "nec-hintikka-scheme"[THEN "≡E"(1), OF a_eq, THEN "&E"(2)]
            "∀E" by blast
    ultimately AOT_have ‹x = a›
      using "→E" by blast
    AOT_hence ‹x = \ιx φ{x}›
      using a_eq "rule=E" by blast
    AOT_hence ‹x = \ιx ψ{x}›
      using 0 "rule=E" by blast
    AOT_thus ‹\Aψ{x}›
      by (metis "actual-desc:3" "vdash-properties:6")
  next
    AOT_assume ‹\Aψ{x}›
    moreover AOT_have ‹\Aψ{x} → x = b› for x
      using "nec-hintikka-scheme"[THEN "≡E"(1), OF b_eq, THEN "&E"(2)]
            "∀E" by blast
    ultimately AOT_have ‹x = b›
      using "→E" by blast
    AOT_hence ‹x = \ιx ψ{x}›
      using b_eq "rule=E" by blast
    AOT_hence ‹x = \ιx φ{x}›
      using 1 "rule=E" by blast
    AOT_thus ‹\Aφ{x}›
      by (metis "actual-desc:3" "vdash-properties:6")
  qed
  AOT_hence ‹\A(φ{x} ≡ ψ{x})› for x
    by (metis "Act-Basic:5" "≡E"(2))
  AOT_hence ‹∀x \A(φ{x} ≡ ψ{x})›
    by (rule "∀I")
  AOT_thus ‹\A∀x (φ{x} ≡ ψ{x})›
    using "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)] by fast
qed    

AOT_theorem "!box-desc:1": ‹∃!x ◻φ{x} → ∀y (y = \ιx φ{x} → φ{y})›
proof(rule "→I")
  AOT_assume ‹∃!x ◻φ{x}›
  AOT_hence ζ: ‹∃x (◻φ{x} & ∀z (◻φ{z} → z = x))›
    using "uniqueness:1"[THEN "≡_dfE"] by blast
  then AOT_obtain b where θ: ‹◻φ{b} & ∀z (◻φ{z} → z = b)›
    using "instantiation"[rotated] by blast
  AOT_show ‹∀y (y = \ιx φ{x} → φ{y})›
  proof(rule GEN; rule "→I")
    fix y
    AOT_assume ‹y = \ιx φ{x}›
    AOT_hence ‹\Aφ{y} & ∀z (\Aφ{z} → z = y)›
      using "nec-hintikka-scheme"[THEN "≡E"(1)] by blast
    AOT_hence ‹\Aφ{b} → b = y›
      using "&E" "∀E" by blast
    moreover AOT_have ‹\Aφ{b}›
      using θ[THEN "&E"(1)]  by (metis "nec-imp-act" "→E")
    ultimately AOT_have ‹b = y›
      using "→E" by blast
    moreover AOT_have ‹φ{b}›
      using θ[THEN "&E"(1)]  by (metis "qml:2"[axiom_inst] "→E") 
    ultimately AOT_show ‹φ{y}›
      using "rule=E" by blast
  qed
qed

AOT_theorem "!box-desc:2":
  ‹∀x (φ{x} → ◻φ{x}) → (∃!x φ{x} → ∀y (y = \ιx φ{x} → φ{y}))›
proof(rule "→I"; rule "→I")
  AOT_assume ‹∀x (φ{x} → ◻φ{x})›
  moreover AOT_assume ‹∃!x φ{x}›
  ultimately AOT_have ‹∃!x ◻φ{x}›
    using "nec-exist-!"[THEN "→E", THEN "→E"] by blast
  AOT_thus ‹∀y (y = \ιx φ{x} → φ{y})›
    using "!box-desc:1" "→E" by blast
qed

(* Note: vacuous in the embedding. *)
AOT_theorem "dr-alphabetic-thm": ‹\ιν φ{ν}↓ → \ιν φ{ν} = \ιμ φ{μ}›
  by (simp add: "rule=I:1" "→I")

subsection‹The Theory of Necessity›
text‹\label{PLM: 9.9}›

AOT_theorem "RM:1[prem]":
  assumes ‹Γ \⊨_\◻ φ → ψ›
  shows ‹◻Γ \⊨_\◻ ◻φ → ◻ψ›
proof -
  AOT_have ‹◻Γ \⊨_\◻ ◻(φ → ψ)›
    using "RN[prem]" assms by blast
  AOT_thus ‹◻Γ \⊨_\◻ ◻φ → ◻ψ›
    by (metis "qml:1"[axiom_inst] "→E")
qed

AOT_theorem "RM:1":
  assumes ‹\⊨_\◻ φ → ψ›
  shows ‹\⊨_\◻ ◻φ → ◻ψ›
  using "RM:1[prem]" assms by blast

lemmas RM = "RM:1"

AOT_theorem "RM:2[prem]":
  assumes ‹Γ \⊨_\◻ φ → ψ›
  shows ‹◻Γ \⊨_\◻ ♢φ → ♢ψ›
proof -
  AOT_have ‹Γ \⊨_\◻ ¬ψ → ¬φ›
    using assms 
    by (simp add: "contraposition:1[1]")
  AOT_hence ‹◻Γ \⊨_\◻ ◻¬ψ → ◻¬φ›
    using "RM:1[prem]" by blast
  AOT_thus ‹◻Γ \⊨_\◻ ♢φ → ♢ψ›
    by (meson "≡_dfE" "≡_dfI" "conventions:5" "→I" "modus-tollens:1")
qed

AOT_theorem "RM:2":
  assumes ‹\⊨_\◻ φ → ψ›
  shows ‹\⊨_\◻ ♢φ → ♢ψ›
  using "RM:2[prem]" assms by blast

lemmas "RM♢" = "RM:2"

AOT_theorem "RM:3[prem]":
  assumes ‹Γ \⊨_\◻ φ ≡ ψ›
  shows ‹◻Γ \⊨_\◻ ◻φ ≡ ◻ψ›
proof -
  AOT_have ‹Γ \⊨_\◻ φ → ψ› and ‹Γ \⊨_\◻ ψ → φ›
    using assms "≡E" "→I" by metis+
  AOT_hence ‹◻Γ \⊨_\◻ ◻φ → ◻ψ› and ‹◻Γ \⊨_\◻ ◻ψ → ◻φ›
    using "RM:1[prem]" by metis+
  AOT_thus ‹◻Γ \⊨_\◻ ◻φ ≡ ◻ψ›
    by (simp add: "≡I")
qed

AOT_theorem "RM:3":
  assumes ‹\⊨_\◻ φ ≡ ψ›
  shows ‹\⊨_\◻ ◻φ ≡ ◻ψ›
  using "RM:3[prem]" assms by blast

lemmas RE = "RM:3"

AOT_theorem "RM:4[prem]":
  assumes ‹Γ \⊨_\◻ φ ≡ ψ›
  shows ‹◻Γ \⊨_\◻ ♢φ ≡ ♢ψ›
proof -
  AOT_have ‹Γ \⊨_\◻ φ → ψ› and ‹Γ \⊨_\◻ ψ → φ›
    using assms "≡E" "→I" by metis+
  AOT_hence ‹◻Γ \⊨_\◻ ♢φ → ♢ψ› and ‹◻Γ \⊨_\◻ ♢ψ → ♢φ›
    using "RM:2[prem]" by metis+
  AOT_thus ‹◻Γ \⊨_\◻ ♢φ ≡ ♢ψ›
    by (simp add: "≡I")
qed

AOT_theorem "RM:4":
  assumes ‹\⊨_\◻ φ ≡ ψ›
  shows ‹\⊨_\◻ ♢φ ≡ ♢ψ›
  using "RM:4[prem]" assms by blast

lemmas "RE♢" = "RM:4"

AOT_theorem "KBasic:1": ‹◻φ → ◻(ψ → φ)›
  by (simp add: RM "pl:1"[axiom_inst])

AOT_theorem "KBasic:2": ‹◻¬φ → ◻(φ → ψ)›
  by (simp add: RM "useful-tautologies:3")

AOT_theorem "KBasic:3": ‹◻(φ & ψ) ≡ (◻φ & ◻ψ)›
proof (rule "≡I"; rule "→I")
  AOT_assume ‹◻(φ & ψ)›
  AOT_thus ‹◻φ & ◻ψ›
    by (meson RM "&I" "Conjunction Simplification"(1, 2) "→E")
next
  AOT_have ‹◻φ → ◻(ψ → (φ & ψ))›
    by (simp add: "RM:1" Adjunction)
  AOT_hence ‹◻φ → (◻ψ → ◻(φ & ψ))›
    by (metis "Hypothetical Syllogism" "qml:1"[axiom_inst])
  moreover AOT_assume ‹◻φ & ◻ψ›
  ultimately AOT_show ‹◻(φ & ψ)›
    using "→E" "&E" by blast
qed

AOT_theorem "KBasic:4": ‹◻(φ ≡ ψ) ≡ (◻(φ → ψ) & ◻(ψ → φ))›
proof -
  AOT_have θ: ‹◻((φ → ψ) & (ψ → φ)) ≡ (◻(φ → ψ) & ◻(ψ → φ))›
    by (fact "KBasic:3")
  AOT_modally_strict {
    AOT_have ‹(φ ≡ ψ) ≡ ((φ → ψ) & (ψ → φ))›
      by (fact "conventions:3"[THEN "≡Df"])
  }
  AOT_hence ξ: ‹◻(φ ≡ ψ) ≡ ◻((φ → ψ) & (ψ → φ))›
    by (rule RE)
  with ξ and θ AOT_show ‹◻(φ ≡ ψ) ≡ (◻(φ → ψ) & ◻(ψ → φ))›
    using "≡E"(5) by blast
qed

AOT_theorem "KBasic:5": ‹(◻(φ → ψ) & ◻(ψ → φ)) → (◻φ ≡ ◻ψ)›
proof -
  AOT_have ‹◻(φ → ψ) → (◻φ → ◻ψ)›
    by (fact "qml:1"[axiom_inst])
  moreover AOT_have ‹◻(ψ → φ) → (◻ψ → ◻φ)›
    by (fact "qml:1"[axiom_inst])
  ultimately AOT_have ‹(◻(φ → ψ) & ◻(ψ → φ)) → ((◻φ → ◻ψ) & (le='font-size: 18px;'>◻ψ → ◻φ))›
    by (metis "&I" MP "Double Composition")
  moreover AOT_have ‹((◻φ → ◻ψ) & (◻ψ → ◻φ)) → (◻φ ≡ ◻ψ)›
    using "conventions:3"[THEN "≡_dfI"] "→I" by blast
  ultimately AOT_show ‹(◻(φ → ψ) & ◻(ψ → φ)) → (◻φ ≡ ◻ψ)›
    by (metis "Hypothetical Syllogism")
qed

AOT_theorem "KBasic:6": ‹◻(φ ≡ ψ) → (◻φ ≡ ◻ψ)›
  using "KBasic:4" "KBasic:5" "deduction-theorem" "≡E"(1) "→E" by blast
AOT_theorem "KBasic:7": ‹((◻φ & ◻ψ) ∨ (◻¬φ & ◻¬ψ)) → ◻(φ ≡ ψ)›
proof (rule "→I"; drule "∨E"(1); (rule "→I")?)
  AOT_assume ‹◻φ & ◻ψ›
  AOT_hence ‹◻φ› and ‹◻ψ› using "&E" by blast+
  AOT_hence ‹◻(φ → ψ)› and ‹◻(ψ → φ)› using "KBasic:1" "→E" by blast+
  AOT_hence ‹◻(φ → ψ) & ◻(ψ → φ)› using "&I" by blast
  AOT_thus ‹◻(φ ≡ ψ)›  by (metis "KBasic:4" "≡E"(2))
next
  AOT_assume ‹◻¬φ & ◻¬ψ›
  AOT_hence 0: ‹◻(¬φ & ¬ψ)› using "KBasic:3"[THEN "≡E"(2)] by blast
  AOT_modally_strict {
    AOT_have ‹(¬φ & ¬ψ) → (φ ≡ ψ)›
      by (metis "&E"(1) "&E"(2) "deduction-theorem" "≡I" "reductio-aa:1")
  }
  AOT_hence ‹◻(¬φ & ¬ψ) → ◻(φ ≡ ψ)›
    by (rule RM)
  AOT_thus ‹◻(φ ≡ ψ)› using 0 "→E" by blast
qed(auto)

AOT_theorem "KBasic:8": ‹◻(φ & ψ) → ◻(φ ≡ ψ)›
  by (meson "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I")
AOT_theorem "KBasic:9": ‹◻(¬φ & ¬ψ) → ◻(φ ≡ ψ)›
  by (metis "RM:1" "&E"(1) "&E"(2) "deduction-theorem" "≡I" "raa-cor:4")
AOT_theorem "KBasic:10": ‹◻φ ≡ ◻¬¬φ›
  by (simp add: "RM:3" "oth-class-taut:3:b")
AOT_theorem "KBasic:11": ‹¬◻φ ≡ ♢¬φ›
proof (rule "≡I"; rule "→I")
  AOT_show ‹♢¬φ› if ‹¬◻φ›
    using that "≡_dfI" "conventions:5" "KBasic:10" "≡E"(3) by blast
next
  AOT_show ‹¬◻φ› if ‹♢¬φ›
    using "≡_dfE" "conventions:5" "KBasic:10" "≡E"(4) that by blast
qed
AOT_theorem "KBasic:12": ‹◻φ ≡ ¬♢¬φ›
proof (rule "≡I"; rule "→I")
  AOT_show ‹¬♢¬φ› if ‹◻φ›
    using "¬¬I" "KBasic:11" "≡E"(3) that by blast
next
  AOT_show ‹◻φ› if ‹¬♢¬φ›
  using "KBasic:11" "≡E"(1) "reductio-aa:1" that by blast
qed
AOT_theorem "KBasic:13": ‹◻(φ → ψ) → (♢φ → ♢ψ)›
proof -
  AOT_have ‹φ → ψ \⊨_\◻ φ → ψ› by blast
  AOT_hence ‹◻(φ → ψ) \⊨_\◻ ♢φ → ♢ψ›
    using "RM:2[prem]" by blast
  AOT_thus ‹◻(φ → ψ) → (♢φ → ♢ψ)› using "→I" by blast
qed
lemmas "K♢" = "KBasic:13"
AOT_theorem "KBasic:14": ‹♢◻φ ≡ ¬◻♢¬φ›
  by (meson "RE♢" "KBasic:11" "KBasic:12" "≡E"(6) "oth-class-taut:3:a")
AOT_theorem "KBasic:15": ‹(◻φ ∨ ◻ψ) → ◻(φ ∨ ψ)›
proof -
  AOT_modally_strict {
    AOT_have ‹φ → (φ ∨ ψ)› and ‹ψ → (φ ∨ ψ)›
      by (auto simp: "Disjunction Addition"(1) "Disjunction Addition"(2))
  }
  AOT_hence ‹◻φ → ◻(φ ∨ ψ)› and ‹◻ψ → ◻(φ ∨ ψ)›
    using RM by blast+
  AOT_thus ‹(◻φ ∨ ◻ψ) → ◻(φ ∨ ψ)›
    by (metis "∨E"(1) "deduction-theorem")
qed

AOT_theorem "KBasic:16": ‹(◻φ & ♢ψ) → ♢(φ & ψ)›
  by (meson "KBasic:13" "RM:1" Adjunction "Hypothetical Syllogism"
            Importation "→E")

AOT_theorem "rule-sub-lem:1:a":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ ¬ψ ≡ ¬χ›
  using "qml:2"[axiom_inst, THEN "→E", OF assms]
        "≡E"(1) "oth-class-taut:4:b" by blast

AOT_theorem "rule-sub-lem:1:b":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ (ψ → Θ) ≡ (χ → Θ)›
  using "qml:2"[axiom_inst, THEN "→E", OF assms]
  using "oth-class-taut:4:c" "vdash-properties:6" by blast

AOT_theorem "rule-sub-lem:1:c":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ (Θ → ψ) ≡ (Θ → χ)›
  using "qml:2"[axiom_inst, THEN "→E", OF assms]
  using "oth-class-taut:4:d" "vdash-properties:6" by blast

AOT_theorem "rule-sub-lem:1:d":
  assumes ‹for arbitrary α: \⊨_\◻ ◻(ψ{α} ≡ χ{α})›
  shows ‹\⊨_\◻ ∀α ψ{α} ≡ ∀α χ{α}›
proof -
  AOT_modally_strict {
    AOT_have ‹∀α (ψ{α} ≡ χ{α})›
      using "qml:2"[axiom_inst, THEN "→E", OF assms] "∀I" by fast
    AOT_hence 0: ‹ψ{α} ≡ χ{α}› for α using "∀E" by blast
    AOT_show ‹∀α ψ{α} ≡ ∀α χ{α}›
    proof (rule "≡I"; rule "→I")
      AOT_assume ‹∀α ψ{α}›
      AOT_hence ‹ψ{α}› for α using "∀E" by blast
      AOT_hence ‹χ{α}› for α using 0 "≡E" by blast
      AOT_thus ‹∀α χ{α}› by (rule "∀I")
    next
      AOT_assume ‹∀α χ{α}›
      AOT_hence ‹χ{α}› for α using "∀E" by blast
      AOT_hence ‹ψ{α}› for α using 0 "≡E" by blast
      AOT_thus ‹∀α ψ{α}› by (rule "∀I")
    qed
  }
qed

AOT_theorem "rule-sub-lem:1:e":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ [λ ψ] ≡ [λ χ]›
  using "qml:2"[axiom_inst, THEN "→E", OF assms]
  using "≡E"(1) "propositions-lemma:6" by blast

AOT_theorem "rule-sub-lem:1:f":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ \Aψ ≡ \Aχ›
  using "qml:2"[axiom_inst, THEN "→E", OF assms, THEN "RA[2]"]
  by (metis "Act-Basic:5" "≡E"(1))

AOT_theorem "rule-sub-lem:1:g":
  assumes ‹\⊨_\◻ ◻(ψ ≡ χ)›
  shows ‹\⊨_\◻ ◻ψ ≡ ◻χ›
  using "KBasic:6" assms "vdash-properties:6" by blast

text‹Note that instead of deriving @{text "rule-sub-lem:2"},
 @{text "rule-sub-lem:3"}, @{text "rule-sub-lem:4"},
 and @{text "rule-sub-nec"}, we construct substitution methods instead.›

class AOT_subst =
  fixes AOT_subst :: "('a ==> o) ==> bool"
    and AOT_subst_cond :: "'a ==> 'a ==> bool"
  assumes AOT_subst:
    "AOT_subst φ ==> AOT_subst_cond ψ χ ==> [v ⊨ «φ ψ¬ ≡ «φ χ¬]"

named_theorems AOT_substI

instantiation o :: AOT_subst
begin

inductive AOT_subst_o where
  AOT_subst_o_id[AOT_substI]:
    ‹AOT_subst_o (λφ. φ)›
  | AOT_subst_o_const[AOT_substI]:
    ‹AOT_subst_o (λφ. ψ)›
  | AOT_subst_o_not[AOT_substI]:
    ‹AOT_subst_o Θ ==> AOT_subst_o (λ φ. «¬Θ{φ}¬)›
  | AOT_subst_o_imp[AOT_substI]:
    ‹AOT_subst_o Θ ==> AOT_subst_o Ξ ==> AOT_subst_o (λ φ. «Θ{φ} → Ξ{φ}¬)›
  | AOT_subst_o_lambda0[AOT_substI]:
    ‹AOT_subst_o Θ ==> AOT_subst_o (λ φ. (AOT_lambda0 (Θ φ)))›
  | AOT_subst_o_act[AOT_substI]:
    ‹AOT_subst_o Θ ==> AOT_subst_o (λ φ. «\AΘ{φ}¬)›
  | AOT_subst_o_box[AOT_substI]:
    ‹AOT_subst_o Θ ==> AOT_subst_o (λ φ. «◻Θ{φ}¬)›
  | AOT_subst_o_by_def[AOT_substI]:
    ‹(∧ ψ . AOT_model_equiv_def (Θ ψ) (Ξ ψ)) ==>
 AOT_subst_o Ξ ==> AOT_subst_o Θ›


definition AOT_subst_cond_o where
  ‹AOT_subst_cond_o ≡ λ ψ χ . ∀ v . [v ⊨ ψ ≡ χ]›

instance
proof
  fix ψ χ :: o and φ :: ‹o ==> o›
  assume cond: ‹AOT_subst_cond ψ χ›
  assume ‹AOT_subst φ›
  moreover AOT_have ‹\⊨_\◻ ψ ≡ χ›
    using cond unfolding AOT_subst_cond_o_def by blast
  ultimately AOT_show ‹\⊨_\◻ φ{ψ} ≡ φ{χ}›
  proof (induct arbitrary: ψ χ)
    case AOT_subst_o_id
    thus ?case
      using "≡E"(2) "oth-class-taut:4:b" "rule-sub-lem:1:a" by blast
  next
    case (AOT_subst_o_const ψ)
    thus ?case
      by (simp add: "oth-class-taut:3:a")
  next
    case (AOT_subst_o_not Θ)
    thus ?case
      by (simp add: RN "rule-sub-lem:1:a")
  next
    case (AOT_subst_o_imp Θ Ξ)
    thus ?case
      by (meson RN "≡E"(5) "rule-sub-lem:1:b" "rule-sub-lem:1:c")
  next
    case (AOT_subst_o_lambda0 Θ)
    thus ?case
      by (simp add: RN "rule-sub-lem:1:e")
  next
    case (AOT_subst_o_act Θ)
    thus ?case
      by (simp add: RN "rule-sub-lem:1:f")
  next
    case (AOT_subst_o_box Θ)
    thus ?case
      by (simp add: RN "rule-sub-lem:1:g")
  next
    case (AOT_subst_o_by_def Θ Ξ)
    AOT_modally_strict {
      AOT_have ‹Ξ{ψ} ≡ Ξ{χ}›
        using AOT_subst_o_by_def by simp
      AOT_thus ‹Θ{ψ} ≡ Θ{χ}›
        using "≡Df"[OF AOT_subst_o_by_def(1), of _ ψ]
              "≡Df"[OF AOT_subst_o_by_def(1), of _ χ]
        by (metis "≡E"(6) "oth-class-taut:3:a")
    }
  qed
qed
end

instantiation "fun" :: (AOT_Term_id_2, AOT_subst) AOT_subst
begin

definition AOT_subst_cond_fun :: ‹('a ==> 'b) ==> ('a ==> 'b) ==> bool› where
  ‹AOT_subst_cond_fun ≡ λ φ ψ . ∀ α . AOT_subst_cond (φ (AOT_term_of_var α))
 (ψ (AOT_term_of_var α))›

inductive AOT_subst_fun :: ‹(('a ==> 'b) ==> o) ==> bool› where
  AOT_subst_fun_const[AOT_substI]:
    ‹AOT_subst_fun (λφ. ψ)›
  | AOT_subst_fun_id[AOT_substI]:
    ‹AOT_subst Ψ ==> AOT_subst_fun (λφ. Ψ (φ (AOT_term_of_var α)))›
  | AOT_subst_fun_all[AOT_substI]:
    ‹AOT_subst Ψ ==> (∧ α . AOT_subst_fun (Θ (AOT_term_of_var α))) ==>
 AOT_subst_fun (λφ :: 'a ==> 'b. Ψ «∀α «Θ (α::'a) φ¬¬)›
  | AOT_subst_fun_not[AOT_substI]:
    ‹AOT_subst Ψ ==> AOT_subst_fun (λφ. «¬«Ψ φ¬¬)›
  | AOT_subst_fun_imp[AOT_substI]:
    ‹AOT_subst Ψ ==> AOT_subst Θ ==> AOT_subst_fun (λφ. ««Ψ φ¬ → «Θ φ¬¬)›
  | AOT_subst_fun_lambda0[AOT_substI]:
    ‹AOT_subst Θ ==> AOT_subst_fun (λ φ. (AOT_lambda0 (Θ φ)))›
  | AOT_subst_fun_act[AOT_substI]:
    ‹AOT_subst Θ ==> AOT_subst_fun (λ φ. «\A«Θ φ¬¬)›
  | AOT_subst_fun_box[AOT_substI]:
    ‹AOT_subst Θ ==> AOT_subst_fun (λ φ. «◻«Θ φ¬¬)›
  | AOT_subst_fun_def[AOT_substI]:
    ‹(∧ φ . AOT_model_equiv_def (Θ φ) (Ψ φ)) ==>
 AOT_subst_fun Ψ ==> AOT_subst_fun Θ›

instance proof
  fix ψ χ :: ‹'a ==> 'b› and φ :: ‹('a ==> 'b) ==> o›
  assume ‹AOT_subst φ›
  moreover assume cond: ‹AOT_subst_cond ψ χ›
  ultimately AOT_show ‹\⊨_\◻ «φ ψ¬ ≡ «φ χ¬›
  proof(induct)
    case (AOT_subst_fun_const ψ)
    then show ?case by (simp add: "oth-class-taut:3:a")
  next
  case (AOT_subst_fun_id Ψ x)
  then show ?case by (simp add: AOT_subst AOT_subst_cond_fun_def) 
  next
  next
  case (AOT_subst_fun_all Ψ Θ)
  AOT_have ‹\⊨_\◻ ◻(Θ{α, «ψ¬} ≡ Θ{α, «χ¬})› for α
    using AOT_subst_fun_all.hyps(3) AOT_subst_fun_all.prems RN by presburger
  thus ?case using AOT_subst[OF AOT_subst_fun_all(1)]
    by (simp add: RN "rule-sub-lem:1:d"
                  AOT_subst_cond_fun_def AOT_subst_cond_o_def)
  next
  case (AOT_subst_fun_not Ψ)
  then show ?case by (simp add: RN "rule-sub-lem:1:a")
  next
  case (AOT_subst_fun_imp Ψ Θ)
  then show ?case 
    unfolding AOT_subst_cond_fun_def AOT_subst_cond_o_def
    by (meson "≡E"(5) "oth-class-taut:4:c" "oth-class-taut:4:d" "→E")
  next
  case (AOT_subst_fun_lambda0 Θ)
  then show ?case by (simp add: RN "rule-sub-lem:1:e")
  next
  case (AOT_subst_fun_act Θ)
  then show ?case by (simp add: RN "rule-sub-lem:1:f")
  next
  case (AOT_subst_fun_box Θ)
  then show ?case by (simp add: RN "rule-sub-lem:1:g")
  next
  case (AOT_subst_fun_def Θ Ψ)
  then show ?case
    by (meson "df-rules-formulas[3]" "df-rules-formulas[4]" "≡I" "≡E"(5))
  qed
qed
end

ML‹
  prove_AOT_subst_tac ctxt = REPEAT (SUBGOAL (fn (trm,_) => let
 fun findHeadConst (Const x) = SOME x
 | findHeadConst (A $ _) = findHeadConst A
 | findHeadConst _ = NONE
 fun findDef (Const (🍋‹AOT_model_equiv_def›, _) $ lhs $ _)
 = findHeadConst lhs
 | findDef (A $ B) = (case findDef A of SOME x => SOME x | _ => findDef B)
 | findDef (Abs (_,_,c)) = findDef c
 | findDef _ = NONE
 val const_opt = (findDef trm)
 val defs = case const_opt of SOME const => List.filter (fn thm => let
 val concl = Thm.concl_of thm
 val thmconst = (findDef concl)
 in case thmconst of SOME (c,_) => fst const = c | _ => false end)
 (AOT_Definitions.get ctxt)
 | _ => []
 val tac = case defs of
 [] => safe_step_tac (ctxt addSIs @{thms AOT_substI}) 1
 | _ => resolve_tac ctxt defs 1
 in tac end) 1)
  getSubstThm ctxt reversed phi p q = let
  p_ty = Term.type_of p
  abs = HOLogic.mk_Trueprop (@{const AOT_subst(_)} $ phi)
  abs = Syntax.check_term ctxt abs
  substThm = Goal.prove ctxt [] [] abs
 (fn {context=ctxt, prems=_} => prove_AOT_subst_tac ctxt)
  substThm = substThm RS @{thm AOT_subst}
  if reversed then let
 val substThm = Drule.instantiate_normalize
 (TVars.empty, Vars.make [((("χ", 0), p_ty), Thm.cterm_of ctxt p),
 ((("ψ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
 val substThm = substThm RS @{thm "≡E"(1)}
 in substThm end
 
 let
 val substThm = Drule.instantiate_normalize
 (TVars.empty, Vars.make [((("ψ", 0), p_ty), Thm.cterm_of ctxt p),
 ((("χ", 0), p_ty), Thm.cterm_of ctxt q)]) substThm
 val substThm = substThm RS @{thm "≡E"(2)}
 in substThm end end
 ›

method_setup AOT_subst = ‹
 .option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
 .lift (Parse.embedded_inner_syntax -- Parse.embedded_inner_syntax) --
 .option (Scan.lift (Args.$$$ "for" -- Args.colon) |--
 .repeat1 (Scan.lift (Parse.embedded_inner_syntax) --
 .option (Scan.lift (Args.$$$ "::" |-- Parse.embedded_inner_syntax))))
 > (fn ((reversed,(raw_p,raw_q)),raw_bounds) => (fn ctxt =>
 Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _,
 prems = prems, asms = asms, concl = concl, schematics = _} =>
 
  thms = prems
  ctxt' = ctxt
  ctxt = Context_Position.set_visible false ctxt
  raw_bounds = case raw_bounds of SOME bounds => bounds | _ => []

  ctxt = (fold (fn (bound, ty) => fn ctxt =>
 let
 val bound = AOT_read_term @{nonterminal τ'} ctxt bound
 val ty = Option.map (Syntax.read_typ ctxt) ty
 val ctxt = case ty of SOME ty => let
 val bound = Const ("_type_constraint_", Type ("fun", [ty,ty])) $ bound
 val bound = Syntax.check_term ctxt bound
 in Variable.declare_term bound ctxt end | _ => ctxt
 in ctxt end)) raw_bounds ctxt

  p = AOT_read_term @{nonterminal φ'} ctxt raw_p
  p = Syntax.check_term ctxt p
  ctxt = Variable.declare_term p ctxt
  q = AOT_read_term @{nonterminal φ'} ctxt raw_q
  q = Syntax.check_term ctxt q
  ctxt = Variable.declare_term q ctxt

  bounds = (map (fn (bound, _) =>
 Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt bound)
 ) raw_bounds
  p = fold (fn bound => fn p =>
 Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,p)))
 bounds p
  p = Syntax.check_term ctxt p
  p_ty = Term.type_of p

  pat = @{const Trueprop} $
 (@{const AOT_model_valid_in} $ Var (("w",0), @{typ w}) $
 (Var (("φ",0), Type (🍋‹fun›, [p_ty, @{typ o}])) $ p))
  univ = Unify.matchers (Context.Proof ctxt) [(pat, Thm.term_of concl)]
val univ = hd (Seq.list_of univ) (* TODO: consider all matches *)
val phi = the (Envir.lookup univ
  (("φ",0), Type (🍋‹fun›, [p_ty, @{typ o}])))

val q = fold (fn bound => fn q =>
  Term.abs ("α", Term.type_of bound) (Term.abstract_over (bound,q))) bounds q
val q = Syntax.check_term ctxt q

(* Reparse to report bounds as fixes. *)
val ctxt = Context_Position.restore_visible ctxt' ctxt
val ctxt' = ctxt
fun unsource str = fst (Input.source_content (Syntax.read_input str))
val (_,ctxt') = Proof_Context.add_fixes (map (fn (str,_) =>
  (Binding.make (unsource str, Position.none), NONE, Mixfix.NoSyn)) raw_bounds)
  ctxt'
val _ = (map (fn (x,_) =>
  Syntax.check_term ctxt (AOT_read_term @{nonterminal τ'} ctxt' x)))
  raw_bounds
val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_p
val _ = AOT_read_term @{nonterminal φ'} ctxt' raw_q
val reversed = case reversed of SOME _ => true | _ => false
val simpThms = [@{thm AOT_subst_cond_o_def}, @{thm AOT_subst_cond_fun_def}]
in
resolve_tac ctxt [getSubstThm ctxt reversed phi p q] 1
THEN simp_tac (ctxt addsimps simpThms) 1
THEN (REPEAT (resolve_tac ctxt [@{thm allI}] 1))
THEN (TRY (resolve_tac ctxt thms 1))
end
) ctxt 1))))
›

method_setup AOT_subst_def = ‹
Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
Attrib.thm
>> (fn (reversed,fact) => (fn ctxt =>
(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _,
  prems = prems, asms = asms, concl = concl, schematics = _} =>
let
val c = Thm.concl_of fact
val (lhs, rhs) = case c of (\<^const>\<open>Trueprop\<close> $
    (\<^const>\<open>AOT_model_equiv_def\<close> $ lhs $ rhs)) => (lhs, rhs)
  | _ => raise Fail "Definition expected."
val substCond = HOLogic.mk_Trueprop
  (Const (\<^const_name>\<open>AOT_subst_cond\<close>, dummyT) $ lhs $ rhs)
val substCond = Syntax.check_term
  (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
  substCond
val simpThms = [@{thm AOT_subst_cond_\<o>_def},
  @{thm AOT_subst_cond_fun_def},
  fact RS @{thm "\<equiv>Df"}]
val substCondThm = Goal.prove ctxt [] [] substCond
  (fn {context=ctxt, prems=prems} =>
      (SUBGOAL (fn (trm,int) =>
        auto_tac (ctxt addsimps simpThms)) 1))
val substThm = substCondThm RSN (2,@{thm AOT_subst})
in
resolve_tac ctxt [substThm RS
  (case reversed of NONE => @{thm "\<equiv>E"(2)} | _ => @{thm "\<equiv>E"(1)})] 1
THEN prove_AOT_subst_tac ctxt
THEN (TRY (resolve_tac ctxt prems 1))
end
) ctxt 1))))
\<close>

method_setup AOT_subst_thm = \<open>
Scan.option (Scan.lift (Args.parens (Args.$$$ "reverse"))) --
Attrib.thm
>> (fn (reversed,fact) => (fn ctxt =>
(Method.SIMPLE_METHOD (Subgoal.FOCUS (fn {context = ctxt, params = _,
  prems = prems, asms = asms, concl = concl, schematics = _} =>
let
val c = Thm.concl_of fact
val (lhs, rhs) = case c of
  (\<^const>\<open>Trueprop\<close> $
   (\<^const>\<open>AOT_model_valid_in\<close> $ _ $
    (\<^const>\<open>AOT_equiv\<close> $ lhs $ rhs))) => (lhs, rhs)
  | _ => raise Fail "Equivalence expected."

val substCond = HOLogic.mk_Trueprop
  (Const (\<^const_name>\<open>AOT_subst_cond\<close>, dummyT) $ lhs $ rhs)
val substCond = Syntax.check_term
  (Proof_Context.set_mode Proof_Context.mode_schematic ctxt)
  substCond
val simpThms = [@{thm AOT_subst_cond_\<o>_def},
  @{thm AOT_subst_cond_fun_def},
  fact]
val substCondThm = Goal.prove ctxt [] [] substCond
  (fn {context=ctxt, prems=prems} =>
      (SUBGOAL (fn (trm,int) => auto_tac (ctxt addsimps simpThms)) 1))
val substThm = substCondThm RSN (2,@{thm AOT_subst})
in
resolve_tac ctxt [substThm RS
  (case reversed of NONE => @{thm "\<equiv>E"(2)} | _ => @{thm "\<equiv>E"(1)})] 1
THEN prove_AOT_subst_tac ctxt
THEN (TRY (resolve_tac ctxt prems 1))
end
) ctxt 1))))
\<close>

AOT_theorem "rule-sub-remark:1[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> A!x \<equiv> \<not>\<diamond>E!x\<close> and \<open>\<not>A!x\<close>
  shows \<open>\<not>\<not>\<diamond>E!x\<close>
  by (AOT_subst (reverse) \<open>\<not>\<diamond>E!x\<close> \<open>A!x\<close>)
     (auto simp: assms) 

AOT_theorem "rule-sub-remark:1[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> A!x \<equiv> \<not>\<diamond>E!x\<close> and  \<open>\<not>\<not>\<diamond>E!x\<close>
  shows \<open>\<not>A!x\<close>
  by (AOT_subst \<open>A!x\<close> \<open>\<not>\<diamond>E!x\<close>)
     (auto simp: assms)

AOT_theorem "rule-sub-remark:2[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> [R]xy \<equiv> ([R]xy & ([Q]a \<or> \<not>[Q]a))\<close>
      and \<open>p \<rightarrow> [R]xy\<close>
  shows \<open>p \<rightarrow> [R]xy & ([Q]a \<or> \<not>[Q]a)\<close>
  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:2[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> [R]xy \<equiv> ([R]xy & ([Q]a \<or> \<not>[Q]a))\<close>
      and \<open>p \<rightarrow> [R]xy & ([Q]a \<or> \<not>[Q]a)\<close>
  shows \<open>p \<rightarrow> [R]xy\<close>
  by (AOT_subst_thm assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:3[1]":
  assumes \<open>for arbitrary x: \<^bold>\<turnstile>\<^sub>\<box> A!x \<equiv> \<not>\<diamond>E!x\<close>
      and \<open>\<exists>x A!x\<close>
    shows \<open>\<exists>x \<not>\<diamond>E!x\<close>
  by (AOT_subst (reverse) \<open>\<not>\<diamond>E!x\<close> \<open>A!x\<close> for: x)
     (auto simp: assms)

AOT_theorem "rule-sub-remark:3[2]":
  assumes \<open>for arbitrary x: \<^bold>\<turnstile>\<^sub>\<box> A!x \<equiv> \<not>\<diamond>E!x\<close>
      and \<open>\<exists>x \<not>\<diamond>E!x\<close>
    shows \<open>\<exists>x A!x\<close>
  by (AOT_subst \<open>A!x\<close> \<open>\<not>\<diamond>E!x\<close> for: x)
     (auto simp: assms)

AOT_theorem "rule-sub-remark:4[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<not>\<not>[P]x \<equiv> [P]x\<close> and \<open>\<^bold>\<A>\<not>\<not>[P]x\<close>
  shows \<open>\<^bold>\<A>[P]x\<close>
  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:4[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<not>\<not>[P]x \<equiv> [P]x\<close> and \<open>\<^bold>\<A>[P]x\<close>
  shows \<open>\<^bold>\<A>\<not>\<not>[P]x\<close>
  by (AOT_subst_thm assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:5[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> (\<phi> \<rightarrow> \<psi>) \<equiv> (\<not>\<psi> \<rightarrow> \<not>\<phi>)\<close> and \<open>\<box>(\<phi> \<rightarrow> \<psi>)\<close>
  shows \<open>\<box>(\<not>\<psi> \<rightarrow> \<not>\<phi>)\<close>
  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:5[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> (\<phi> \<rightarrow> \<psi>) \<equiv> (\<not>\<psi> \<rightarrow> \<not>\<phi>)\<close> and \<open>\<box>(\<not>\<psi> \<rightarrow> \<not>\<phi>)\<close>
  shows \<open>\<box>(\<phi> \<rightarrow> \<psi>)\<close> 
  by (AOT_subst_thm assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:6[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<psi> \<equiv> \<chi>\<close> and \<open>\<box>(\<phi> \<rightarrow> \<psi>)\<close>
  shows \<open>\<box>(\<phi> \<rightarrow> \<chi>)\<close> 
  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:6[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<psi> \<equiv> \<chi>\<close> and \<open>\<box>(\<phi> \<rightarrow> \<chi>)\<close>
  shows \<open>\<box>(\<phi> \<rightarrow> \<psi>)\<close>
  by (AOT_subst_thm assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:7[1]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi> \<equiv> \<not>\<not>\<phi>\<close> and \<open>\<box>(\<phi> \<rightarrow> \<phi>)\<close>
  shows \<open>\<box>(\<not>\<not>\<phi> \<rightarrow> \<phi>)\<close> 
  by (AOT_subst_thm (reverse) assms(1)) (simp add: assms(2))

AOT_theorem "rule-sub-remark:7[2]":
  assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi> \<equiv> \<not>\<not>\<phi>\<close> and \<open>\<box>(\<not>\<not>\<phi> \<rightarrow> \<phi>)\<close>
  shows \<open>\<box>(\<phi> \<rightarrow> \<phi>)\<close>
  by (AOT_subst_thm assms(1)) (simp add: assms(2))

AOT_theorem "KBasic2:1": \<open>\<box>\<not>\<phi> \<equiv> \<not>\<diamond>\<phi>\<close>
  by (meson "conventions:5" "contraposition:2"
            "Hypothetical Syllogism" "df-rules-formulas[3]"
            "df-rules-formulas[4]" "\<equiv>I" "useful-tautologies:1")

AOT_theorem "KBasic2:2": \<open>\<diamond>(\<phi> \<or> \<psi>) \<equiv> (\<diamond>\<phi> \<or> \<diamond>\<psi>)\<close>
proof -
  AOT_have \<open>\<diamond>(\<phi> \<or> \<psi>) \<equiv> \<diamond>\<not>(\<not>\<phi> & \<not>\<psi>)\<close>
    by (simp add: "RE\<diamond>" "oth-class-taut:5:b")
  also AOT_have \<open>\<dots> \<equiv> \<not>\<box>(\<not>\<phi> & \<not>\<psi>)\<close>
    using "KBasic:11" "\<equiv>E"(6) "oth-class-taut:3:a" by blast
  also AOT_have \<open>\<dots> \<equiv> \<not>(\<box>\<not>\<phi> & \<box>\<not>\<psi>)\<close>
    using "KBasic:3" "\<equiv>E"(1) "oth-class-taut:4:b" by blast
  also AOT_have \<open>\<dots> \<equiv> \<not>(\<not>\<diamond>\<phi> & \<not>\<diamond>\<psi>)\<close>
    using "KBasic2:1"
    by (AOT_subst \<open>\<box>\<not>\<phi>\<close> \<open>\<not>\<diamond>\<phi>\<close>; AOT_subst \<open>\<box>\<not>\<psi>\<close> \<open>\<not>\<diamond>\<psi>\<close>;
        auto simp: "oth-class-taut:3:a")
  also AOT_have \<open>\<dots> \<equiv> \<not>\<not>(\<diamond>\<phi> \<or> \<diamond>\<psi>)\<close>
    using "\<equiv>E"(6) "oth-class-taut:3:b" "oth-class-taut:5:b" by blast
  also AOT_have \<open>\<dots> \<equiv> \<diamond>\<phi> \<or> \<diamond>\<psi>\<close>
    by (simp add: "\<equiv>I" "useful-tautologies:1" "useful-tautologies:2")
  finally show ?thesis .
qed

AOT_theorem "KBasic2:3": \<open>\<diamond>(\<phi> & \<psi>) \<rightarrow> (\<diamond>\<phi> &n> \<diamond>\<psi>)\<close>
  by (metis "RM\<diamond>" "&I" "Conjunction Simplification"(1,2)
            "\<rightarrow>I" "modus-tollens:1" "reductio-aa:1")

AOT_theorem "KBasic2:4": \<open>\<diamond>(\<phi> \<rightarrow> \<psi>) \<equiv> (\<box>\<phi> \<rightarrow> \<diamond>\<psi>)\<close>
proof -
  AOT_have \<open>\<diamond>(\<phi> \<rightarrow> \<psi>) \<equiv> \<diamond>(\<not>\<phi> \<or> \<psi>)\<close>
    by (AOT_subst \<open>\<phi> \<rightarrow> \<psi>\<close> \<open>\<not>\<phi> \<or> \<psi>\<close>)
       (auto simp: "oth-class-taut:1:c" "oth-class-taut:3:a")
  also AOT_have \<open>... \<equiv> \<diamond>\<not>\<phi> \<or> \<diamond>\<psi>\<close>
    by (simp add: "KBasic2:2")
  also AOT_have \<open>... \<equiv> \<not>\<box>\<phi> \<or> \<diamond>\<psi>\<close>
    by (AOT_subst \<open>\<not>\<box>\<phi>\<close> \<open>\<diamond>\<not>\<phi>\<close>)
       (auto simp: "KBasic:11" "oth-class-taut:3:a")
  also AOT_have \<open>... \<equiv> \<box>\<phi> \<rightarrow> \<diamond>\<psi>\<close>
    using "\<equiv>E"(6) "oth-class-taut:1:c" "oth-class-taut:3:a" by blast
  finally show ?thesis .
qed

AOT_theorem "KBasic2:5": \<open>\<diamond>\<diamond>\<phi> \<equiv> \<not>\<box>\<box>\<not>\<phi>\<close>
  using "conventions:5"[THEN "\<equiv>Df"]
  by (AOT_subst \<open>\<diamond>\<phi>\<close> \<open>\<not>\<box>\<not>\<phi>\<close>;
      AOT_subst \<open>\<diamond>\<not>\<box>\<not>\<phi>\<close> \<open>\<not>\<box>\<not>\<not>\<box>\<not>\<phi>\<close>;
      AOT_subst (reverse) \<open>\<not>\<not>\<box>\<not>\<phi>\<close> \<open>\<box>\<not>\<phi>\<close>)
     (auto simp: "oth-class-taut:3:b" "oth-class-taut:3:a")


AOT_theorem "KBasic2:6": \<open>\<box>(\<phi> \<or> \<psi>) \<rightarrow> (\<box>\<phi> \<or> \<diamond>\<psi>)\<close>
proof(rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<box>(\<phi> \<or> \<psi>)\<close>
  AOT_hence \<open>\<box>(\<not>\<phi> \<rightarrow> \<psi>)\<close>
    using "conventions:2"[THEN "\<equiv>Df"]
    by (AOT_subst (reverse) \<open>\<not>\<phi> \<rightarrow> \<psi>\<close> \<open>\<phi> \<or> \<psi>\<close>) simp
  AOT_hence 1: \<open>\<diamond>\<not>\<phi> \<rightarrow> \<diamond>\<psi>\<close>
    using "KBasic:13" "vdash-properties:10" by blast
  AOT_assume \<open>\<not>(\<box>\<phi> \<or> \<diamond>\<psi>)\<close>
  AOT_hence \<open>\<not>\<box>\<phi>\<close> and \<open>\<not>\<diamond>\<psi>\<close>
    using "&E" "\<equiv>E"(1) "oth-class-taut:5:d" by blast+
  AOT_thus \<open>\<diamond>\<psi> & \<not>\<diamond>\<psi>\<close>
    using "&I"(1) 1[THEN "\<rightarrow>E"] "KBasic:11" "\<equiv>E"(4) "raa-cor:3" by blast
qed

AOT_theorem "KBasic2:7": \<open>(\<box>(\<phi> \<or> \<psi>) & \<diamond>\<not>\<phi>) \<rightarrow> \<diamond>\<psi>\<close>
proof(rule "\<rightarrow>I"; frule "&E"(1); drule "&E"(2))
  AOT_assume \<open>\<box>(\<phi> \<or> \<psi>)\<close>
  AOT_hence 1: \<open>\<box>\<phi> \<or> \<diamond>\<psi>\<close>
    using "KBasic2:6" "\<or>I"(2) "\<or>E"(1) by blast
  AOT_assume \<open>\<diamond>\<not>\<phi>\<close>
  AOT_hence \<open>\<not>\<box>\<phi>\<close> using "KBasic:11" "\<equiv>E"(2) by blast
  AOT_thus \<open>\<diamond>\<psi>\<close> using 1 "\<or>E"(2) by blast
qed

AOT_theorem "T-S5-fund:1": \<open>\<phi> \<rightarrow> \<diamond>\<phi>\<close>
  by (meson "\<equiv>\<^sub>d\<^sub>fI" "conventions:5" "contraposition:2"
            "Hypothetical Syllogism" "\<rightarrow>I" "qml:2"[axiom_inst])
lemmas "T\<diamond>" = "T-S5-fund:1"

AOT_theorem "T-S5-fund:2": \<open>\<diamond>\<box>\<phi> \<rightarrow> \<box>\<phi>\<close>
proof(rule "\<rightarrow>I")
  AOT_assume \<open>\<diamond>\<box>\<phi>\<close>
  AOT_hence \<open>\<not>\<box>\<diamond>\<not>\<phi>\<close>
    using "KBasic:14" "\<equiv>E"(4) "raa-cor:3" by blast
  moreover AOT_have \<open>\<diamond>\<not>\<phi> \<rightarrow> \<box>\<diamond>\<not>\<phi>\<close>
    by (fact "qml:3"[axiom_inst])
  ultimately AOT_have \<open>\<not>\<diamond>\<not>\<phi>\<close>
    using "modus-tollens:1" by blast
  AOT_thus \<open>\<box>\<phi>\<close> using "KBasic:12" "\<equiv>E"(2) by blast
qed
lemmas "5\<diamond>" = "T-S5-fund:2"

AOT_theorem "Act-Sub:1": \<open>\<^bold>\<A>\<phi> \<equiv> \<not>\<^bold>\<A>\<not>\<phi>\<close>
  by (AOT_subst \<open>\<^bold>\<A>\<not>\<phi>\<close> \<open>\<not>\<^bold>\<A>\<phi>\<close>)
     (auto simp: "logic-actual-nec:1"[axiom_inst] "oth-class-taut:3:b")

AOT_theorem "Act-Sub:2": \<open>\<diamond>\<phi> \<equiv> \<^bold>\<A>\<diamond>\<phi>\<close>
  using "conventions:5"[THEN "\<equiv>Df"]
  by (AOT_subst \<open>\<diamond>\<phi>\<close> \<open>\<not>\<box>\<not>\<phi>\<close>)
     (metis "deduction-theorem" "\<equiv>I" "\<equiv>E"(1) "\<equiv>E"(2) "\<equiv>E"(3)
            "logic-actual-nec:1"[axiom_inst] "qml-act:2"[axiom_inst])

AOT_theorem "Act-Sub:3": \<open>\<^bold>\<A>\<phi> \<rightarrow> \<diamond>\<phi>\<close>
  using "conventions:5"[THEN "\<equiv>Df"]
  by (AOT_subst \<open>\<diamond>\<phi>\<close> \<open>\<not>\<box>\<not>\<phi>\<close>)
     (metis "Act-Sub:1" "\<rightarrow>I" "\<equiv>E"(4) "nec-imp-act" "reductio-aa:2" "\<rightarrow>E")

AOT_theorem "Act-Sub:4": \<open>\<^bold>\<A>\<phi> \<equiv> \<diamond>\<^bold>\<A>\<phi>\<close>
proof (rule "\<equiv>I"; rule "\<rightarrow>I")
  AOT_assume \<open>\<^bold>\<A>\<phi>\<close>
  AOT_thus \<open>\<diamond>\<^bold>\<A>\<phi>\<close> using "T\<diamond>" "vdash-properties:10" by blast
next
  AOT_assume \<open>\<diamond>\<^bold>\<A>\<phi>\<close>
  AOT_hence \<open>\<not>\<box>\<not>\<^bold>\<A>\<phi>\<close>
    using "\<equiv>\<^sub>d\<^sub>fE" "conventions:5" by blast
  AOT_hence \<open>\<not>\<box>\<^bold>\<A>\<not>\<phi>\<close>
    by (AOT_subst \<open>\<^bold>\<A>\<not>\<phi>\<close> \<open>\<not>\<^bold>\<A>\<phi>\<close>)
       (simp add: "logic-actual-nec:1"[axiom_inst])
  AOT_thus \<open>\<^bold>\<A>\<phi>\<close>
    using "Act-Basic:1" "Act-Basic:6" "\<or>E"(3) "\<equiv>E"(4)
          "reductio-aa:1" by blast
qed

AOT_theorem "Act-Sub:5": \<open>\<diamond>\<^bold>\<A>\<phi> \<rightarrow> \<^bold>\<A>\<diamond>\<phi>\<close>
  by (metis "Act-Sub:2" "Act-Sub:3" "Act-Sub:4" "\<rightarrow>I" "\<equiv>E"(1) "\<equiv>E"(2) "\<rightarrow>E")

AOT_theorem "S5Basic:1": \<open>\<diamond>\<phi> \<equiv> \<box>\<diamond>\<phi>\<close>
  by (simp add: "\<equiv>I" "qml:2"[axiom_inst] "qml:3"[axiom_inst])

AOT_theorem "S5Basic:2": \<open>\<box>\<phi> \<equiv> \<diamond>\<box>\<phi>\<close>
  by (simp add: "T\<diamond>" "5\<diamond>" "\<equiv>I")

AOT_theorem "S5Basic:3": \<open>\<phi> \<rightarrow> \<box>\<diamond>\<phi>\<close>
  using "T\<diamond>" "Hypothetical Syllogism" "qml:3"[axiom_inst] by blast
lemmas "B" = "S5Basic:3"

AOT_theorem "S5Basic:4": \<open>\<diamond>\<box>\<phi> \<rightarrow> \<phi>\<close>
  using "5\<diamond>" "Hypothetical Syllogism" "qml:2"[axiom_inst] by blast
lemmas "B\<diamond>" = "S5Basic:4"

AOT_theorem "S5Basic:5": \<open>\<box>\<phi> \<rightarrow> \<box>\<box>\<phi>\<close>
  using "RM:1" "B" "5\<diamond>" "Hypothetical Syllogism" by blast
lemmas "4" = "S5Basic:5"

AOT_theorem "S5Basic:6": \<open>\<box>\<phi> \<equiv> \<box>\<box>\<phi>\<close>
  by (simp add: "4" "\<equiv>I" "qml:2"[axiom_inst])

AOT_theorem "S5Basic:7": \<open>\<diamond>\<diamond>\<phi> \<rightarrow> \<diamond>\<phi>\<close>
  using "conventions:5"[THEN "\<equiv>Df"] "oth-class-taut:3:b"
  by (AOT_subst \<open>\<diamond>\<diamond>\<phi>\<close> \<open>\<not>\<box>\<not>\<diamond>\<phi>\<close>;
      AOT_subst \<open>\<diamond>\<phi>\<close> \<open>\<not>\<box>\<not>\<phi>\<close>;
      AOT_subst (reverse) \<open>\<not>\<not>\<box>\<not>\<phi>\<close> \<open>\<box>\<not>\<phi>\<close>;
      AOT_subst (reverse) \<open>\<box>\<box>\<not>\<phi>\<close> \<open>\<box>\<not>\<phi>\<close>)
     (auto simp: "S5Basic:6" "if-p-then-p")

lemmas "4\<diamond>" = "S5Basic:7"

AOT_theorem "S5Basic:8": \<open>\<diamond>\<diamond>\<phi> \<equiv> \<diamond>\<phi>\<close>
  by (simp add: "4\<diamond>" "T\<diamond>" "\<equiv>I")

AOT_theorem "S5Basic:9": \<open>\<box>(\<phi> \<or> \<box>\<psi>) \<equiv> (\<box>\<phi> \<or> \<box>\<psi>)\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "KBasic2:6" "5\<diamond>" "\<or>I"(3) "if-p-then-p" "vdash-properties:10"
   apply blast
  by (meson "KBasic:15" "4" "\<or>I"(3) "\<or>E"(1) "Disjunction Addition"(1)
            "con-dis-taut:7" "intro-elim:1" "Commutativity of \<or>")

AOT_theorem "S5Basic:10": \<open>\<box>(\<phi> \<or> \<diamond>\<psi>) \<equiv> (\<box>\<phi> \<or> \<diamond>\<psi>)\<close>
proof(rule "\<equiv>I"; rule "\<rightarrow>I")
  AOT_assume \<open>\<box>(\<phi> \<or> \<diamond>\<psi>)\<close>
  AOT_hence \<open>\<box>\<phi> \<or> \<diamond>\<diamond>\<psi>\<close>
    by (meson "KBasic2:6" "\<or>I"(2) "\<or>E"(1))
  AOT_thus \<open>\<box>\<phi> \<or> \<diamond>\<psi>\<close>
    by (meson "B\<diamond>" "4" "4\<diamond>" "T\<diamond>" "\<or>I"(3))
next
  AOT_assume \<open>\<box>\<phi> \<or> \<diamond>\<psi>\<close>
  AOT_hence \<open>\<box>\<phi> \<or> \<box>\<diamond>\<psi>\<close>
    by (meson "S5Basic:1" "B\<diamond>" "S5Basic:6" "T\<diamond>" "5\<diamond>" "\<or>I"(3) "intro-elim:1")
  AOT_thus \<open>\<box>(\<phi> \<or> \<diamond>\<psi>)\<close>
    by (meson "KBasic:15" "\<or>I"(3) "\<or>E"(1) "Disjunction Addition"(1,2))
qed

AOT_theorem "S5Basic:11": \<open>\<diamond>(\<phi> & \<diamond>\<psi>) \<equiv> (\<diamond>\<phi> &</span> \<diamond>\<psi>)\<close>
proof -
  AOT_have \<open>\<diamond>(\<phi> & \<diamond>\<psi>) \<equiv> \<diamond>\<not>(\<not>\<phi> \<or> \<not>\<diamond>\<psi>)\<close>
    by (AOT_subst \<open>\<phi> & \<diamond>\<psi>\<close> \<open>\<not>(\<not>\<phi> \<or> \<not>\<diamond>\<psi>)\<close>)
       (auto simp: "oth-class-taut:5:a" "oth-class-taut:3:a")
  also AOT_have \<open>\<dots> \<equiv> \<diamond>\<not>(\<not>\<phi> \<or> \<box>\<not>\<psi>)\<close>
    by (AOT_subst \<open>\<box>\<not>\<psi>\<close> \<open>\<not>\<diamond>\<psi>\<close>)
       (auto simp: "KBasic2:1" "oth-class-taut:3:a")
  also AOT_have \<open>\<dots> \<equiv> \<not>\<box>(\<not>\<phi> \<or> \<box>\<not>\<psi>)\<close>
    using "KBasic:11" "\<equiv>E"(6) "oth-class-taut:3:a" by blast
  also AOT_have \<open>\<dots> \<equiv> \<not>(\<box>\<not>\<phi> \<or> \<box>\<not>\<psi>)\<close>
    using "S5Basic:9" "\<equiv>E"(1) "oth-class-taut:4:b" by blast
  also AOT_have \<open>\<dots> \<equiv> \<not>(\<not>\<diamond>\<phi> \<or> \<not>\<diamond>\<psi>)\<close>
    using "KBasic2:1"
    by (AOT_subst \<open>\<box>\<not>\<phi>\<close> \<open>\<not>\<diamond>\<phi>\<close>; AOT_subst \<open>\<box>\<not>\<psi>\<close> \<open>\<not>\<diamond>\<psi>\<close>)
       (auto simp:  "oth-class-taut:3:a")
  also AOT_have \<open>\<dots> \<equiv> \<diamond>\<phi> & \<diamond>\<psi>\<close>
    using "\<equiv>E"(6) "oth-class-taut:3:a" "oth-class-taut:5:a" by blast
  finally show ?thesis .
qed

AOT_theorem "S5Basic:12": \<open>\<diamond>(\<phi> & \<box>\<psi>) \<equiv> (\<diamond>\<phi> &n> \<box>\<psi>)\<close>
proof (rule "\<equiv>I"; rule "\<rightarrow>I")
  AOT_assume \<open>\<diamond>(\<phi> & \<box>\<psi>)\<close>
  AOT_hence \<open>\<diamond>\<phi> & \<diamond>\<box>\<psi>\<close>
    using "KBasic2:3" "vdash-properties:6" by blast
  AOT_thus \<open>\<diamond>\<phi> & \<box>\<psi>\<close>
    using "5\<diamond>" "&I" "&E"(1) "&E"(2) "vdash-properties:6" by blast
next
  AOT_assume \<open>\<diamond>\<phi> & \<box>\<psi>\<close>
  moreover AOT_have \<open>(\<box>\<box>\<psi> & \<diamond>\<phi>) \<rightarrow> \<diamond>(\<phi> &an> \<box>\<psi>)\<close>
    by (AOT_subst \<open>\<phi> & \<box>\<psi>\<close> \<open>\<box>\<psi> & \<phi>\<close>)
       (auto simp: "Commutativity of &" "KBasic:16")
  ultimately AOT_show \<open>\<diamond>(\<phi> & \<box>\<psi>)\<close>
    by (metis "4" "&I" "Conjunction Simplification"(1,2) "\<rightarrow>E")
qed

AOT_theorem "S5Basic:13": \<open>\<box>(\<phi> \<rightarrow> \<box>\<psi>) \<equiv> \<box>(\<diamond>\<phi> \<rightarrow> \<psi>)\<close>
proof (rule "\<equiv>I")
  AOT_modally_strict {
    AOT_have \<open>\<box>(\<phi> \<rightarrow> \<box>\<psi>) \<rightarrow> (\<diamond>\<phi> \<rightarrow> \<psi>)\<close>
      by (meson "KBasic:13" "B\<diamond>" "Hypothetical Syllogism" "\<rightarrow>I")
  }
  AOT_hence \<open>\<box>\<box>(\<phi> \<rightarrow> \<box>\<psi>) \<rightarrow> \<box>(\<diamond>\<phi> \<rightarrow> \<psi>)\<close>
    by (rule RM)
  AOT_thus  \<open>\<box>(\<phi> \<rightarrow> \<box>\<psi>) \<rightarrow> \<box>(\<diamond>\<phi> \<rightarrow> \<psi>)\<close>
    using "4" "Hypothetical Syllogism" by blast
next
  AOT_modally_strict {
    AOT_have \<open>\<box>(\<diamond>\<phi> \<rightarrow> \<psi>) \<rightarrow> (\<phi> \<rightarrow> \<box>\<psi>)\<close>
      by (meson "B" "Hypothetical Syllogism" "\<rightarrow>I" "qml:1"[axiom_inst])
  }
  AOT_hence  \<open>\<box>\<box>(\<diamond>\<phi> \<rightarrow> \<psi>) \<rightarrow> \<box>(\<phi> \<rightarrow> \<box>\<psi>)\<close>
    by (rule RM)
  AOT_thus \<open>\<box>(\<diamond>\<phi> \<rightarrow> \<psi>) \<rightarrow> \<box>(\<phi> \<rightarrow> \<box>\<psi>)\<close>
    using "4" "Hypothetical Syllogism" by blast
qed

AOT_theorem "derived-S5-rules:1":
  assumes \<open>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<diamond>\<phi> \<rightarrow> \<psi>\<close>
  shows \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<phi> \<rightarrow> \<box>\<psi>\<close>
proof -
  AOT_have \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<box>\<diamond>\<phi> \<rightarrow> \<box>\<psi>\<close>
    using assms by (rule "RM:1[prem]")
  AOT_thus \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<phi> \<rightarrow> \<box>\<psi>\<close>
    using "B" "Hypothetical Syllogism" by blast
qed

AOT_theorem "derived-S5-rules:2":
  assumes \<open>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<phi> \<rightarrow> \<box>\<psi>\<close>
  shows \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<diamond>\<phi> \<rightarrow> \<psi>\<close>
proof -
  AOT_have \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<diamond>\<phi> \<rightarrow> \<diamond>\<box>\<psi>\<close>
    using assms by (rule "RM:2[prem]")
  AOT_thus \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<diamond>\<phi> \<rightarrow> \<psi>\<close>
    using "B\<diamond>" "Hypothetical Syllogism" by blast
qed

AOT_theorem "BFs:1": \<open>\<forall>\<alpha> \<box>\<phi>{\<alpha>} \<rightarrow> \<box>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
proof -
  AOT_modally_strict {
    AOT_have \<open>\<diamond>\<forall>\<alpha> \<box>\<phi>{\<alpha>} \<rightarrow> \<diamond>\<box>\<phi>{\<alpha>}\<close> for \<alpha>
      using "cqt-orig:3" by (rule "RM\<diamond>")
    AOT_hence \<open>\<diamond>\<forall>\<alpha> \<box>\<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> \<phi>{\<alpha>}\<close>
      using "B\<diamond>" "\<forall>I" "\<rightarrow>E" "\<rightarrow>I" by metis
  }
  thus ?thesis
    using "derived-S5-rules:1" by blast
qed
lemmas "BF" = "BFs:1"

AOT_theorem "BFs:2": \<open>\<box>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> \<box>\<phi>{\<alpha>}\<close>
proof -
  AOT_have \<open>\<box>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<box>\<phi>{\<alpha>}\<close> for \<alpha>
    using RM "cqt-orig:3" by metis
  thus ?thesis
    using "cqt-orig:2"[THEN "\<rightarrow>E"] "\<forall>I" by metis
qed
lemmas "CBF" = "BFs:2"

AOT_theorem "BFs:3": \<open>\<diamond>\<exists>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<exists>\<alpha> \<diamond>\<phi>{\<alpha>}\<close>
proof(rule "\<rightarrow>I")
  AOT_modally_strict {
    AOT_have \<open>\<box>\<forall>\<alpha> \<not>\<phi>{\<alpha>} \<equiv> \<forall>\<alpha> \<box>\<not>\<phi>{\<alpha>}\<close>
      using BF CBF "\<equiv>I" by blast
  } note \<theta> = this

  AOT_assume \<open>\<diamond>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
  AOT_hence \<open>\<not>\<box>\<not>(\<exists>\<alpha> \<phi>{\<alpha>})\<close>
    using "\<equiv>\<^sub>d\<^sub>fE" "conventions:5" by blast
  AOT_hence \<open>\<not>\<box>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>
    apply (AOT_subst \<open>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close> \<open>\<not>(\<exists>\<alpha> \<phi>{\<alpha>})\<close>)
    using "\<equiv>\<^sub>d\<^sub>fI" "conventions:3" "conventions:4" "&I"
          "contraposition:2" "cqt-further:4"
          "df-rules-formulas[3]" by blast
  AOT_hence \<open>\<not>\<forall>\<alpha> \<box>\<not>\<phi>{\<alpha>}\<close>
    apply (AOT_subst (reverse) \<open>\<forall>\<alpha> \<box>\<not>\<phi>{\<alpha>}\<close> \<open>\<box>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>)
    using \<theta> by blast
  AOT_hence \<open>\<not>\<forall>\<alpha> \<not>\<not>\<box>\<not>\<phi>{\<alpha>}\<close>
    by (AOT_subst (reverse) \<open>\<not>\<not>\<box>\<not>\<phi>{\<alpha>}\<close> \<open>\<box>\<not>\<phi>{\<alpha>}\<close> for: \<alpha>)
       (simp add: "oth-class-taut:3:b")
  AOT_hence \<open>\<exists>\<alpha> \<not>\<box>\<not>\<phi>{\<alpha>}\<close>
    by (rule "conventions:4"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
  AOT_thus \<open>\<exists>\<alpha> \<diamond>\<phi>{\<alpha>}\<close>
    using "conventions:5"[THEN "\<equiv>Df"]
    by (AOT_subst \<open>\<diamond>\<phi>{\<alpha>}\<close> \<open>\<not>\<box>\<not>\<phi>{\<alpha>}\<close> for: \<alpha>)
qed
lemmas "BF\<diamond>" = "BFs:3"

AOT_theorem "BFs:4": \<open>\<exists>\<alpha> \<diamond>\<phi>{\<alpha>} \<rightarrow> \<diamond>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
proof(rule "\<rightarrow>I")
  AOT_assume \<open>\<exists>\<alpha> \<diamond>\<phi>{\<alpha>}\<close>
  AOT_hence \<open>\<not>\<forall>\<alpha> \<not>\<diamond>\<phi>{\<alpha>}\<close>
    using "conventions:4"[THEN "\<equiv>\<^sub>d\<^sub>fE"] by blast
  AOT_hence \<open>\<not>\<forall>\<alpha> \<box>\<not>\<phi>{\<alpha>}\<close>
    using "KBasic2:1"
    by (AOT_subst \<open>\<box>\<not>\<phi>{\<alpha>}\<close> \<open>\<not>\<diamond>\<phi>{\<alpha>}\<close> for: \<alpha>)
  moreover AOT_have \<open>\<forall>\<alpha> \<box>\<not>\<phi>{\<alpha>} \<equiv> \<box>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>
    using "\<equiv>I" "BF" "CBF" by metis
  ultimately AOT_have 1: \<open>\<not>\<box>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>
    using "\<equiv>E"(3) by blast
  AOT_show \<open>\<diamond>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
    apply (rule "conventions:5"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
    apply (AOT_subst \<open>\<exists>\<alpha> \<phi>{\<alpha>}\<close> \<open>\<not>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>)
     apply (simp add: "conventions:4" "\<equiv>Df")
    apply (AOT_subst \<open>\<not>\<not>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close> \<open>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>)
    by (auto simp: 1 "\<equiv>I" "useful-tautologies:1" "useful-tautologies:2")
qed
lemmas "CBF\<diamond>" = "BFs:4"

AOT_theorem "sign-S5-thm:1": \<open>\<exists>\<alpha> \<box>\<phi>{\<alpha>} \<rightarrow> \<box>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
proof(rule "\<rightarrow>I")
  AOT_assume \<open>\<exists>\<alpha> \<box>\<phi>{\<alpha>}\<close>
  then AOT_obtain \<alpha> where \<open>\<box>\<phi>{\<alpha>}\<close> using "\<exists>E" by metis
  moreover AOT_have \<open>\<box>\<alpha>\<down>\<close>
    by (simp add: "ex:1:a" "rule-ui:2[const_var]" RN)
  moreover AOT_have \<open>\<box>\<phi>{\<tau>}, \<box>\<tau>\<down> \<^bold>\<turnstile>\<^sub>\<box> \<box>\<exists>\<alpha> \<phi>{\<alpha>}\<close> for \<tau>
  proof -
    AOT_have \<open>\<phi>{\<tau>}, \<tau>\<down> \<^bold>\<turnstile>\<^sub>\<box> \<exists>\<alpha> \<phi>{\<alpha>}\<close> using "existential:1" by blast
    AOT_thus \<open>\<box>\<phi>{\<tau>}, \<box>\<tau>\<down> \<^bold>\<turnstile>\<^sub>\<box> \<box>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
      using "RN[prem]"[where \<Gamma>="{\<phi> \<tau>, \<guillemotleft>\<tau>\<down>\<guillemotright>}", simplified] by blast
  qed
  ultimately AOT_show \<open>\<box>\<exists>\<alpha> \<phi>{\<alpha>}\<close> by blast
qed
lemmas Buridan = "sign-S5-thm:1"

AOT_theorem "sign-S5-thm:2": \<open>\<diamond>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> \<diamond>\<phi>{\<alpha>}\<close>
proof -
  AOT_have \<open>\<forall>\<alpha> (\<diamond>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<diamond>\<phi>{\<alpha>})\<close>
    by (simp add: "RM\<diamond>" "cqt-orig:3" "\<forall>I")
  AOT_thus \<open>\<diamond>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> \<diamond>\<phi>{\<alpha>}\<close>
    using "\<forall>E"(4) "\<forall>I" "\<rightarrow>E" "\<rightarrow>I" by metis
qed
lemmas "Buridan\<diamond>" = "sign-S5-thm:2"

AOT_theorem "sign-S5-thm:3":
  \<open>\<diamond>\<exists>\<alpha> (\<phi>{\<alpha>} & \<psi>{\<alpha>}) \<rightarrow> \<diamond>(\<exists>\<alpha> \<phi>{\<alpha>} & \<exists>\<alpha> \<psi>{\<alpha>})\<close>
  apply (rule "RM:2")
  by (metis (no_types, lifting) "\<exists>E" "&I" "&E"(1) "&E"(2) "\<rightarrow>I" "\<exists>I"(2))

AOT_theorem "sign-S5-thm:4": \<open>\<diamond>\<exists>\<alpha> (\<phi>{\<alpha>} & \<psi>{\<alpha>}) \<rightarrow> \<diamond>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
  apply (rule "RM:2")
  by (meson "instantiation" "&E"(1) "\<rightarrow>I" "\<exists>I"(2))

AOT_theorem "sign-S5-thm:5":
  \<open>(\<box>\<forall>\<alpha> (\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}) & \<box>\<forall>\<alpha> (\<psi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})) \<rightarrow> \<box>\<forall>\<alpha> (\<phi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})\<close>
proof -
  {
    fix \<phi>' \<psi>' \<chi>'
    AOT_assume \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>' & \<psi>' \<rightarrow> \<chi>'\<close>
    AOT_hence \<open>\<box>\<phi>' & \<box>\<psi>' \<rightarrow> \<box>\<chi>'\<close>
      using "RN[prem]"[where \<Gamma>="{\<phi>', \<psi>'}"] apply simp
      using "&E" "&I" "\<rightarrow>E" "\<rightarrow>I" by metis
  } note R = this
  show ?thesis by (rule R; fact AOT)
qed

AOT_theorem "sign-S5-thm:6":
  \<open>(\<box>\<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) & \<box>\<forall>\<alpha>(\<psi>{\<alpha>} \<equiv> \<chi>{\<alpha>})) \<rightarrow> \<box>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<chi>{\<alpha>})\<close>
proof -
  {
    fix \<phi>' \<psi>' \<chi>'
    AOT_assume \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>' & \<psi>' \<rightarrow> \<chi>'\<close>
    AOT_hence \<open>\<box>\<phi>' & \<box>\<psi>' \<rightarrow> \<box>\<chi>'\<close>
      using "RN[prem]"[where \<Gamma>="{\<phi>', \<psi>'}"] apply simp
      using "&E" "&I" "\<rightarrow>E" "\<rightarrow>I" by metis
  } note R = this
  show ?thesis by (rule R; fact AOT)
qed

AOT_theorem "exist-nec2:1": \<open>\<diamond>\<tau>\<down> \<rightarrow> \<tau>\<down>\<close>
  using "B\<diamond>" "RM\<diamond>" "Hypothetical Syllogism" "exist-nec" by blast

AOT_theorem "exists-nec2:2": \<open>\<diamond>\<tau>\<down> \<equiv> \<box>\<tau>\<down>\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism" "exist-nec"
            "exist-nec2:1" "\<equiv>I" "nec-imp-act")

AOT_theorem "exists-nec2:3": \<open>\<not>\<tau>\<down> \<rightarrow> \<box>\<not>\<tau>\<down>\<close>
  using "KBasic2:1" "\<rightarrow>I" "exist-nec2:1" "\<equiv>E"(2) "modus-tollens:1" by blast

AOT_theorem "exists-nec2:4": \<open>\<diamond>\<not>\<tau>\<down> \<equiv> \<box>\<not>\<tau>\<down>\<close>
  by (metis "Act-Sub:3" "KBasic:12" "\<rightarrow>I" "exist-nec" "exists-nec2:3"
            "\<equiv>I" "\<equiv>E"(4) "nec-imp-act" "reductio-aa:1")

AOT_theorem "id-nec2:1": \<open>\<diamond>\<alpha> = \<beta> \<rightarrow> \<alpha> = \<beta>\<close>
  using "B\<diamond>" "RM\<diamond>" "Hypothetical Syllogism" "id-nec:1" by blast

AOT_theorem "id-nec2:2": \<open>\<alpha> \<noteq> \<beta> \<rightarrow> \<box>\<alpha> \<noteq> \<beta>\<close>
  apply (AOT_subst \<open>\<alpha> \<noteq> \<beta>\<close> \<open>\<not>(\<alpha> = \<beta>)\<close>)
  using "=-infix"[THEN "\<equiv>Df"] apply blast
  using "KBasic2:1" "\<rightarrow>I" "id-nec2:1" "\<equiv>E"(2) "modus-tollens:1" by blast

AOT_theorem "id-nec2:3": \<open>\<diamond>\<alpha> \<noteq> \<beta> \<rightarrow> \<alpha> \<noteq> \<beta>\<close>
  apply (AOT_subst \<open>\<alpha> \<noteq> \<beta>\<close> \<open>\<not>(\<alpha> = \<beta>)\<close>)
  using "=-infix"[THEN "\<equiv>Df"] apply blast
  by (metis "KBasic:11" "\<rightarrow>I" "id-nec:2" "\<equiv>E"(3) "reductio-aa:2" "\<rightarrow>E")

AOT_theorem "id-nec2:4": \<open>\<diamond>\<alpha> = \<beta> \<rightarrow> \<box>\<alpha> = \<beta>\<close>
  using "Hypothetical Syllogism" "id-nec2:1" "id-nec:1" by blast

AOT_theorem "id-nec2:5": \<open>\<diamond>\<alpha> \<noteq> \<beta> \<rightarrow> \<box>\<alpha> \<noteq> \<beta>\<close>
  using "id-nec2:3" "id-nec2:2" "\<rightarrow>I" "\<rightarrow>E" by metis

AOT_theorem "sc-eq-box-box:1": \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<equiv> (\<diamond>\<phi> \<rightarrow> \<box>\<phi>)\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "KBasic:13" "5\<diamond>" "Hypothetical Syllogism" "\<rightarrow>E" apply blast
  by (metis "KBasic2:1" "KBasic:1" "KBasic:2" "S5Basic:13" "\<equiv>E"(2)
            "raa-cor:5" "\<rightarrow>E")

AOT_theorem "sc-eq-box-box:2": \<open>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<or> (\<diamond>\<phi> \<rightarrow> \<box>\<phi>)) \<rightarrow> (\<diamond>\<phi> \<equiv> \<box>\<phi>)\<close>
  by (metis "Act-Sub:3" "KBasic:13" "5\<diamond>" "\<or>E"(2) "\<rightarrow>I" "\<equiv>I"
            "nec-imp-act" "raa-cor:2" "\<rightarrow>E")

AOT_theorem "sc-eq-box-box:3": \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<rightarrow> (\<not>\<box>\<phi> \<equiv> \<box>\<not>\<phi>)\<close>
proof (rule "\<rightarrow>I"; rule "\<equiv>I"; rule "\<rightarrow>I")
  AOT_assume \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>)\<close>
  AOT_hence \<open>\<diamond>\<phi> \<rightarrow> \<box>\<phi>\<close> using "sc-eq-box-box:1" "\<equiv>E" by blast
  moreover AOT_assume \<open>\<not>\<box>\<phi>\<close>
  ultimately AOT_have \<open>\<not>\<diamond>\<phi>\<close>
    using "modus-tollens:1" by blast
  AOT_thus \<open>\<box>\<not>\<phi>\<close>
    using "KBasic2:1" "\<equiv>E"(2) by blast
next
  AOT_assume \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>)\<close>
  moreover AOT_assume \<open>\<box>\<not>\<phi>\<close>
  ultimately AOT_show \<open>\<not>\<box>\<phi>\<close>
    using "modus-tollens:1" "qml:2"[axiom_inst] "\<rightarrow>E" by blast
qed

AOT_theorem "sc-eq-box-box:4":
  \<open>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>)) \<rightarrow> ((\<box>\<phi> \<equiv> \<box>\<psi>) \<rightarrow> \<box>(\<phi> \<equiv> \<psi>))\<close>
proof(rule "\<rightarrow>I"; rule "\<rightarrow>I")
  AOT_assume \<theta>: \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>)\<close>
  AOT_assume \<xi>: \<open>\<box>\<phi> \<equiv> \<box>\<psi>\<close>
  AOT_hence \<open>(\<box>\<phi> & \<box>\<psi>) \<or> (\<not>\<box>\<phi> & \<not>\<box>\<psi>)\<close>
    using "\<equiv>E"(4) "oth-class-taut:4:g" "raa-cor:3" by blast
  moreover {
    AOT_assume \<open>\<box>\<phi> & \<box>\<psi>\<close>
    AOT_hence \<open>\<box>(\<phi> \<equiv> \<psi>)\<close>
      using "KBasic:3" "KBasic:8" "\<equiv>E"(2) "vdash-properties:10" by blast
  }
  moreover {
    AOT_assume \<open>\<not>\<box>\<phi> & \<not>\<box>\<psi>\<close>
    moreover AOT_have \<open>\<not>\<box>\<phi> \<equiv> \<box>\<not>\<phi>\<close> and \<open>\<not>\<box>\<psi> \<equiv> \<box>\<not>\<psi>\<close>
      using \<theta> "Conjunction Simplification"(1,2)
            "sc-eq-box-box:3" "\<rightarrow>E" by metis+
    ultimately AOT_have \<open>\<box>\<not>\<phi> & \<box>\<not>\<psi>\<close>
      by (metis "&I" "Conjunction Simplification"(1,2)
                "\<equiv>E"(4) "modus-tollens:1" "raa-cor:3")
    AOT_hence \<open>\<box>(\<phi> \<equiv> \<psi>)\<close>
      using "KBasic:3" "KBasic:9" "\<equiv>E"(2) "\<rightarrow>E" by blast
  }
  ultimately AOT_show \<open>\<box>(\<phi> \<equiv> \<psi>)\<close>
    using "\<or>E"(2) "reductio-aa:1" by blast
qed

AOT_theorem "sc-eq-box-box:5":
  \<open>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>)) \<rightarrow> \<box>((\<phi> \<equiv> \<psi>) \<rightarrow> \<box>(\<phi> \<equiv> \<psi>))\<close>
proof (rule "\<rightarrow>I")
  AOT_assume \<open>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>))\<close>
  AOT_hence \<open>\<box>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>))\<close>
    using 4[THEN "\<rightarrow>E"] "&E" "&I" "KBasic:3" "\<equiv>E"(2) by metis
  moreover AOT_have \<open>\<box>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>)) \<rightarrow> \<box>((\<phi> \<equiv> \<psi>) \<rightarrow> \<box>(\<phi> \<equiv> \<psi>))\<close>
  proof (rule RM; rule "\<rightarrow>I"; rule "\<rightarrow>I")
    AOT_modally_strict {
      AOT_assume A: \<open>(\<box>(\<phi> \<rightarrow> \<box>\<phi>) & \<box>(\<psi> \<rightarrow> \<box>\<psi>))\<close>
      AOT_hence \<open>\<phi> \<rightarrow> \<box>\<phi>\<close> and \<open>\<psi> \<rightarrow> \<box>\<psi>\<close>
        using "&E" "qml:2"[axiom_inst] "\<rightarrow>E" by blast+
      moreover AOT_assume \<open>\<phi> \<equiv> \<psi>\<close>
      ultimately AOT_have \<open>\<box>\<phi> \<equiv> \<box>\<psi>\<close>
        using "\<rightarrow>E" "qml:2"[axiom_inst] "\<equiv>E" "\<equiv>I" by meson
      moreover AOT_have \<open>(\<box>\<phi> \<equiv> \<box>\<psi>) \<rightarrow> \<box>(\<phi> \<equiv> \<psi>)\<close>
        using A "sc-eq-box-box:4" "\<rightarrow>E" by blast
      ultimately AOT_show \<open>\<box>(\<phi> \<equiv> \<psi>)\<close> using "\<rightarrow>E" by blast
    }
  qed
  ultimately AOT_show \<open>\<box>((\<phi> \<equiv> \<psi>) \<rightarrow> \<box>(\<phi> \<equiv> \<psi>))\<close> using "\<rightarrow>E" by blast
qed

AOT_theorem "sc-eq-box-box:6": \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<rightarrow> ((\<phi> \<rightarrow> \<box>\<psi>) \<rightarrow> \<box>(\<phi> \<rightarrow> \<psi>))\<close>
proof (rule "\<rightarrow>I"; rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<box>(\<phi> \<rightarrow> \<psi>)\<close>
  AOT_hence \<open>\<diamond>\<not>(\<phi> \<rightarrow> \<psi>)\<close>
    by (metis "KBasic:11" "\<equiv>E"(1))
  AOT_hence \<open>\<diamond>(\<phi> & \<not>\<psi>)\<close>
    by (AOT_subst \<open>\<phi> & \<not>\<psi>\<close> \<open>\<not>(\<phi> \<rightarrow> \<psi>)\<close>)
       (meson "Commutativity of \<equiv>" "\<equiv>E"(1) "oth-class-taut:1:b")
  AOT_hence \<open>\<diamond>\<phi>\<close> and 2: \<open>\<diamond>\<not>\<psi>\<close>
    using "KBasic2:3"[THEN "\<rightarrow>E"] "&E" by blast+
  moreover AOT_assume \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>)\<close>
  ultimately AOT_have \<open>\<box>\<phi>\<close>
    by (metis "\<equiv>E"(1) "sc-eq-box-box:1" "\<rightarrow>E")
  AOT_hence \<phi>
    using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] by blast
  moreover AOT_assume \<open>\<phi> \<rightarrow> \<box>\<psi>\<close>
  ultimately AOT_have \<open>\<box>\<psi>\<close>
    using "\<rightarrow>E" by blast
  moreover AOT_have \<open>\<not>\<box>\<psi>\<close>
    using 2 "KBasic:12" "\<not>\<not>I" "intro-elim:3:d" by blast
  ultimately AOT_show \<open>\<box>\<psi> & \<not>\<box>\<psi>\<close>
    using "&I" by blast
qed

AOT_theorem "sc-eq-box-box:7": \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<rightarrow> ((\<phi> \<rightarrow> \<^bold>\<A>\<psi>) \<rightarrow> \<^bold>\<A>(\<phi> \<rightarrow> \<psi>))\<close>
proof (rule "\<rightarrow>I"; rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<^bold>\<A>(\<phi> \<rightarrow> \<psi>)\<close>
  AOT_hence \<open>\<^bold>\<A>\<not>(\<phi> \<rightarrow> \<psi>)\<close>
    by (metis "Act-Basic:1" "\<or>E"(2))
  AOT_hence \<open>\<^bold>\<A>(\<phi> & \<not>\<psi>)\<close>
    by (AOT_subst \<open>\<phi> & \<not>\<psi>\<close> \<open>\<not>(\<phi> \<rightarrow> \<psi>)\<close>)
       (meson "Commutativity of \<equiv>" "\<equiv>E"(1) "oth-class-taut:1:b")
  AOT_hence \<open>\<^bold>\<A>\<phi>\<close> and 2: \<open>\<^bold>\<A>\<not>\<psi>\<close>
    using "Act-Basic:2"[THEN "\<equiv>E"(1)] "&E" by blast+
  AOT_hence \<open>\<diamond>\<phi>\<close>
    by (metis "Act-Sub:3" "\<rightarrow>E")
  moreover AOT_assume \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>)\<close>
  ultimately AOT_have \<open>\<box>\<phi>\<close>
    by (metis "\<equiv>E"(1) "sc-eq-box-box:1" "\<rightarrow>E")
  AOT_hence \<phi>
    using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] by blast
  moreover AOT_assume \<open>\<phi> \<rightarrow> \<^bold>\<A>\<psi>\<close>
  ultimately AOT_have \<open>\<^bold>\<A>\<psi>\<close>
    using "\<rightarrow>E" by blast
  moreover AOT_have \<open>\<not>\<^bold>\<A>\<psi>\<close>
    using 2 by (meson "Act-Sub:1" "\<equiv>E"(4) "raa-cor:3")
  ultimately AOT_show \<open>\<^bold>\<A>\<psi> & \<not>\<^bold>\<A>\<psi>\<close>
    using "&I" by blast
qed

AOT_theorem "sc-eq-fur:1": \<open>\<diamond>\<^bold>\<A>\<phi> \<equiv> \<box>\<^bold>\<A>\<phi>\<close>
  using "Act-Basic:6" "Act-Sub:4" "\<equiv>E"(6) by blast

AOT_theorem "sc-eq-fur:2": \<open>\<box>(\<phi> \<rightarrow> \<box>\<phi>) \<rightarrow> (\<^bold>\<A>\<phi> \<equiv> \<phi>)\<close>
  by (metis "B\<diamond>" "Act-Sub:3" "KBasic:13" "T\<diamond>" "Hypothetical Syllogism"
            "\<rightarrow>I" "\<equiv>I" "nec-imp-act")

AOT_theorem "sc-eq-fur:3":
  \<open>\<box>\<forall>x (\<phi>{x} \<rightarrow> \<box>\<phi>{x}) \<rightarrow> (\<exists>!x \<phi>{x} \<rightarrow> \<^bold>\<iota>x \<phi>{x}\<down>)\<close>
proof (rule "\<rightarrow>I"; rule "\<rightarrow>I")
  AOT_assume \<open>\<box>\<forall>x (\<phi>{x} \<rightarrow> \<box>\<phi>{x})\<close>
  AOT_hence A: \<open>\<forall>x \<box>(\<phi>{x} \<rightarrow> \<box>\<phi>{x})\<close>
    using CBF "\<rightarrow>E" by blast
  AOT_assume \<open>\<exists>!x \<phi>{x}\<close>
  then AOT_obtain a where a_def: \<open>\<phi>{a} & \<forall>y (\<phi>{y} \<rightarrow> y = a)\<close>
    using "\<exists>E"[rotated 1, OF "uniqueness:1"[THEN "\<equiv>\<^sub>d\<^sub>fE"]] by blast
  moreover AOT_have \<open>\<box>\<phi>{a}\<close>
    using calculation A "\<forall>E"(2) "qml:2"[axiom_inst] "\<rightarrow>E" "&E"(1) by blast
  AOT_hence \<open>\<^bold>\<A>\<phi>{a}\<close>
    using "nec-imp-act" "\<rightarrow>E" by blast
  moreover AOT_have \<open>\<forall>y (\<^bold>\<A>\<phi>{y} \<rightarrow> y = a)\<close>
  proof (rule "\<forall>I"; rule "\<rightarrow>I")
    fix b
    AOT_assume \<open>\<^bold>\<A>\<phi>{b}\<close>
    AOT_hence \<open>\<diamond>\<phi>{b}\<close>
      using "Act-Sub:3" "\<rightarrow>E" by blast
    moreover {
      AOT_have \<open>\<box>(\<phi>{b} \<rightarrow> \<box>\<phi>{b})\<close>
        using A "\<forall>E"(2) by blast
      AOT_hence \<open>\<diamond>\<phi>{b} \<rightarrow> \<box>\<phi>{b}\<close>
        using "KBasic:13" "5\<diamond>" "Hypothetical Syllogism" "\<rightarrow>E" by blast
    }
    ultimately AOT_have \<open>\<box>\<phi>{b}\<close>
      using "\<rightarrow>E" by blast
    AOT_hence \<open>\<phi>{b}\<close>
      using "qml:2"[axiom_inst] "\<rightarrow>E" by blast
    AOT_thus \<open>b = a\<close>
      using a_def[THEN "&E"(2)] "\<forall>E"(2) "\<rightarrow>E" by blast
  qed
  ultimately AOT_have \<open>\<^bold>\<A>\<phi>{a} & \<forall>y (\<^bold>\<A>\<phi>{y} \<rightarrow> y = a)\<close>
    using "&I" by blast
  AOT_hence \<open>\<exists>x (\<^bold>\<A>\<phi>{x} & \<forall>y (\<^bold>\<A>\<phi>{y} \<rightarrow> y = x))\<close>
    using "\<exists>I" by fast
  AOT_hence \<open>\<exists>!x \<^bold>\<A>\<phi>{x}\<close>
    using "uniqueness:1"[THEN "\<equiv>\<^sub>d\<^sub>fI"] by fast
  AOT_thus \<open>\<^bold>\<iota>x \<phi>{x}\<down>\<close>
    using "actual-desc:1"[THEN "\<equiv>E"(2)] by blast
qed

AOT_theorem "sc-eq-fur:4":
\<open>\<box>\<forall>x (\<phi>{x} \<rightarrow> \<box>\<phi>{x}) \<rightarrow> (x = \<^bold>\<iota>x \<phi>{x} \<equiv> (\<phi>{x} & \<forall>z (\<phi>{z} \<rightarrow> z = x)))\<close>
proof (rule "\<rightarrow>I")
  AOT_assume \<open>\<box>\<forall>x (\<phi>{x} \<rightarrow> \<box>\<phi>{x})\<close>
  AOT_hence \<open>\<forall>x \<box>(\<phi>{x} \<rightarrow> \<box>\<phi>{x})\<close>
    using CBF "\<rightarrow>E" by blast
  AOT_hence A: \<open>\<^bold>\<A>\<phi>{\<alpha>} \<equiv> \<phi>{\<alpha>}\<close> for \<alpha>
    using "sc-eq-fur:2" "\<forall>E" "\<rightarrow>E" by fast
  AOT_show \<open>x = \<^bold>\<iota>x \<phi>{x} \<equiv> (\<phi>{x} & \<forall>z (\<phi>{z} \<rightarrow> z = x))\<close>
  proof (rule "\<equiv>I"; rule "\<rightarrow>I")
    AOT_assume \<open>x = \<^bold>\<iota>x \<phi>{x}\<close>
    AOT_hence B: \<open>\<^bold>\<A>\<phi>{x} & \<forall>z (\<^bold>\<A>\<phi>{z} \<rightarrow> z = x)\<close>
      using "nec-hintikka-scheme"[THEN "\<equiv>E"(1)] by blast
    AOT_show \<open>\<phi>{x} & \<forall>z (\<phi>{z} \<rightarrow> z = x)\<close>
    proof (rule "&I"; (rule "\<forall>I"; rule "\<rightarrow>I")?)
      AOT_show \<open>\<phi>{x}\<close>
        using A B[THEN "&E"(1)] "\<equiv>E"(1) by blast
    next
      AOT_show \<open>z = x\<close> if \<open>\<phi>{z}\<close> for z
        using that B[THEN "&E"(2)] "\<forall>E"(2) "\<rightarrow>E" A[THEN "\<equiv>E"(2)] by blast
    qed
  next
    AOT_assume B: \<open>\<phi>{x} & \<forall>z (\<phi>{z} \<rightarrow> z = x)\<close>
    AOT_have \<open>\<^bold>\<A>\<phi>{x} & \<forall>z (\<^bold>\<A>\<phi>{z} \<rightarrow> z = x)\<close>
    proof(rule "&I"; (rule "\<forall>I"; rule "\<rightarrow>I")?)
      AOT_show \<open>\<^bold>\<A>\<phi>{x}\<close>
        using B[THEN "&E"(1)] A[THEN "\<equiv>E"(2)] by blast
    next
      AOT_show \<open>b = x\<close> if \<open>\<^bold>\<A>\<phi>{b}\<close> for b
        using A[THEN "\<equiv>E"(1)] that
              B[THEN "&E"(2), THEN "\<forall>E"(2), THEN "\<rightarrow>E"] by blast
    qed
    AOT_thus \<open>x = \<^bold>\<iota>x \<phi>{x}\<close>
      using "nec-hintikka-scheme"[THEN "\<equiv>E"(2)] by blast
  qed
qed

AOT_theorem "id-act:1": \<open>\<alpha> = \<beta> \<equiv> \<^bold>\<A>\<alpha> = \<beta>\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism"
            "id-nec2:1" "id-nec:2" "\<equiv>I" "nec-imp-act")

AOT_theorem "id-act:2": \<open>\<alpha> \<noteq> \<beta> \<equiv> \<^bold>\<A>\<alpha> \<noteq> \<beta>\<close>
proof (AOT_subst \<open>\<alpha> \<noteq> \<beta>\<close> \<open>\<not>(\<alpha> = \<beta>)\<close>)
  AOT_modally_strict {
    AOT_show \<open>\<alpha> \<noteq> \<beta> \<equiv> \<not>(\<alpha> = \<beta>)\<close>
      by (simp add: "=-infix" "\<equiv>Df")
  }
next
  AOT_show \<open>\<not>(\<alpha> = \<beta>) \<equiv> \<^bold>\<A>\<not>(\<alpha> = \<beta>)\<close>
  proof (safe intro!: "\<equiv>I" "\<rightarrow>I")
    AOT_assume \<open>\<not>\<alpha> = \<beta>\<close>
    AOT_hence \<open>\<not>\<^bold>\<A>\<alpha> = \<beta>\<close> using "id-act:1" "\<equiv>E"(3) by blast
    AOT_thus \<open>\<^bold>\<A>\<not>\<alpha> = \<beta>\<close>
      using "\<not>\<not>E" "Act-Sub:1" "\<equiv>E"(3) by blast
  next
    AOT_assume \<open>\<^bold>\<A>\<not>\<alpha> = \<beta>\<close>
    AOT_hence \<open>\<not>\<^bold>\<A>\<alpha> = \<beta>\<close>
      using "\<not>\<not>I" "Act-Sub:1" "\<equiv>E"(4) by blast
    AOT_thus \<open>\<not>\<alpha> = \<beta>\<close>
      using "id-act:1" "\<equiv>E"(4) by blast
  qed
qed

AOT_theorem "A-Exists:1": \<open>\<^bold>\<A>\<exists>!\<alpha> \<phi>{\<alpha>} \<equiv> \<exists>!\<alpha> \<^bold>\<A>\<phi>{\<alpha>}\<close>
proof -
  AOT_have \<open>\<^bold>\<A>\<exists>!\<alpha> \<phi>{\<alpha>} \<equiv> \<^bold>\<A>\<exists>\<alpha>\<forall>\<beta> (\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close>
    by (AOT_subst \<open>\<exists>!\<alpha> \<phi>{\<alpha>}\<close> \<open>\<exists>\<alpha>\<forall>\<beta> (\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close>)
       (auto simp add: "oth-class-taut:3:a" "uniqueness:2")
  also AOT_have \<open>\<dots> \<equiv> \<exists>\<alpha> \<^bold>\<A>\<forall>\<beta> (\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close>
    by (simp add: "Act-Basic:10")
  also AOT_have \<open>\<dots> \<equiv> \<exists>\<alpha>\<forall>\<beta> \<^bold>\<A>(\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close>
    by (AOT_subst \<open>\<^bold>\<A>\<forall>\<beta> (\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close> \<open>\<forall>\<beta> \<^bold>\<A>(\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close> for: \<alpha>)
       (auto simp: "logic-actual-nec:3"[axiom_inst] "oth-class-taut:3:a")
  also AOT_have \<open>\<dots> \<equiv> \<exists>\<alpha>\<forall>\<beta> (\<^bold>\<A>\<phi>{\<beta>} \<equiv> \<^bold>\<A>\<beta> = \<alpha>)\<close>
    by (AOT_subst (reverse) \<open>\<^bold>\<A>\<phi>{\<beta>} \<equiv> \<^bold>\<A>\<beta> = \<alpha>\<close>
                            \<open>\<^bold>\<A>(\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close> for: \<alpha> \<beta> :: 'a)
       (auto simp: "Act-Basic:5" "cqt-further:7")
  also AOT_have \<open>\<dots> \<equiv> \<exists>\<alpha>\<forall>\<beta> (\<^bold>\<A>\<phi>{\<beta>} \<equiv> \<beta> = \<alpha>)\<close>
    by (AOT_subst (reverse) \<open>\<^bold>\<A>\<beta> = \<alpha>\<close> \<open>\<beta> = \<alpha>\<close> for: \<alpha> \<beta> :: 'a)
       (auto simp: "id-act:1" "cqt-further:7")
  also AOT_have \<open>... \<equiv> \<exists>!\<alpha> \<^bold>\<A>\<phi>{\<alpha>}\<close>
    using "uniqueness:2" "Commutativity of \<equiv>"[THEN "\<equiv>E"(1)] by fast
  finally show ?thesis.
qed

AOT_theorem "A-Exists:2": \<open>\<^bold>\<iota>x \<phi>{x}\<down> \<equiv> \<^bold>\<A>\<exists>!x \<phi>{x}\<close>
  by (AOT_subst \<open>\<^bold>\<A>\<exists>!x \<phi>{x}\<close> \<open>\<exists>!x \<^bold>\<A>\<phi>{x}\<close>)
     (auto simp: "actual-desc:1" "A-Exists:1")

AOT_theorem "id-act-desc:1": \<open>\<^bold>\<iota>x (x = y)\<down>\<close>
proof(rule "existence:1"[THEN "\<equiv>\<^sub>d\<^sub>fI"]; rule "\<exists>I")
  AOT_show \<open>[\<lambda>x E!x \<rightarrow> E!x]\<^bold>\<iota>x (x = y)\<close>
  proof (rule "russell-axiom[exe,1].nec-russell-axiom"[THEN "\<equiv>E"(2)];
         rule "\<exists>I"; (rule "&I")+)
    AOT_show \<open>\<^bold>\<A>y = y\<close> by (simp add: "RA[2]" "id-eq:1")
  next
    AOT_show \<open>\<forall>z (\<^bold>\<A>z = y \<rightarrow> z = y)\<close>
      apply (rule "\<forall>I")
      using "id-act:1"[THEN "\<equiv>E"(2)] "\<rightarrow>I" by blast
  next
    AOT_show \<open>[\<lambda>x E!x \<rightarrow> E!x]y\<close>
    proof (rule "lambda-predicates:2"[axiom_inst, THEN "\<rightarrow>E", THEN "\<equiv>E"(2)])
      AOT_show \<open>[\<lambda>x E!x \<rightarrow> E!x]\<down>\<close>
        by "cqt:2[lambda]"
    next
      AOT_show \<open>E!y \<rightarrow> E!y\<close> 
        by (simp add: "if-p-then-p")
    qed
  qed
next
  AOT_show \<open>[\<lambda>x E!x \<rightarrow> E!x]\<down>\<close>
    by "cqt:2[lambda]"
qed

AOT_theorem "id-act-desc:2": \<open>y = \<^bold>\<iota>x (x = y)\<close>
  by (rule descriptions[axiom_inst, THEN "\<equiv>E"(2)];
      rule "\<forall>I"; rule "id-act:1"[symmetric])

AOT_theorem "pre-en-eq:1[1]": \<open>x\<^sub>1[F] \<rightarrow> \<box>x\<^sub>1[F]\<close>
  by (simp add: encoding "vdash-properties:1[2]")

AOT_theorem "pre-en-eq:1[2]": \<open>x\<^sub>1x\<^sub>2[F] \<rightarrow> \<box>x\<^sub>1x\<^sub>2[F]\<close>
proof (rule "\<rightarrow>I")
  AOT_assume \<open>x\<^sub>1x\<^sub>2[F]\<close>
  AOT_hence \<open>x\<^sub>1[\<lambda>y [F]yx\<^sub>2]\<close> and \<open>x\<^sub>2[\<lambda>y [F]x\<^sub>1y]\<close>
    using "nary-encoding[2]"[axiom_inst, THEN "\<equiv>E"(1)] "&E" by blast+
  moreover AOT_have \<open>[\<lambda>y [F]yx\<^sub>2]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1y]\<down>\<close> by "cqt:2"
  ultimately AOT_have \<open>\<box>x\<^sub>1[\<lambda>y [F]yx\<^sub>2]\<close> and \<open>\<box>x\<^sub>2[\<lambda>y [F]x\<^sub>1y]\<close>
    using encoding[axiom_inst, unvarify F] "\<rightarrow>E" "&I" by blast+
  note A = this
  AOT_hence \<open>\<box>(x\<^sub>1[\<lambda>y [F]yx\<^sub>2] & x\<^sub>2[\<lambda>y [F]x\<^sub>1y])\<close>
    using "KBasic:3"[THEN "\<equiv>E"(2)] "&I" by blast
  AOT_thus \<open>\<box>x\<^sub>1x\<^sub>2[F]\<close>
    by (rule "nary-encoding[2]"[axiom_inst, THEN RN,
                                THEN "KBasic:6"[THEN "\<rightarrow>E"],
                                THEN "\<equiv>E"(2)])
qed

AOT_theorem "pre-en-eq:1[3]": \<open>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<rightarrow> \<box>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
proof (rule "\<rightarrow>I")
  AOT_assume \<open>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  AOT_hence \<open>x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3]\<close>
        and \<open>x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3]\<close>
        and \<open>x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2y]\<close>
    using "nary-encoding[3]"[axiom_inst, THEN "\<equiv>E"(1)] "&E" by blast+
  moreover AOT_have \<open>[\<lambda>y [F]yx\<^sub>2x\<^sub>3]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1yx\<^sub>3]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1x\<^sub>2y]\<down>\<close> by "cqt:2"
  ultimately AOT_have \<open>\<box>x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3]\<close>
                  and \<open>\<box>x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3]\<close>
                  and \<open>\<box>x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2y]\<close>
    using encoding[axiom_inst, unvarify F] "\<rightarrow>E" by blast+
  note A = this
  AOT_have B: \<open>\<box>(x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3] & x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3] & x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2y])\<close>
    by (rule "KBasic:3"[THEN "\<equiv>E"(2)] "&I" A)+
  AOT_thus \<open>\<box>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
    by (rule "nary-encoding[3]"[axiom_inst, THEN RN,
                THEN "KBasic:6"[THEN "\<rightarrow>E"], THEN "\<equiv>E"(2)])
qed

AOT_theorem "pre-en-eq:1[4]": \<open>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<rightarrow> \<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
proof (rule "\<rightarrow>I")
  AOT_assume \<open>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  AOT_hence \<open>x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3x\<^sub>4]\<close>
        and \<open>x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3x\<^sub>4]\<close>
        and \<open>x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2yx\<^sub>4]\<close>
        and \<open>x\<^sub>4[\<lambda>y [F]x\<^sub>1x\<^sub>2x\<^sub>3y]\<close>
    using "nary-encoding[4]"[axiom_inst, THEN "\<equiv>E"(1)] "&E" by metis+
  moreover AOT_have \<open>[\<lambda>y [F]yx\<^sub>2x\<^sub>3x\<^sub>4]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1yx\<^sub>3x\<^sub>4]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1x\<^sub>2yx\<^sub>4]\<down>\<close> by "cqt:2"
  moreover AOT_have \<open>[\<lambda>y [F]x\<^sub>1x\<^sub>2x\<^sub>3y]\<down>\<close> by "cqt:2"
  ultimately AOT_have \<open>\<box>x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3x\<^sub>4]\<close>
                  and \<open>\<box>x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3x\<^sub>4]\<close>
                  and \<open>\<box>x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2yx\<^sub>4]\<close>
                  and \<open>\<box>x\<^sub>4[\<lambda>y [F]x\<^sub>1x\<^sub>2x\<^sub>3y]\<close>
    using "\<rightarrow>E" encoding[axiom_inst, unvarify F] by blast+
  note A = this
  AOT_have B: \<open>\<box>(x\<^sub>1[\<lambda>y [F]yx\<^sub>2x\<^sub>3x\<^sub>4] &
                  x\<^sub>2[\<lambda>y [F]x\<^sub>1yx\<^sub>3x\<^sub>4] &
                  x\<^sub>3[\<lambda>y [F]x\<^sub>1x\<^sub>2yx\<^sub>4] &
                  x\<^sub>4[\<lambda>y [F]x\<^sub>1x\<^sub>2x\<^sub>3y])\<close>
    by (rule "KBasic:3"[THEN "\<equiv>E"(2)] "&I" A)+
  AOT_thus \<open>\<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
    by (rule "nary-encoding[4]"[axiom_inst, THEN RN,
              THEN "KBasic:6"[THEN "\<rightarrow>E"], THEN "\<equiv>E"(2)])
qed

AOT_theorem "pre-en-eq:2[1]": \<open>\<not>x\<^sub>1[F] \<rightarrow> \<box>\<not>x\<^sub>1[F]\<close>
proof (rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<box>\<not>x\<^sub>1[F]\<close>
  AOT_hence \<open>\<diamond>x\<^sub>1[F]\<close>
    by (rule "conventions:5"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
  AOT_hence \<open>x\<^sub>1[F]\<close>
    by(rule "S5Basic:13"[THEN "\<equiv>E"(1), OF  "pre-en-eq:1[1]"[THEN RN],
              THEN "qml:2"[axiom_inst, THEN "\<rightarrow>E"], THEN "\<rightarrow>E"])
  moreover AOT_assume \<open>\<not>x\<^sub>1[F]\<close>
  ultimately AOT_show \<open>x\<^sub>1[F] & \<not>x\<^sub>1[F]\<close> by (rule "&I")
qed
AOT_theorem "pre-en-eq:2[2]": \<open>\<not>x\<^sub>1x\<^sub>2[F] \<rightarrow> \<box>\<not>x\<^sub>1x\<^sub>2[F]\<close>
proof (rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<box>\<not>x\<^sub>1x\<^sub>2[F]\<close>
  AOT_hence \<open>\<diamond>x\<^sub>1x\<^sub>2[F]\<close>
    by (rule "conventions:5"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
  AOT_hence \<open>x\<^sub>1x\<^sub>2[F]\<close>
    by(rule "S5Basic:13"[THEN "\<equiv>E"(1), OF  "pre-en-eq:1[2]"[THEN RN],
              THEN "qml:2"[axiom_inst, THEN "\<rightarrow>E"], THEN "\<rightarrow>E"])
  moreover AOT_assume \<open>\<not>x\<^sub>1x\<^sub>2[F]\<close>
  ultimately AOT_show \<open>x\<^sub>1x\<^sub>2[F] & \<not>x\<^sub>1x\<^sub>2[F]\<close> by (rule "&I")
qed

AOT_theorem "pre-en-eq:2[3]": \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<rightarrow> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
proof (rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  AOT_hence \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
    by (rule "conventions:5"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
  AOT_hence \<open>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
    by(rule "S5Basic:13"[THEN "\<equiv>E"(1), OF  "pre-en-eq:1[3]"[THEN RN],
              THEN "qml:2"[axiom_inst, THEN "\<rightarrow>E"], THEN "\<rightarrow>E"])
  moreover AOT_assume \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  ultimately AOT_show \<open>x\<^sub>1x\<^sub>2x\<^sub>3[F] & \<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close> by (rule "&I")
qed

AOT_theorem "pre-en-eq:2[4]": \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<rightarrow> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
proof (rule "\<rightarrow>I"; rule "raa-cor:1")
  AOT_assume \<open>\<not>\<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  AOT_hence \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
    by (rule "conventions:5"[THEN "\<equiv>\<^sub>d\<^sub>fI"])
  AOT_hence \<open>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
    by(rule "S5Basic:13"[THEN "\<equiv>E"(1), OF  "pre-en-eq:1[4]"[THEN RN],
                         THEN "qml:2"[axiom_inst, THEN "\<rightarrow>E"], THEN "\<rightarrow>E"])
  moreover AOT_assume \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  ultimately AOT_show \<open>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] & \<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close> by (rule "&I")
qed

AOT_theorem "en-eq:1[1]": \<open>\<diamond>x\<^sub>1[F] \<equiv> \<box>x\<^sub>1[F]\<close>
  using "pre-en-eq:1[1]"[THEN RN] "sc-eq-box-box:2" "\<or>I" "\<rightarrow>E" by metis
AOT_theorem "en-eq:1[2]": \<open>\<diamond>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>x\<^sub>1x\<^sub>2[F]\<close>
  using "pre-en-eq:1[2]"[THEN RN] "sc-eq-box-box:2" "\<or>I" "\<rightarrow>E" by metis
AOT_theorem "en-eq:1[3]": \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  using "pre-en-eq:1[3]"[THEN RN] "sc-eq-box-box:2" "\<or>I" "\<rightarrow>E" by fast
AOT_theorem "en-eq:1[4]": \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  using "pre-en-eq:1[4]"[THEN RN] "sc-eq-box-box:2" "\<or>I" "\<rightarrow>E" by fast

AOT_theorem "en-eq:2[1]": \<open>x\<^sub>1[F] \<equiv> \<box>x\<^sub>1[F]\<close>
  by (simp add: "\<equiv>I" "pre-en-eq:1[1]" "qml:2"[axiom_inst])
AOT_theorem "en-eq:2[2]": \<open>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>x\<^sub>1x\<^sub>2[F]\<close>
  by (simp add: "\<equiv>I" "pre-en-eq:1[2]" "qml:2"[axiom_inst])
AOT_theorem "en-eq:2[3]": \<open>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  by (simp add: "\<equiv>I" "pre-en-eq:1[3]" "qml:2"[axiom_inst])
AOT_theorem "en-eq:2[4]": \<open>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  by (simp add: "\<equiv>I" "pre-en-eq:1[4]" "qml:2"[axiom_inst])

AOT_theorem "en-eq:3[1]": \<open>\<diamond>x\<^sub>1[F] \<equiv> x\<^sub>1[F]\<close>
  using "T\<diamond>" "derived-S5-rules:2"[OF "pre-en-eq:1[1]"] "\<equiv>I" by blast
AOT_theorem "en-eq:3[2]": \<open>\<diamond>x\<^sub>1x\<^sub>2[F] \<equiv> x\<^sub>1x\<^sub>2[F]\<close>
  using "T\<diamond>" "derived-S5-rules:2"[OF "pre-en-eq:1[2]"] "\<equiv>I" by blast
AOT_theorem "en-eq:3[3]": \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  using "T\<diamond>" "derived-S5-rules:2"[OF "pre-en-eq:1[3]"] "\<equiv>I" by blast
AOT_theorem "en-eq:3[4]": \<open>\<diamond>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  using "T\<diamond>" "derived-S5-rules:2"[OF "pre-en-eq:1[4]"] "\<equiv>I" by blast

AOT_theorem "en-eq:4[1]":
  \<open>(x\<^sub>1[F] \<equiv> y\<^sub>1[G]) \<equiv> (\<box>x\<^sub>1[F] \<equiv> \<box>y\<^sub>1[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I"; rule "\<equiv>I"; rule "\<rightarrow>I")
  using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] "\<equiv>E"(1,2) "en-eq:2[1]" by blast+
AOT_theorem "en-eq:4[2]":
  \<open>(x\<^sub>1x\<^sub>2[F] \<equiv> y\<^sub>1y\<^sub>2[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>y\<^sub>1y\<^sub>2[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I"; rule "\<equiv>I"; rule "\<rightarrow>I")
  using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] "\<equiv>E"(1,2) "en-eq:2[2]" by blast+
AOT_theorem "en-eq:4[3]":
  \<open>(x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>y\<^sub>1y\<^sub>2y\<^sub>3[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I"; rule "\<equiv>I"; rule "\<rightarrow>I")
  using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] "\<equiv>E"(1,2) "en-eq:2[3]" by blast+
AOT_theorem "en-eq:4[4]":
  \<open>(x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I"; rule "\<equiv>I"; rule "\<rightarrow>I")
  using "qml:2"[axiom_inst, THEN "\<rightarrow>E"] "\<equiv>E"(1,2) "en-eq:2[4]" by blast+

AOT_theorem "en-eq:5[1]":
  \<open>\<box>(x\<^sub>1[F] \<equiv> y\<^sub>1[G]) \<equiv> (\<box>x\<^sub>1[F] \<equiv> \<box>y\<^sub>1[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "en-eq:4[1]"[THEN "\<equiv>E"(1)] "qml:2"[axiom_inst, THEN "\<rightarrow>E"]
   apply blast
  using "sc-eq-box-box:4"[THEN "\<rightarrow>E", THEN "\<rightarrow>E"]
        "&I"[OF "pre-en-eq:1[1]"[THEN RN], OF "pre-en-eq:1[1]"[THEN RN]]
  by blast
AOT_theorem "en-eq:5[2]":
  \<open>\<box>(x\<^sub>1x\<^sub>2[F] \<equiv> y\<^sub>1y\<^sub>2[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>y\<^sub>1y\<^sub>2[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "en-eq:4[2]"[THEN "\<equiv>E"(1)] "qml:2"[axiom_inst, THEN "\<rightarrow>E"]
   apply blast
  using "sc-eq-box-box:4"[THEN "\<rightarrow>E", THEN "\<rightarrow>E"]
        "&I"[OF "pre-en-eq:1[2]"[THEN RN], OF "pre-en-eq:1[2]"[THEN RN]]
  by blast
AOT_theorem "en-eq:5[3]":
  \<open>\<box>(x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>y\<^sub>1y\<^sub>2y\<^sub>3[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "en-eq:4[3]"[THEN "\<equiv>E"(1)] "qml:2"[axiom_inst, THEN "\<rightarrow>E"]
   apply blast
  using "sc-eq-box-box:4"[THEN "\<rightarrow>E", THEN "\<rightarrow>E"]
        "&I"[OF "pre-en-eq:1[3]"[THEN RN], OF "pre-en-eq:1[3]"[THEN RN]]
  by blast
AOT_theorem "en-eq:5[4]":
  \<open>\<box>(x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G]) \<equiv> (\<box>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G])\<close>
  apply (rule "\<equiv>I"; rule "\<rightarrow>I")
  using "en-eq:4[4]"[THEN "\<equiv>E"(1)] "qml:2"[axiom_inst, THEN "\<rightarrow>E"]
   apply blast
  using "sc-eq-box-box:4"[THEN "\<rightarrow>E", THEN "\<rightarrow>E"]
        "&I"[OF "pre-en-eq:1[4]"[THEN RN], OF "pre-en-eq:1[4]"[THEN RN]]
  by blast

AOT_theorem "en-eq:6[1]":
  \<open>(x\<^sub>1[F] \<equiv> y\<^sub>1[G]) \<equiv> \<box>(x\<^sub>1[F] \<equiv> y\<^sub>1[G])\<close>
  using "en-eq:5[1]"[symmetric] "en-eq:4[1]" "\<equiv>E"(5) by fast
AOT_theorem "en-eq:6[2]":
  \<open>(x\<^sub>1x\<^sub>2[F] \<equiv> y\<^sub>1y\<^sub>2[G]) \<equiv> \<box>(x\<^sub>1x\<^sub>2[F] \<equiv> y\<^sub>1y\<^sub>2[G])\<close>
  using "en-eq:5[2]"[symmetric] "en-eq:4[2]" "\<equiv>E"(5) by fast
AOT_theorem "en-eq:6[3]":
  \<open>(x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3[G]) \<equiv> \<box>(x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3[G])\<close>
  using "en-eq:5[3]"[symmetric] "en-eq:4[3]" "\<equiv>E"(5) by fast
AOT_theorem "en-eq:6[4]":
  \<open>(x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G]) \<equiv> \<box>(x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> y\<^sub>1y\<^sub>2y\<^sub>3y\<^sub>4[G])\<close>
  using "en-eq:5[4]"[symmetric] "en-eq:4[4]" "\<equiv>E"(5) by fast

AOT_theorem "en-eq:7[1]": \<open>\<not>x\<^sub>1[F] \<equiv> \<box>\<not>x\<^sub>1[F]\<close>
  using "pre-en-eq:2[1]" "qml:2"[axiom_inst] "\<equiv>I" by blast
AOT_theorem "en-eq:7[2]": \<open>\<not>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2[F]\<close>
  using "pre-en-eq:2[2]" "qml:2"[axiom_inst] "\<equiv>I" by blast
AOT_theorem "en-eq:7[3]": \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  using "pre-en-eq:2[3]" "qml:2"[axiom_inst] "\<equiv>I" by blast
AOT_theorem "en-eq:7[4]": \<open>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  using "pre-en-eq:2[4]" "qml:2"[axiom_inst] "\<equiv>I" by blast

AOT_theorem "en-eq:8[1]": \<open>\<diamond>\<not>x\<^sub>1[F] \<equiv> \<not>x\<^sub>1[F]\<close>
  using "en-eq:2[1]"[THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)]]
        "KBasic:11" "\<equiv>E"(5)[symmetric] by blast
AOT_theorem "en-eq:8[2]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2[F] \<equiv> \<not>x\<^sub>1x\<^sub>2[F]\<close>
  using "en-eq:2[2]"[THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)]]
        "KBasic:11" "\<equiv>E"(5)[symmetric] by blast
AOT_theorem "en-eq:8[3]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  using "en-eq:2[3]"[THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)]]
        "KBasic:11" "\<equiv>E"(5)[symmetric] by blast
AOT_theorem "en-eq:8[4]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  using "en-eq:2[4]"[THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)]]
        "KBasic:11" "\<equiv>E"(5)[symmetric] by blast

AOT_theorem "en-eq:9[1]": \<open>\<diamond>\<not>x\<^sub>1[F] \<equiv> \<box>\<not>x\<^sub>1[F]\<close>
  using "en-eq:7[1]" "en-eq:8[1]" "\<equiv>E"(5) by blast
AOT_theorem "en-eq:9[2]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2[F]\<close>
  using "en-eq:7[2]" "en-eq:8[2]" "\<equiv>E"(5) by blast
AOT_theorem "en-eq:9[3]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  using "en-eq:7[3]" "en-eq:8[3]" "\<equiv>E"(5) by blast
AOT_theorem "en-eq:9[4]": \<open>\<diamond>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> \<box>\<not>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  using "en-eq:7[4]" "en-eq:8[4]" "\<equiv>E"(5) by blast

AOT_theorem "en-eq:10[1]": \<open>\<^bold>\<A>x\<^sub>1[F] \<equiv> x\<^sub>1[F]\<close>
  by (metis "Act-Sub:3" "deduction-theorem" "\<equiv>I" "\<equiv>E"(1)
            "nec-imp-act" "en-eq:3[1]" "pre-en-eq:1[1]")
AOT_theorem "en-eq:10[2]": \<open>\<^bold>\<A>x\<^sub>1x\<^sub>2[F] \<equiv> x\<^sub>1x\<^sub>2[F]\<close>
  by (metis "Act-Sub:3" "deduction-theorem" "\<equiv>I" "\<equiv>E"(1)
            "nec-imp-act" "en-eq:3[2]" "pre-en-eq:1[2]")
AOT_theorem "en-eq:10[3]": \<open>\<^bold>\<A>x\<^sub>1x\<^sub>2x\<^sub>3[F] \<equiv> x\<^sub>1x\<^sub>2x\<^sub>3[F]\<close>
  by (metis "Act-Sub:3" "deduction-theorem" "\<equiv>I" "\<equiv>E"(1)
            "nec-imp-act" "en-eq:3[3]" "pre-en-eq:1[3]")
AOT_theorem "en-eq:10[4]": \<open>\<^bold>\<A>x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F] \<equiv> x\<^sub>1x\<^sub>2x\<^sub>3x\<^sub>4[F]\<close>
  by (metis "Act-Sub:3" "deduction-theorem" "\<equiv>I" "\<equiv>E"(1)
            "nec-imp-act" "en-eq:3[4]" "pre-en-eq:1[4]")

AOT_theorem "oa-facts:1": \<open>O!x \<rightarrow> \<box>O!x\<close>
proof(rule "\<rightarrow>I")
  AOT_modally_strict {
    AOT_have \<open>[\<lambda>x \<diamond>E!x]x \<equiv> \<diamond>E!x\<close>
      by (rule "lambda-predicates:2"[axiom_inst, THEN "\<rightarrow>E"]) "cqt:2"
  } note \<theta> = this
  AOT_assume \<open>O!x\<close>
  AOT_hence \<open>[\<lambda>x \<diamond>E!x]x\<close>
    by (rule "=\<^sub>d\<^sub>fE"(2)[OF AOT_ordinary, rotated 1]) "cqt:2"
  AOT_hence \<open>\<diamond>E!x\<close> using \<theta>[THEN "\<equiv>E"(1)] by blast
  AOT_hence \<open>\<box>\<diamond>E!x\<close> using "qml:3"[axiom_inst, THEN "\<rightarrow>E"] by blast
  AOT_hence \<open>\<box>[\<lambda>x \<diamond>E!x]x\<close>
    by (AOT_subst \<open>[\<lambda>x \<diamond>E!x]x\<close> \<open>\<diamond>E!x\<close>)
       (auto simp: \<theta>)
  AOT_thus \<open>\<box>O!x\<close>
    by (rule "=\<^sub>d\<^sub>fI"(2)[OF AOT_ordinary, rotated 1]) "cqt:2"
qed

AOT_theorem "oa-facts:2": \<open>A!x \<rightarrow> \<box>A!x\<close>
proof(rule "\<rightarrow>I")
  AOT_modally_strict {
    AOT_have \<open>[\<lambda>x \<not>\<diamond>E!x]x \<equiv> \<not>\<diamond>E!x\<close>
      by (rule "lambda-predicates:2"[axiom_inst, THEN "\<rightarrow>E"]) "cqt:2"
  } note \<theta> = this
  AOT_assume \<open>A!x\<close>
  AOT_hence \<open>[\<lambda>x \<not>\<diamond>E!x]x\<close>
    by (rule "=\<^sub>d\<^sub>fE"(2)[OF AOT_abstract, rotated 1]) "cqt:2"
  AOT_hence \<open>\<not>\<diamond>E!x\<close> using \<theta>[THEN "\<equiv>E"(1)] by blast
  AOT_hence \<open>\<box>\<not>E!x\<close> using "KBasic2:1"[THEN "\<equiv>E"(2)] by blast
  AOT_hence \<open>\<box>\<box>\<not>E!x\<close> using "4"[THEN "\<rightarrow>E"] by blast
  AOT_hence \<open>\<box>\<not>\<diamond>E!x\<close>
    using "KBasic2:1"
    by (AOT_subst (reverse) \<open>\<not>\<diamond>E!x\<close> \<open>\<box>\<not>E!x\<close>) blast
  AOT_hence \<open>\<box>[\<lambda>x \<not>\<diamond>E!x]x\<close>
    by (AOT_subst \<open>[\<lambda>x \<not>\<diamond>E!x]x\<close> \<open>\<not>\<diamond>E!x\<close>)
       (auto simp: \<theta>)
  AOT_thus \<open>\<box>A!x\<close>
    by (rule "=\<^sub>d\<^sub>fI"(2)[OF AOT_abstract, rotated 1]) "cqt:2[lambda]"
qed

AOT_theorem "oa-facts:3": \<open>\<diamond>O!x \<rightarrow> O!x\<close>
  using "oa-facts:1" "B\<diamond>" "RM\<diamond>" "Hypothetical Syllogism" by blast
AOT_theorem "oa-facts:4": \<open>\<diamond>A!x \<rightarrow> A!x\<close>
  using "oa-facts:2" "B\<diamond>" "RM\<diamond>" "Hypothetical Syllogism" by blast

AOT_theorem "oa-facts:5": \<open>\<diamond>O!x \<equiv> \<box>O!x\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism" "\<equiv>I" "nec-imp-act"
            "oa-facts:1" "oa-facts:3")

AOT_theorem "oa-facts:6": \<open>\<diamond>A!x \<equiv> \<box>A!x\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism" "\<equiv>I" "nec-imp-act"
            "oa-facts:2" "oa-facts:4")

AOT_theorem "oa-facts:7": \<open>O!x \<equiv> \<^bold>\<A>O!x\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism" "\<equiv>I" "nec-imp-act"
            "oa-facts:1" "oa-facts:3")

AOT_theorem "oa-facts:8": \<open>A!x \<equiv> \<^bold>\<A>A!x\<close>
  by (meson "Act-Sub:3" "Hypothetical Syllogism" "\<equiv>I" "nec-imp-act"
            "oa-facts:2" "oa-facts:4")

subsection\<open>The Theory of Relations\<close>
text\<open>\label{PLM: 9.10}\<close>

AOT_theorem "beta-C-meta":
  \<open>[\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n}]\<down> \<rightarrow>
   ([\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n}]\<nu>\<^sub>1...\<nu>\<^sub>n \<equiv> \<phi>{\<nu>\<^sub>1...\<nu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n})\<close>
  using "lambda-predicates:2"[axiom_inst] by blast

AOT_theorem "beta-C-cor:1":
  \<open>(\<forall>\<nu>\<^sub>1...\<forall>\<nu>\<^sub>n([\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n}]\<down>)) \<rightarrow>
   \<forall>\<nu>\<^sub>1...\<forall>\<nu>\<^sub>n ([\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n}]\<nu>\<^sub>1...\<nu>\<^sub>n \<equiv> \<phi>{\<nu>\<^sub>1...\<nu>\<^sub>n, \<nu>\<^sub>1...\<nu>\<^sub>n})\<close>
  apply (rule "cqt-basic:14"[where 'a='a, THEN "\<rightarrow>E"])
  using "beta-C-meta" "\<forall>I" by fast

AOT_theorem "beta-C-cor:2":
  \<open>[\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n}]\<down> \<rightarrow>
   \<forall>\<nu>\<^sub>1...\<forall>\<nu>\<^sub>n ([\<lambda>\<mu>\<^sub>1...\<mu>\<^sub>n \<phi>{\<mu>\<^sub>1...\<mu>\<^sub>n}]\<nu>\<^sub>1...\<nu>\<^sub>n \<equiv> \<phi>{\<nu>\<^sub>1...\<nu>\<^sub>n})\<close>
  apply (rule "\<rightarrow>I"; rule "\<forall>I")
  using "beta-C-meta"[THEN "\<rightarrow>E"] by fast

(* TODO: add better syntax parsing for INSTANCE_OF_CQT_2 *)
theorem "beta-C-cor:3":
  assumes ‹∧ν₁ν_n. AOT_instance_of_cqt_2 (φ (AOT_term_of_var ν₁ν_n))›
  shows ‹[v ⊨ ∀ν₁...∀ν_n ([λμ₁...μ_n φ{ν₁...ν_n, μ₁...μ_n}]ν₁...ν_n ≡
 φ{ν₁...ν_n, ν₁...ν_n})]›
  using "cqt:2[lambda]"[axiom_inst, OF assms]
        "beta-C-cor:1"[THEN "→E"] "∀I" by fast

AOT_theorem "betaC:1:a": ‹[λμ₁...μ_n φ{μ₁...μ_n}]κ₁...κ_n \⊨_\◻ φ{κ₁...κ_n}›
proof -
  AOT_modally_strict {
    AOT_assume ‹[λμ₁...μ_n φ{μ₁...μ_n}]κ₁...κ_n›
    moreover AOT_have ‹[λμ₁...μ_n φ{μ₁...μ_n}]↓› and ‹κ₁...κ_n↓›
      using calculation "cqt:5:a"[axiom_inst, THEN "→E"] "&E" by blast+
    ultimately AOT_show ‹φ{κ₁...κ_n}›
      using "beta-C-cor:2"[THEN "→E", THEN "∀E"(1), THEN "≡E"(1)] by blast
  }
qed

AOT_theorem "betaC:1:b": ‹¬φ{κ₁...κ_n} \⊨_\◻ ¬[λμ₁...μ_n φ{μ₁...μ_n}]κ₁...κ_n›
  using "betaC:1:a" "raa-cor:3" by blast

lemmas "β→C" = "betaC:1:a" "betaC:1:b"

AOT_theorem "betaC:2:a":
  ‹[λμ₁...μ_n φ{μ₁...μ_n}]↓, κ₁...κ_n↓, φ{κ₁...κ_n} \⊨_\◻
 [λμ₁...μ_n φ{μ₁...μ_n}]κ₁...κ_n›
proof -
  AOT_modally_strict {
    AOT_assume 1: ‹[λμ₁...μ_n φ{μ₁...μ_n}]↓›
           and 2: ‹κ₁...κ_n↓›
           and 3: ‹φ{κ₁...κ_n}›
    AOT_hence ‹[λμ₁...μ_n φ{μ₁...μ_n}]κ₁...κ_n›
      using "beta-C-cor:2"[THEN "→E", OF 1, THEN "∀E"(1), THEN "≡E"(2)]
      by blast
  }
   ‹mu>_{μ1...μ_, κ_n↓, φκ1...κ_\⊨>◻
java.lang.NullPointerException
 by blast
 

  "betaC:2:b":
 en>[\lambda {μ1...μ_, κsub>1...κ_<down>, ¬[λμ_\^bn\phi>{\mu_n}]κ_n⊨^u>◻
java.lang.NullPointerException
 byforce simp: Weaks_def deductions.simps)

  "<><

  "eta-conversion-lemma1:1": ‹ [λ1...x_]x_n] = Π
 using "lambda-predicates:3"[axiom_inst] "∀🚫

(* Note: generalized alphabetic variant of the last theorem *)
AOT_theorem "eta-conversion-lemma1:2": ‹Π↓ → [λν₁...ν_n [Π]ν₁...ν_n] = Π›
  using "eta-conversion-lemma1:1".

text‹
  id_sym:
 assumes ‹τ = τ'›
 shows ‹τ' = τ›
 using "rule=E"[where φ="λ τ' . «τ' = τ¬", rotated 1, OF assms]
 "=I"(1)[OF "t=t-proper:1"[THEN "→E", OF assms]] by auto
  id_sym[sym]

 ‹Note: not explicitly part of PLM.›
  id_trans:
 assumes ‹τ = τ'› and ‹τ' = τ''›
 shows ‹τ = τ''›
 using "rule=E" assms by blast
  id_trans[trans]

  "ηC" for Π :: ‹<'a::{AOT_Term_id_2,AOT_κs}>› =
 (match conclusion in "[v ⊨ τ{Π} = τ'{Π}]" for v τ τ' ==> ‹
 rule "rule=E"[rotated 1, OF "eta-conversion-lemma1:2"
 [THEN "→E", of v "«[Π]¬", symmetric]]›)

  "sub-des-lam:1":
 ‹[λz₁...z_n χ{z₁...z_n, \ιx φ{x}}]↓ & \ιx φ{x} = \ιx ψ{x} →
 [λz₁...z_n χ{z₁...z_n, \ιx φ{x}}] = [λz₁...z_n χ{z₁...z_n, \ιx ψ{x}}]›
 (rule "→I")
 AOT_assume A: ‹[λz₁...z_n χ{z₁...z_n, \ιx φ{x}}]↓ & \ιx φ{x} = \ιx ψ{x}›
 AOT_show ‹[λz₁...z_n χ{z₁...z_n, \ιx φ{x}}] = [λz₁...z_n χ{z₁...z_n, \ιx ψ{x}}]›
 using "rule=E"[where φ="λ τ . «[λz₁...z_n χ{z₁...z_n, \ιx φ{x}}] =
 [λz₁...z_n χ{z₁...z_n, τ}]¬",
 OF "=I"(1)[OF A[THEN "&E"(1)]], OF A[THEN "&E"(2)]]
 by blast
 

  "sub-des-lam:2":
 ‹\ιx φ{x} = \ιx ψ{x} → χ{\ιx φ{x}} = χ{\ιx ψ{x}}› for χ :: ‹κ ==> o›
 using "rule=E"[where φ="λ τ . «χ{\ιx φ{x}} = χ{τ}¬",
 OF "=I"(1)[OF "log-prop-prop:2"]] "→I" by blast

  "prop-equiv": ‹F = G ≡ ∀x (x[F] ≡ x[G])›
 (rule "≡I"; rule "→I")
 AOT_assume ‹F = G›
 AOT_thus ‹∀x (x[F] ≡ x[G])›
 by (rule "rule=E"[rotated]) (fact "oth-class-taut:3:a"[THEN GEN])
 
 AOT_assume ‹∀x (x[F] ≡ x[G])›
 AOT_hence ‹x[F] ≡ x[G]› for x
 using "∀E" by blast
 AOT_hence ‹◻(x[F] ≡ x[G])› for x
 using "en-eq:6[1]"[THEN "≡E"(1)] by blast
 AOT_hence ‹∀x ◻(x[F] ≡ x[G])›
 by (rule GEN)
 AOT_hence ‹◻∀x (x[F] ≡ x[G])›
 using BF[THEN "→E"] by fast
 AOT_thus "F = G"
 using "p-identity-thm2:1"[THEN "≡E"(2)] by blast
 

  "relations:1":
 assumes ‹INSTANCE_OF_CQT_2(φ)›
 shows ‹∃F ◻∀x₁...∀x_n ([F]x₁...x_n ≡ φ{x₁...x_n})›
 apply (rule "∃I"(1)[where τ="«[λx₁...x_n φ{x₁...x_n}]¬"])
 using "cqt:2[lambda]"[OF assms, axiom_inst]
 "beta-C-cor:2"[THEN "→E", THEN RN] by blast+

  "relations:2":
 assumes ‹INSTANCE_OF_CQT_2(φ)›
 shows ‹∃F ◻∀x ([F]x ≡ φ{x})›
 using "relations:1" assms by blast

  "block-paradox:1": ‹¬[λx ∃G (x[G] & ¬[G]x)]↓›
 (rule "raa-cor:2")
 let ?K="«[λx ∃G (x[G] & ¬[G]x)]¬"
 AOT_assume A: ‹«?K¬↓›
 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ F = «?K¬))›
 using "A-objects"[axiom_inst] by fast
 then AOT_obtain a where ξ: ‹A!a & ∀F (a[F] ≡ F = «?K¬)›
 using "∃E"[rotated] by blast
 AOT_show ‹p & ¬p› for p
 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
 AOT_assume B: ‹[«?K¬]a›
 AOT_hence ‹∃G (a[G] & ¬[G]a)›
 using "β→C" A by blast
 then AOT_obtain P where ‹a[P] & ¬[P]a›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹P = [«?K¬]›
 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)]
 calculation[THEN "&E"(1)] by blast
 ultimately AOT_have ‹¬[«?K¬]a›
 using "rule=E" "&E"(2) by fast
 AOT_thus ‹p & ¬p›
 using B RAA by blast
 next
 AOT_assume B: ‹¬[«?K¬]a›
 AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
 using "β←C" "cqt:2[const_var]"[of a, axiom_inst] A by blast
 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)›
 using "cqt-further:4"[THEN "→E"] by blast
 AOT_have ‹∀G (a[G] → [G]a)›
 by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› for: G)
 (auto simp: "oth-class-taut:1:a" C)
 AOT_hence ‹a[«?K¬] → [«?K¬]a›
 using "∀E" A by blast
 moreover AOT_have ‹a[«?K¬]›
 using ξ[THEN "&E"(2), THEN "∀E"(1), OF A, THEN "≡E"(2)]
 using "=I"(1)[OF A] by blast
 ultimately AOT_show ‹p & ¬p›
 using B "→E" RAA by blast
 qed
 

  "block-paradox:2": ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
 (rule RAA(2))
 AOT_assume ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
 then AOT_obtain F where F_prop: ‹∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
 using "∃E"[rotated] by blast
 AOT_have ‹∃x (A!x & ∀G (x[G] ≡ G = F))›
 using "A-objects"[axiom_inst] by fast
 then AOT_obtain a where ξ: ‹A!a & ∀G (a[G] ≡ G = F)›
 using "∃E"[rotated] by blast
 AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
 AOT_assume B: ‹[F]a›
 AOT_hence ‹∃G (a[G] & ¬[G]a)›
 using F_prop[THEN "∀E"(2), THEN "≡E"(1)] by blast
 then AOT_obtain P where ‹a[P] & ¬[P]a›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹P = F›
 using ξ[THEN "&E"(2), THEN "∀E"(2), THEN "≡E"(1)]
 calculation[THEN "&E"(1)] by blast
 ultimately AOT_have ‹¬[F]a›
 using "rule=E" "&E"(2) by fast
 AOT_thus ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
 using B RAA by blast
 next
 AOT_assume B: ‹¬[F]a›
 AOT_hence ‹¬∃G (a[G] & ¬[G]a)›
 using "oth-class-taut:4:b"[THEN "≡E"(1),
 OF F_prop[THEN "∀E"(2)[of _ _ a]], THEN "≡E"(1)]
 by simp
 AOT_hence C: ‹∀G ¬(a[G] & ¬[G]a)›
 using "cqt-further:4"[THEN "→E"] by blast
 AOT_have ‹∀G (a[G] → [G]a)›
 by (AOT_subst ‹a[G] → [G]a› ‹¬(a[G] & ¬[G]a)› for: G)
 (auto simp: "oth-class-taut:1:a" C)
 AOT_hence ‹a[F] → [F]a›
 using "∀E" by blast
 moreover AOT_have ‹a[F]›
 using ξ[THEN "&E"(2), THEN "∀E"(2), of F, THEN "≡E"(2)]
 using "=I"(2) by blast
 ultimately AOT_show ‹¬∃F ∀x([F]x ≡ ∃G(x[G] & ¬[G]x))›
 using B "→E" RAA by blast
 qed
 (simp)

  "block-paradox:3": ‹¬∀y [λz z = y]↓›
 (rule RAA(2))
 AOT_assume θ: ‹∀y [λz z = y]↓›
 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃y(F = [λz z = y] & style='font-size: 18px;'>¬y[F])))›
 using "A-objects"[axiom_inst] by force
 then AOT_obtain a where
 a_prop: ‹A!a & ∀F (a[F] ≡ ∃y (F = [λz z = y] & le='font-size: 18px;'>¬y[F]))›
 using "∃E"[rotated] by blast
 AOT_have ζ: ‹a[λz z = a] ≡ ∃y ([λz z = a] = [λz z = y] & >¬y[λz z = a])›
 using θ[THEN "∀E"(2)] a_prop[THEN "&E"(2), THEN "∀E"(1)] by blast
 AOT_show ‹¬∀y [λz z = y]↓›
 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
 AOT_assume A: ‹a[λz z = a]›
 AOT_hence ‹∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
 using ζ[THEN "≡E"(1)] by blast
 then AOT_obtain b where b_prop: ‹[λz z = a] = [λz z = b] & ¬b[λz z = a]›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹a = a› by (rule "=I")
 moreover AOT_have ‹[λz z = a]↓› using θ "∀E" by blast
 moreover AOT_have ‹a↓› using "cqt:2[const_var]"[axiom_inst] .
 ultimately AOT_have ‹[λz z = a]a› using "β←C" by blast
 AOT_hence ‹[λz z = b]a› using "rule=E" b_prop[THEN "&E"(1)] by fast
 AOT_hence ‹a = b› using "β→C" by blast
 AOT_hence ‹b[λz z = a]› using A "rule=E" by fast
 AOT_thus ‹¬∀y [λz z = y]↓› using b_prop[THEN "&E"(2)] RAA by blast
 next
 AOT_assume A: ‹¬a[λz z = a]›
 AOT_hence ‹¬∃y ([λz z = a] = [λz z = y] & ¬y[λz z = a])›
 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
 AOT_hence ‹∀y ¬([λz z = a] = [λz z = y] & ¬y[λz z = a])›
 using "cqt-further:4"[THEN "→E"] by blast
 AOT_hence ‹¬([λz z = a] = [λz z = a] & ¬a[λz z = a])›
 using "∀E" by blast
 AOT_hence ‹[λz z = a] = [λz z = a] → a[λz z = a]›
 by (metis "&I" "deduction-theorem" "raa-cor:4")
 AOT_hence ‹a[λz z = a]› using "=I"(1) θ[THEN "∀E"(2)] "→E" by blast
 AOT_thus ‹¬∀y [λz z = y]↓› using A RAA by blast
 qed
 (simp)

  "block-paradox:4": ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
 (rule RAA(2))
 AOT_assume θ: ‹∀y ∃F ∀x([F]x ≡ x = y)›
 AOT_have ‹∃x (A!x & ∀F (x[F] ≡ ∃z (∀y([F]y ≡ y = z) & <span style='font-size: 18px;'>¬z[F])))›
 using "A-objects"[axiom_inst] by force
 then AOT_obtain a where
 a_prop: ‹A!a & ∀F (a[F] ≡ ∃z (∀y([F]y ≡ y = z) & style='font-size: 18px;'>¬z[F]))›
 using "∃E"[rotated] by blast
 AOT_obtain F where F_prop: ‹∀x ([F]x ≡ x = a)›
 using θ[THEN "∀E"(2)] "∃E"[rotated] by blast
 AOT_have ζ: ‹a[F] ≡ ∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
 using a_prop[THEN "&E"(2), THEN "∀E"(2)] by blast
 AOT_show ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
 proof (rule "∨E"(1)[OF "exc-mid"]; rule "→I")
 AOT_assume A: ‹a[F]›
 AOT_hence ‹∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
 using ζ[THEN "≡E"(1)] by blast
 then AOT_obtain b where b_prop: ‹∀y ([F]y ≡ y = b) & >¬b[F]›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹[F]a›
 using F_prop[THEN "∀E"(2), THEN "≡E"(2)] "=I"(2) by blast
 ultimately AOT_have ‹a = b›
 using "∀E"(2) "≡E"(1) "&E" by fast
 AOT_hence ‹a = b›
 using "β→C" by blast
 AOT_hence ‹b[F]›
 using A "rule=E" by fast
 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
 using b_prop[THEN "&E"(2)] RAA by blast
 next
 AOT_assume A: ‹¬a[F]›
 AOT_hence ‹¬∃z (∀y ([F]y ≡ y = z) & ¬z[F])›
 using ζ "oth-class-taut:4:b"[THEN "≡E"(1), THEN "≡E"(1)] by blast
 AOT_hence ‹∀z ¬(∀y ([F]y ≡ y = z) & ¬z[F])›
 using "cqt-further:4"[THEN "→E"] by blast
 AOT_hence ‹¬(∀y ([F]y ≡ y = a) & ¬a[F])›
 using "∀E" by blast
 AOT_hence ‹∀y ([F]y ≡ y = a) → a[F]›
 by (metis "&I" "deduction-theorem" "raa-cor:4")
 AOT_hence ‹a[F]› using F_prop "→E" by blast
 AOT_thus ‹¬∀y ∃F ∀x([F]x ≡ x = y)›
 using A RAA by blast
 qed
 (simp)

  "block-paradox:5": ‹¬∃F∀x∀y([F]xy ≡ y = x)›
 (rule "raa-cor:2")
 AOT_assume ‹∃F∀x∀y([F]xy ≡ y = x)›
 then AOT_obtain F where F_prop: ‹∀x∀y([F]xy ≡ y = x)›
 using "∃E"[rotated] by blast
 {
 fix x
 AOT_have 1: ‹∀y([F]xy ≡ y = x)›
 using F_prop "∀E" by blast
 AOT_have 2: ‹[λz [F]xz]↓› by "cqt:2"
 moreover AOT_have ‹∀y([λz [F]xz]y ≡ y = x)›
 proof(rule "∀I")
 fix y
 AOT_have ‹[λz [F]xz]y ≡ [F]xy›
 using "beta-C-meta"[THEN "→E"] 2 by fast
 also AOT_have ‹... ≡ y = x›
 using 1 "∀E" by fast
 finally AOT_show ‹[λz [F]xz]y ≡ y = x›.
 qed
 ultimately AOT_have ‹∃F∀y([F]y ≡ y = x)›
 using "∃I" by fast
 }
 AOT_hence ‹∀x∃F∀y([F]y ≡ y = x)›
 by (rule GEN)
 AOT_thus ‹∀x∃F∀y([F]y ≡ y = x) & ¬∀x∃F∀y([F]y ≡ y = x)›
 using "&I" "block-paradox:4" by blast
 

  "block-paradox2:1":
 ‹∀x [G]x → ¬[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 (rule "→I"; rule "raa-cor:2")
 AOT_assume antecedant: ‹∀x [G]x›
 AOT_have Lemma: ‹∀x ([G]\ιy(y = x & ∃H (x[H] & le='font-size: 18px;'>¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
 proof(rule GEN)
 fix x
 AOT_have A: ‹[G]\ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡
 ∃!y (y = x & ∃H (x[H] & ¬[H]x))›
 proof(rule "≡I"; rule "→I")
 AOT_assume ‹[G]\ιy (y = x & ∃H (x[H] & ¬[H]x))›
 AOT_hence ‹\ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
 using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
 AOT_thus ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
 using "!-exists:1"[THEN "≡E"(1)] by blast
 next
 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
 AOT_obtain a where a_1: ‹a = x & ∃H (x[H] & ='font-size: 18px;'>¬[H]x)›
 and a_2: ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = a)›
 using "uniqueness:1"[THEN "≡_dfE", OF A] "&E" "∃E"[rotated] by blast
 AOT_have a_3: ‹[G]a›
 using antecedant "∀E" by blast
 AOT_show ‹[G]\ιy (y = x & ∃H (x[H] & ¬[H]x))›
 apply (rule "russell-axiom[exe,1].russell-axiom"[THEN "≡E"(2)])
 apply (rule "∃I"(2))
 using a_1 a_2 a_3 "&I" by blast
 qed
 also AOT_have B: ‹... ≡ ∃H (x[H] & ¬[H]x)›
 proof (rule "≡I"; rule "→I")
 AOT_assume A: ‹∃!y (y = x & ∃H (x[H] & ¬[H]x))›
 AOT_obtain a where ‹a = x & ∃H (x[H] & ¬[H]x)›
 using "uniqueness:1"[THEN "≡_dfE", OF A] "&E" "∃E"[rotated] by blast
 AOT_thus ‹∃H (x[H] & ¬[H]x)› using "&E" by blast
 next
 AOT_assume ‹∃H (x[H] & ¬[H]x)›
 AOT_hence ‹x = x & ∃H (x[H] & ¬[H]x)›
 using "id-eq:1" "&I" by blast
 moreover AOT_have ‹∀z (z = x & ∃H (x[H] & ¬[H]x) → z = x)›
 by (simp add: "Conjunction Simplification"(1) "universal-cor")
 ultimately AOT_show ‹∃!y (y = x & ∃H (x[H] & e='font-size: 18px;'>¬[H]x))›
 using "uniqueness:1"[THEN "≡_dfI"] "&I" "∃I"(2) by fast
 qed
 finally AOT_show ‹([G]\ιy(y = x & ∃H (x[H] & ='font-size: 18px;'>¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))› .
 qed

 AOT_assume A: ‹[λx [G]\ιy (y = x & ∃H (x[H] & ='font-size: 18px;'>¬[H]x))]↓›
 AOT_have θ: ‹∀x ([λx [G]\ιy (y = x & ∃H (x[H] & e='font-size: 18px;'>¬[H]x))]x ≡
 [G]\ιy(y = x & ∃H (x[H] & >¬[H]x)))›
 using "beta-C-meta"[THEN "→E", OF A] "∀I" by fast
 AOT_have ‹∀x ([λx [G]\ιy (y = x & ∃H (x[H] & 'font-size: 18px;'>¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
 using θ Lemma "cqt-basic:10"[THEN "→E"] "&I" by fast
 AOT_hence ‹∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))›
 using "∃I"(1) A by fast
 AOT_thus ‹(∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x))) &
 (¬∃F ∀x ([F]x ≡ ∃H (x[H] & ¬[H]x)))›
 using "block-paradox:2" "&I" by blast
 

 ‹Note: Strengthens the above to a modally-strict theorem.
 Not explicitly part of PLM.›
  "block-paradox2:1[strict]":
 ‹∀x \A[G]x → ¬[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 (rule "→I"; rule "raa-cor:2")
 AOT_assume antecedant: ‹∀x \A[G]x›
 AOT_have Lemma: ‹\A∀x ([G]\ιy(y = x & ∃H (x[H] & tyle='font-size: 18px;'>¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
 proof(safe intro!: GEN "Act-Basic:5"[THEN "≡E"(2)]
 "logic-actual-nec:3"[axiom_inst, THEN "≡E"(2)])
 fix x
 AOT_have A: ‹\A[G]\ιy (y = x & ∃H (x[H] & ¬[H]x)) ≡
 ∃!y \A(y = x & ∃H (x[H] & ¬[H]x))›
 proof(rule "≡I"; rule "→I")
 AOT_assume ‹\A[G]\ιy (y = x & ∃H (x[H] & ¬[H]x))›
 moreover AOT_have ‹◻([G]\ιy (y = x & ∃H (x[H] & yle='font-size: 18px;'>¬[H]x)) →
 ◻\ιy (y = x & ∃H (x[H] & ¬[H]x))↓)›
 proof(rule RN; rule "→I")
 AOT_modally_strict {
 AOT_assume ‹[G]\ιy (y = x & ∃H (x[H] & ¬[H]x))›
 AOT_hence ‹\ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
 using "cqt:5:a"[axiom_inst, THEN "→E", THEN "&E"(2)] by blast
 AOT_thus ‹◻\ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
 using "exist-nec"[THEN "→E"] by blast
 }
 qed
 ultimately AOT_have ‹\A◻\ιy (y = x & ∃H (x[H] & yle='font-size: 18px;'>¬[H]x))↓›
 using "act-cond"[THEN "→E", THEN "→E"] "nec-imp-act"[THEN "→E"] by blast
 AOT_hence ‹\ιy (y = x & ∃H (x[H] & ¬[H]x))↓›
 using "Act-Sub:3" "B♢" "vdash-properties:10" by blast
 AOT_thus ‹∃!y \A(y = x & ∃H (x[H] & ¬[H]x))›
 using "actual-desc:1"[THEN "≡E"(1)] by blast
 next
 AOT_assume A: ‹∃!y \A(y = x & ∃H (x[H] & ¬[H]x))›
 AOT_obtain a where a_1: ‹\A(a = x & ∃H (x[H] & yle='font-size: 18px;'>¬[H]x))›
 and a_2: ‹∀z (\A(z = x & ∃H (x[H] & ¬[H]x)) → z = a)›
 using "uniqueness:1"[THEN "≡_dfE", OF A] "&E" "∃E"[rotated] by blast
 AOT_have a_3: ‹\A[G]a›
 using antecedant "∀E" by blast
 moreover AOT_have ‹a = \ιy(y = x & ∃H (x[H] & e='font-size: 18px;'>¬[H]x))›
 using "nec-hintikka-scheme"[THEN "≡E"(2), OF "&I"] a_1 a_2 by auto
 ultimately AOT_show ‹\A[G]\ιy (y = x & ∃H (x[H] & style='font-size: 18px;'>¬[H]x))›
 using "rule=E" by fast
 qed
 also AOT_have B: ‹... ≡ \A∃H (x[H] & ¬[H]x)›
 proof (rule "≡I"; rule "→I")
 AOT_assume A: ‹∃!y \A(y = x & ∃H (x[H] & ¬[H]x))›
 AOT_obtain a where ‹\A(a = x & ∃H (x[H] & ¬[H]x))›
 using "uniqueness:1"[THEN "≡_dfE", OF A] "&E" "∃E"[rotated] by blast
 AOT_thus ‹\A∃H (x[H] & ¬[H]x)›
 using "Act-Basic:2"[THEN "≡E"(1), THEN "&E"(2)] by blast
 next
 AOT_assume ‹\A∃H (x[H] & ¬[H]x)›
 AOT_hence ‹\Ax = x & \A∃H (x[H] & ¬[H]x)›
 using "id-eq:1" "&I" "RA[2]" by blast
 AOT_hence ‹\A(x = x & ∃H (x[H] & ¬[H]x))›
 using "act-conj-act:3" "Act-Basic:2" "≡E" by blast
 moreover AOT_have ‹∀z (\A(z = x & ∃H (x[H] & e='font-size: 18px;'>¬[H]x)) → z = x)›
 proof(safe intro!: GEN "→I")
 fix z
 AOT_assume ‹\A(z = x & ∃H (x[H] & ¬[H]x))›
 AOT_hence ‹\A(z = x)›
 using "Act-Basic:2"[THEN "≡E"(1), THEN "&E"(1)] by blast
 AOT_thus ‹z = x›
 by (metis "id-act:1" "intro-elim:3:b")
 qed
 ultimately AOT_show ‹∃!y \A(y = x & ∃H (x[H] & yle='font-size: 18px;'>¬[H]x))›
 using "uniqueness:1"[THEN "≡_dfI"] "&I" "∃I"(2) by fast
 qed
 finally AOT_show ‹(\A[G]\ιy(y = x & ∃H (x[H] & le='font-size: 18px;'>¬[H]x)) ≡ \A∃H (x[H] & ¬[H]x))›.
 qed

 AOT_assume A: ‹[λx [G]\ιy (y = x & ∃H (x[H] & ='font-size: 18px;'>¬[H]x))]↓›
 AOT_hence ‹\A[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 using "exist-nec" "→E" "nec-imp-act"[THEN "→E"] by blast
 AOT_hence ‹\A([λx [G]\ιy (y = x & ∃H (x[H] & 'font-size: 18px;'>¬[H]x))]↓ &
 ∀x ([G]\ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x)))›
 using Lemma "Act-Basic:2"[THEN "≡E"(2)] "&I" by blast
 moreover AOT_have ‹\A([λx [G]\ιy (y = x & ∃H (x[H] & n style='font-size: 18px;'>¬[H]x))]↓ &
 ∀x ([G]\ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x)))
 → \A∃p (p & ¬p)›
 proof (rule "logic-actual-nec:2"[axiom_inst, THEN "≡E"(1)];
 rule "RA[2]"; rule "→I")
 AOT_modally_strict {
 AOT_assume 0: ‹[λx [G]\ιy (y = x & ∃H (x[H] & ='font-size: 18px;'>¬[H]x))]↓ &
 ∀x ([G]\ιy(y = x & ∃H (x[H] & ¬[H]x)) ≡ ∃H (x[H] & ¬[H]x))›
 AOT_have ‹∃F ∀x ([F]x ≡ ∃G (x[G] & ¬[G]x))›
 proof(rule "∃I"(1))
 AOT_show ‹∀x ([λx [G]\ιy (y = x & ∃H (x[H] & 'font-size: 18px;'>¬[H]x))]x ≡ ∃H (x[H] & ¬[H]x))›
 proof(safe intro!: GEN "≡I" "→I" "β←C" dest!: "β→C")
 fix x
 AOT_assume ‹[G]\ιy(y = x & ∃H (x[H] & ¬[H]x))›
 AOT_thus ‹∃H (x[H] & ¬[H]x)›
 using 0 "&E" "∀E"(2) "≡E"(1) by blast
 next
 fix x
 AOT_assume ‹∃H (x[H] & ¬[H]x)›
 AOT_thus ‹[G]\ιy(y = x & ∃H (x[H] & ¬[H]x))›
 using 0 "&E" "∀E"(2) "≡E"(2) by blast
 qed(auto intro!: 0[THEN "&E"(1)] "cqt:2")
 next
 AOT_show ‹[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 using 0 "&E"(1) by blast
 qed
 AOT_thus ‹∃p (p & ¬p)›
 using "block-paradox:2" "reductio-aa:1" by blast
 }
 qed
 ultimately AOT_have ‹\A∃p (p & ¬p)›
 using "→E" by blast
 AOT_hence ‹∃p \A(p & ¬p)›
 by (metis "Act-Basic:10" "intro-elim:3:a")
 then AOT_obtain p where ‹\A(p & ¬p)›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹¬\A(p & ¬p)›
 using "non-contradiction"[THEN "RA[2]"]
 by (meson "Act-Sub:1" "¬¬I" "intro-elim:3:d")
 ultimately AOT_show ‹p & ¬p› for p
 by (metis "raa-cor:3")
 

  "block-paradox2:2":
 ‹∃G ¬[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 (rule "∃I"(1))
 AOT_have 0: ‹[λx ∀p (p →p)]↓›
 by "cqt:2[lambda]"
 moreover AOT_have ‹∀x [λx ∀p (p →p)]x›
 apply (rule GEN)
 apply (rule "beta-C-cor:2"[THEN "→E", OF 0, THEN "∀E"(2), THEN "≡E"(2)])
 using "if-p-then-p" GEN by fast
 moreover AOT_have ‹∀G (∀x [G]x → ¬[λx [G]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓)›
 using "block-paradox2:1" "∀I" by fast
 ultimately AOT_show ‹¬[λx [λx ∀p (p →p)]\ιy (y = x & ∃H (x[H] & ¬[H]x))]↓›
 using "∀E"(1) "→E" by blast
 ("cqt:2[lambda]")

  propositions: ‹∃p ◻(p ≡ φ)›
 (rule "∃I"(1))
 AOT_show ‹◻(φ ≡ φ)›
 by (simp add: RN "oth-class-taut:3:a")
 
 AOT_show ‹φ↓›
 by (simp add: "log-prop-prop:2")
 

  "pos-not-equiv-ne:1":
 ‹(♢¬∀x₁...∀x_n ([F]x₁...x_n ≡ [G]x₁...x_n)) → F ≠ G›
  (rule "→I")
 AOT_assume ‹♢¬∀x₁...∀x_n ([F]x₁...x_n ≡ [G]x₁...x_n)›
 AOT_hence ‹¬◻∀x₁...∀x_n ([F]x₁...x_n ≡ [G]x₁...x_n)›
 using "KBasic:11"[THEN "≡E"(2)] by blast
 AOT_hence ‹¬(F = G)›
 using "id-rel-nec-equiv:1" "modus-tollens:1" by blast
 AOT_thus ‹F ≠ G›
 using "=-infix"[THEN "≡_dfI"] by blast
 

  "pos-not-equiv-ne:2": ‹(♢¬(φ{F} ≡ φ{G})) → F ≠ G›
  (rule "→I")
 AOT_modally_strict {
 AOT_have ‹¬(φ{F} ≡ φ{G}) → ¬(F = G)›
 proof (rule "→I"; rule "raa-cor:2")
 AOT_assume 1: ‹F = G›
 AOT_hence ‹φ{F} → φ{G}›
 using "l-identity"[axiom_inst, THEN "→E"] by blast
 moreover {
 AOT_have ‹G = F›
 using 1 id_sym by blast
 AOT_hence ‹φ{G} → φ{F}›
 using "l-identity"[axiom_inst, THEN "→E"] by blast
 }
 ultimately AOT_have ‹φ{F} ≡ φ{G}›
 using "≡I" by blast
 moreover AOT_assume ‹¬(φ{F} ≡ φ{G})›
 ultimately AOT_show ‹(φ{F} ≡ φ{G}) & ¬(φ{F} ≡ φ{G})›
 using "&I" by blast
 qed
 }
 AOT_hence ‹♢¬(φ{F} ≡ φ{G}) → ♢¬(F = G)›
 using "RM:2[prem]" by blast
 moreover AOT_assume ‹♢¬(φ{F} ≡ φ{G})›
 ultimately AOT_have 0: ‹♢¬(F = G)› using "→E" by blast
 AOT_have ‹♢(F ≠ G)›
 by (AOT_subst ‹F ≠ G› ‹¬(F = G)›)
 (auto simp: "=-infix" "≡Df" 0)
 AOT_thus ‹F ≠ G›
 using "id-nec2:3"[THEN "→E"] by blast
 

  "pos-not-equiv-ne:2[zero]": ‹(♢¬(φ{p} ≡ φ{q})) → p ≠ q›
  (rule "→I")
 AOT_modally_strict {
 AOT_have ‹¬(φ{p} ≡ φ{q}) → ¬(p = q)›
 proof (rule "→I"; rule "raa-cor:2")
 AOT_assume 1: ‹p = q›
 AOT_hence ‹φ{p} → φ{q}›
 using "l-identity"[axiom_inst, THEN "→E"] by blast
 moreover {
 AOT_have ‹q = p›
 using 1 id_sym by blast
 AOT_hence ‹φ{q} → φ{p}›
 using "l-identity"[axiom_inst, THEN "→E"] by blast
 }
 ultimately AOT_have ‹φ{p} ≡ φ{q}›
 using "≡I" by blast
 moreover AOT_assume ‹¬(φ{p} ≡ φ{q})›
 ultimately AOT_show ‹(φ{p} ≡ φ{q}) & ¬(φ{p} ≡ φ{q})›
 using "&I" by blast
 qed
 }
 AOT_hence ‹♢¬(φ{p} ≡ φ{q}) → ♢¬(p = q)›
 using "RM:2[prem]" by blast
 moreover AOT_assume ‹♢¬(φ{p} ≡ φ{q})›
 ultimately AOT_have 0: ‹♢¬(p = q)› using "→E" by blast
 AOT_have ‹♢(p ≠ q)›
 by (AOT_subst ‹p ≠ q› ‹¬(p = q)›)
 (auto simp: 0 "=-infix" "≡Df")
 AOT_thus ‹p ≠ q›
 using "id-nec2:3"[THEN "→E"] by blast
 

  "pos-not-equiv-ne:3":
 ‹(¬∀x₁...∀x_n ([F]x₁...x_n ≡ [G]x₁...x_n)) → F ≠ G›
 using "→I" "pos-not-equiv-ne:1"[THEN "→E"] "T♢"[THEN "→E"] by blast

  "pos-not-equiv-ne:4": ‹(¬(φ{F} ≡ φ{G})) → F ≠ G›
 using "→I" "pos-not-equiv-ne:2"[THEN "→E"] "T♢"[THEN "→E"] by blast

  "pos-not-equiv-ne:4[zero]": ‹(¬(φ{p} ≡ φ{q})) → p ≠ q›
 using "→I" "pos-not-equiv-ne:2[zero]"[THEN "→E"]
 "T♢"[THEN "→E"] by blast

  relation_negation :: "Π ==> Π" (‹_^-›)
 "df-relation-negation": "[F]^- =_df [λx₁...x_n ¬[F]x₁...x_n]"

  φneg
  "" :: "φneg ==> τ" (‹_›)
  "" :: "φneg ==> φ" (‹'(_')›)

  relation_negation_0 :: ‹φ ==> φneg› (‹'(_')^-›)
 "df-relation-negation[zero]": "(p)^- =_df [λ ¬p]"

  "rel-neg-T:1": ‹[λx₁...x_n ¬[Π]x₁...x_n]↓›
 by "cqt:2[lambda]"

java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
 using "cqt:2[lambda0]"[axiom_inst] b blast

java.lang.NullPointerException
 using "=I"(1)[OF "rel-neg-T:1"]
java.lang.NullPointerException

 2":\openp>\<> 
 using "=I"(1)[OF "rel-neg-T:1[zero]"]
 by (rule "=_dfI"(1)[OF "df-relation-negation[zero]", O et ? == {(x,y).xn0\and(u x) \le (uy)}"

  "rel-neg-T:3": ‹[Πau si iint, goal_c)
 using "=_dfI"(1)[OF "df-relation
 "rel-neg-T:1" by blast

  "rel-neg-T:3[zero]": ‹
 using "log-prop-prop:2" by blast

  "thm-relation-negation:1": ‹
  -
 AOT_have ‹
 using "rule=E"[rotated, OF "rel-neg-T:2"]
 "rule=E"[rotated, OF "rel-neg-T:2"[THEN id_sym]]
 "→I" "≡I" by fast
  c
 using "beta-C-meta"[THEN "→E", OF "rel-neg-T:1"] by fast
 finally show ?thesis.
 

  "thm-relation-negation:2": ‹¬[F]^-x₁...x_n ≡ [F]x₁...x_n›
 apply (AOT_subst ‹[F]x₁...x_n› ‹¬¬[F]x₁...x_n›)
 apply (simp add: "oth-class-taut:3:b")
 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
 using "thm-relation-negation:1".

  "thm-relation-negation:3": ‹4
  -
 AOT_have ‹(p)^- = [λ ¬p]› using "rel-neg-T:2[zero]" by blast
 AOT_hence ‹((p)^-) ≡ [λ ¬p]›
 using "df-relation-negation[zero]" "log-prop-prop:2"
 "oth-class-taut:3:a" "rule-id-df:2:a" by blast
 also AOT_have ‹[λ ¬p] ≡ ¬p›
 by (simp add: "propositions-lemma:2")
 finally show ?thesis.
 

  \R
 using "thm-relation-negation:3"[THEN "≡E"(1)]
 "thm-relation-negation:3"[THEN "≡E"(2)]
 "≡I" "→I" RAA by metis

  "thm-relation-negation:5": ‹[F] ≠ [F]^-›. \<forallx⟶ ?I) ∧\<longrightarrow 
  -
 AOT_have ‹¬([F] = [F]^-)›
 proof (rule RAA(2))
 AOT_show ‹[F]x₁...x_n → [F]x have "fnite {(I, r). valid_re X k II r∧
 using "if-p-then-p".
 next
 AOT_assume \<then 
 AOT_hence ‹[F]^- = [F]› using id_sym by blast
 AOT_hence ‹[F]x₁...x_n ≡ ¬[F]x₁...x_n› for x₁x_n
 using "rule=E" "thm-relation-negation:1" by fast
 AOT_thus ‹¬([F]x₁...x_n → [F]x₁...x_n)› for x₁x_>fX›(auto intr Max_ge)
 using "≡E" RAA by metis
 qed
 thus qe
 using "≡_dfI" "=-infix" by blast
 

  "thm-relation-negation:6": ‹p ≠ (p)^-›
  -
 AOT_have ‹¬(p = (p)^-)›
 ( RAA(22))
 AOT_show ‹p → p›
 then-.
 next
 AOT_assume ‹p = (p)^-›
 AOT_hence ‹(p)^- = p› using id_sym by blast
 AOT_hence \\ ≡
 using "rule=E" "thm-relation-negation:3" by fast
 AOT_thus ‹¬(p → p)›
 
 qed
 thus ?thesis
 using "≡_dfI" "=-infix" by blast
 

java.lang.NullPointerException
 apply (rule "df-relation-negation[zero]"[THEN "=_dfE"(1)])
 using "cqt:2[lambda0]"[axiom_inst] "rel-neg-T:2[zero]"
 propositions-lemma:1" i" id_trans bby blast+

  "thm-relation-negation:8": ‹ qed
 (rule "→
 AOT_assume ‹p = q›
 moreover AOT_have ‹(¬p)↓› using "log-prop-prop:2".
 moreover AOT_have ‹(¬p) = (¬p)› using calculation(2) "=I" by blast
 ultimately AOT_show ‹(¬p) = (¬q)›
 using "rule=E" by fast
 

  "thm-relation-negation:9": ‹p = q → by auto
 (rule "→I")
 AOT_assume ‹p = q›
 AOT_hence ‹
java.lang.NullPointerException
 using "thm-relation-negation:7" id_sym id_trans by metis
 

  Necessary :: ‹
 "contingent-properties:1":
 ‹Necessary([F]) ≡_df ◻∀x₁

  Necessary0 :: ‹φ ==> φ› (‹Necessary0'(_')›)
 "contingent-properties:1[zero]":
 ‹Necessary0(p) ≡_{R_def}

 Π
 "contingent-properties:2":
java.lang.NullPointerException

 next
 "contingent-properties:2[zero]":
 ‹

  NonContingent :: ‹Π ==> φ› (‹NonContingent'(_')›)
 "contingent-properties:3":
 ‹NonContingent([F]) ≡_df Necessary([F]) ∨ Impossible([F])›

  NonContingent0 :: ‹φ ==> φ› (‹NonContingent0'(_')›)
 "contingent-properties:3[zero]":
 ‹NonContingent0(p) \<equivnext

  Contingent :: ‹Π ==> φ› (‹Contingent'(_')›)
 "contingent-properties:4":
 ‹

  Contingent0 :: ‹φ ==> φ› (‹Contingent0'(_')›)
 "contingent-properties:4[zero]":
 ‹ from A have "?S


  "thm-cont-prop:1": ‹frac_lt_1)
  (rule "≡I"; rule "→I")
 AOT_assume ‹NonContingent([F])›
 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
 using "≡_dfE"[OF "contingent-properties:3"] by blast
 moreover {
 AOT_assume ‹Necessary([F])›
 AOT_hence
 using "≡_dfE"[OF "contingent-properties:1"] by blast
 moreover AOT_modally_strict {
java.lang.NullPointerException
 AOT_hence ‹
 AOT_hence ‹¬[F]^-x ith ‹
 by (meson "≡E"(6) "oth-class-taut:3:a"
java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
 AOT_hence ‹∀
 }
 ultimately AOT_have ‹◻(∀x₁...∀x_n ¬[F]^-x₁...x_n)›
 prem" \Gammagu>\<>\
 AOT_hence ‹Impossible([F]^-)›
 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1),
 OF "rel-neg-T:3", THEN "≡E"(2)]
 by blast
 }
 moreover {
 AOT_assume ‹
 AOT_hence ‹
 <-v
 OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)]
 by blast
 moreover AOT_modally_strict {
 AOT_assume ‹
 AOT_hence ‹
java.lang.NullPointerException
 by (meson "≡E"(6) "oth-class-taut:3:a"
 "thm-relation-negation:1" "≡E"(1))
 AOT_hence ‹∀x_^
 }
 <(x
 using "RN[prem]"[where \<Gamma        
 AOT_hence ‹
 using "≡_dfI"[OF "contingent-properties:1"] by blast
 }
java.lang.NullPointerException
 using "∨E"(1) "∨I" "→I" by metis
 AOT_thus ‹
java.lang.NullPointerException
 
 AOT_assume ‹NonContingent([F]^-)›
 AOT_hence ‹Necessary([F]^-) ∨ Impossible([F]^-)›
 using "\< \
 moreover {
 AOT_assume ‹
 AOT_hence ‹◻(∀x₁...∀x_n [F]^-x₁...x_n)›
 using "≡_dfE"[OF "contingent-properties:1"] by blast
 moreover AOT_modally_strict {
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
java.lang.NullPointerException
 AOT_hence ‹
 by (meson "≡E"(6) "oth-class-taut:3:a"
 "thm-relation-negation:1" "≡E"(2))
 AOT_hence ‹∀x₁...∀x_n ¬[F]x₁...x_n›assms(3,4) obxc herx:: "I x =Coc" ""x \\ "x \in X" b a
 }
 ultimately AOT_have ‹
 using "RN[prem]"[where Γ="{«∀x₁
 AOT_hence apply (auto simp: Isimp: I'_de Z_def)
 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1),
 OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(2)]
  "r(c + 11)) \lev \oplus t) x + (t - t')')" unfol cval_adddef by auto
 }
 moreover {
 AOT_assume ‹
 AOT_hence ‹◻(∀x₁...∀x_n ¬[F]^-x₁...x_n)›
 using "≡Df"[OF "contingent-properties:2", THEN "≡S"(1),
 OF "rel-neg-T:3", THEN "≡E"(1)]
 by blast
 moreover AOT_modally_strict {
 open\>^sub>...\forallx\<^> 
 AOT_hence ‹¬[F]^-x₁...x_n› for x₁x_n using "∀E" by blast
 AOT_hence ‹[F]x\<^      AOT_hence
  ^su>0 "{x \inexid. I' x = Intv d}"
 "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡E"(1)]
 "useful-tautologies:1"[THEN "→E"] by blast
java.lang.NullPointerException
 }
 ultimately AOT_have ‹◻(∀x₁...∀x_n [F]x₁...x_n)›
java.lang.NullPointerException
 AOT_hence ‹
 using "≡_dfI"[OF "contingent-properties:1"] by blast
 }
 ultimately AOT_have ‹Necessary([F]) ∨ Impossible([F])›
 using "∨E"(1) "∨I" "→I" by metis
 AOT_thus ‹
 using "≡_dfI"[OF "contingent-properties:3"] by blast
 

  "thm-cont-prop:2": ‹Contingent([F]) ≡ ♢∃x [F]x & ♢∃x ¬[F]x›
  -
 AOT_have ‹
 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1),
 OF "cqt:2[const_var]"[axiom_inst]]
 by blast
 also AOT_have ‹... ≡ ¬Necessary([F]) & ¬
 using "oth-class-taut:5:d" by fastforce
 also AOT_have ‹... ≡ ¬Impossible([F]) & ¬Necessary([F])›
 by (simp add: "Commutativity of &")
 also AOT_have ‹... ≡ ♢∃x [F]x & ¬Necessary([F])›
 proof (rule "oth-class-taut:4:e"[THEN "→E"])
 AOT_have ‹¬Impossible([F]) ≡ ¬◻¬ ∃
 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
 apply (AOT_subst \      trans r' unfoldin trans
 equivD")
 apply (AOT_subst (reverse) ‹¬¬∀>r'"
 apply (simp add: "oth-class-taut:3:b")
 using "contingent-properties:2"[THEN "≡Df", THEN "≡S"(1),
 OF "cqt:2[const_var]"[axiom_inst]]
 by blast
 also AOT_have ‹... ≡ ♢∃x [F]x›
 using "conventions:5"[THEN "≡Df", symmetric] by blast
 finally AOT_show ‹¬Impossible([F]) ≡ ♢∃x [F]x› .
 qed
 also AOT_have ‹
 proof (rule "oth-class-taut:4:f"[THEN "→E"])
 AOT_have ‹¬Necessary([F]) ≡ ¬◻¬∃x ¬[F]x›
 apply (rule "oth-class-taut:4:b"[THEN "≡E"(1)])
 apply (AOT_subst \<open 
 apply (simp add: "conventions:4" "≡Df")
 apply (AOT_subst (reverse) ‹¬¬[F]x› ‹
 apply (simp add: "oth-class-taut:3:b")
 apply (AOT_subst (reverse) ‹¬¬∀x [F]x› ‹∀x [F]x›)
 by (auto simp: "oth-class-taut:3:b" "contingent-properties:1" "≡Df")
 \> ≡
 using "conventions:5"[THEN "≡Df", symmetric] by blast
 finally AOT_show ‹<>x
 qed
 finally show ?thesis.
 

  "thm-cont-prop:3":
 ‹Contingent([F]) ≡ Contingent([F]^-)› for F::‹<\<kappa>> AOT_var›
  -
 {
 Π
 AOT_assume ‹Π↓›
 moreover AOT_have ‹∀F (Contingent([F]) ≡ ♢∃x [F]x & ♢∃x ¬[F]x)›
 using "thm-cont-prop:2" GEN by fast
 ultimately AOT_have ‹Contingent([Π]) ≡ ♢∃x [Π]x & ♢∃x ¬[Π]x›
 using "thm-cont-prop:2" "∀E" by fast
 } note 1 = this
 AOT_have ‹Contingent([F]) ≡ ♢∃x [F]x & ♢∃x ¬[F]x›
 using "thm-cont-prop:2" by blast
 also AOT_have ‹... ≡ ♢∃x ¬[F]x & ♢∃x [F]x›
 by (simp add: "Commutativity of &")
 also AOT_have ‹
 by (AOT_subst ‹[F]^-x› ‹¬[F]x› for: x)
 (auto simp: "thm-relation-negation:1" "oth-class-taut:3:a")
 also AOT_have ‹... ≡ ♢∃x [F]^-x & ♢∃x ¬[F]^-x›
 by (AOT_subst (reverse) ‹[F]x› ‹¬[F]^-x› for: x)
 (auto simp: "thm-relation-negation:2" "oth-class-taut:3:a")
 also AOT_have ‹... ≡ Contingent([F]^-)›
 using 1[OF "rel-neg-T:3", symmetric] by blast
 finally show ?thesis.
 

 open>(\open)
 L_def: ‹L =_d

  "thm-noncont-e-e:1": ‹Necessary(L)›
  -
 AOT_modally_strict {
 fix x
 AOT_have ‹
 moreover AOT_have ‹x↓› using "cqt:2[const_var]"[axiom_inst] by blast
 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
 ultimately AOT_have ‹[λx E!x → E!x]x›
 using "β←C" by blast
 }
 AOT_hence 0: ‹
 using RN GEN by blast
 show ?thesis
 apply (rule "=_dfI"(2)[OF L_def])
 apply "cqt:2[lambda]"
 by (rule "contingent-properties:1"[THEN "≡_dfI", OF 0])
 

  "thm-noncont-e-e:2": ‹Impossible([L]^-)› auto
  -
 AOT_modally_strict {
 fix x

 AOT_have 0: ‹∀F (¬[F]^-x ≡ [F]x)›
 using "thm-relation-negation:2" GEN by fast
java.lang.NullPointerException
 by (rule 0[THEN "∀E"(1)]) "cqt:2[lambda]"
java.lang.NullPointerException
 AOT_have ‹[λx E!x → E!x]↓› by "cqt:2[lambda]"
 moreover show "∀\lefra ((v ⊕
 moreover AOT_have ‹E!x → E!x› using "if-p-then-p" by blast
 ultimately AOT_have ‹[λx E!x → E!x]x›
 using "β←C" by blast
 }
 ultimately AOT_have ‹
 using "≡E" by blast
 }
 AOT_hence 0: ‹◻∀x ¬[λx E!x → E!x]^-x›
 using RN GEN by fast
 show ?thesis
 apply (rule "=_dr X I'r'"
 apply "cqt:2[lambda]"
 apply (rule "contingent-properties:2"[THEN "≡
 using "rel-neg-T:3"
 apply blast
 using 0
 by blast
 

  "thm-noncont-e-e:3": ‹NonContingent(L)›
 using "thm-noncont-e-e:1"
 by (rule "contingent-properties:3"[THEN "≡ :cval_a2)

  "thm-noncont-e-e:4": ‹
  -
 AOT_have 0: ‹∀F (NonContingent([F]) ≡ NonContingent([F]^-))›
 using "thm-cont-prop:1" "∀I" by fast
 moreover AOT_have 1: ‹L↓›
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 AOT_show ‹
 using "∀E"(1)[OF 0, OF 1, THEN "≡E"(1), OF "thm-noncont-e- with * have "(vy + ?t) = c ++ 1" unf frac_de b au
 

  "thm-noncont-e-e:5":
 ‹
  (rule "∃I")+
 {
 AOT_have ‹
 using "thm-relation-negation:5" GEN by fast
 ‹
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 ultimately AOT_have ‹
 using "∀E" by blast
 }
 AOT_thus ‹
 using "thm-noncont-e-e:3" "thm-noncont-e-e:4" "&I" by metis
 
 AOT_show ‹[L]^-↓›
 using "rel-neg-T:3" by blast
 
 AOT_show ‹L↓›
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 

  "lem-cont-e:1": ‹♢∃x ([F]x & ♢¬[F]x) ≡ ♢∃x (¬[F]x & pan style='font-size: 18px;'>♢[F]x)›
  -
 AOT_have ‹then have "I' x = Intv c" " unfolding I' I'_defby auto
 using "BF♢" "CBF♢" "≡I" by blast
 also AOT_have ‹… ≡ ∃x (♢[F]x & ♢¬[F]x)›
 by (AOT_subst ‹♢([F]x & ♢¬[F]x)› ‹♢
 (auto simp: "S5Basic:11" "cqt-further:7")
 also AOT_have \<open  
 by (AOT_subst ‹♢¬[F]x & ♢[F]x› ‹♢[F]x & ♢¬[F]x› for: x)
 (auto simp: "Commutativity of &" "cqt-further:7")
 also AOT_have ‹… ≡ ∃x ♢(¬[F]x & ♢[F]x)›
 by (AOT_subst ‹♢(¬[F]x & ♢[F]x)› ‹♢¬[F]x & 'font-size: 18px;'>♢[F]x› for: x)
 (auto simp: "S5Basic:11" "oth-class-taut:3:a")
 also AOT_have ‹… ≡ ♢∃x (¬[F]x & ♢[F]x)›
 X> = "{x\>. I = In d}"
 finally show ?thesis.
 

  "lem-cont-e:2":
 ‹
  -
 AOT_have ‹♢∃x ([F]x & ♢¬[F]x) ≡ ♢∃x (¬[F]x & e='font-size: 18px;'>♢[F]x)›
 using "lem-cont-e:1".
 also AOT_have ‹… ≡ ♢∃x ([F]^-x & ♢¬[F]^-x)›
 apply (AOT_subst ‹¬[F]^-x› ‹[F]x› for: x)
 apply (simp add: "thm-relation-negation:2")
 apply (AOT_subst ‹[F]^-x› ‹¬[F]x› for: x)
 apply (simp add: "thm-relation-negation:1")
 by (simp add: "oth-class-taut:3:a")
 finally show ?thesis.
 

  "thm-cont-e:1": ‹♢∃x (E!x & ♢¬E!x)›
  (rule "CBF♢"[THEN "→E"])
 AOT_have ‹∃x ♢(E!x & ¬\AE!x)›
 using "qml:4"[axiom_inst] "BF♢"[THEN "→E"] by blast
 then AOT_obtain a where ‹♢(E!a & ¬\AE!a)›
 using "∃E"[rotated] by blast
 AOT_hence θ: ‹♢E!a & ♢¬
  🚫
 AOT_have ξ: ‹ t:
 by (AOT_subst ‹\A¬E!a› ‹¬\AE!a›)
 (auto simp: "logic-actual-nec:1"[axiom_inst] θ)
 AOT_have ζ: ‹♢E!a & \A¬E!a›
 by (AOT_subst ‹\A¬E!a› ‹♢ (cases "n = 0)
 (auto simp add: "Act-Sub:4" ξ)
 AOT_hence ‹♢E!a & ♢¬E!a›
 using "&E" "&I" "Act-Sub:3"[THEN "→E"] by blast
 AOT_hence 🚫
 using "S5Basic:11"[THEN "≡E"(2)] by simp
 AOT_thus ‹∃x ♢(E!x & ♢¬E!x)›
 using "∃2) ‹
 

  "thm-cont-e:2": ‹ by auto auto
  -
 AOT_have ‹∀F (♢∃x ([F]x & ♢¬[F]x) ≡ auto
 using "lem-cont-e:1" GEN by fast
  with y sshow False auto
 using "∀E"(2) by blast
 thus ?thesis using "thm-cont-e:1" "≡E" by blast
 

  "thm-cont-e:3": ‹♢∃x E!x›
  (rule "CBF♢
 AOT_obtain a where ‹♢(E!a & ♢¬E!a)›
 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF♢"[THEN "→E"]]] by blast
 AOT_hence ‹♢E!a›
 3TH "\rightarrowE()]by bla
 AOT_thus ‹∃x \<diamond 
 

  "thm-cont-e:4": ‹
  (rule "CBF♢"[THEN "→E"])
 AOT_obtain a where ‹♢(E!a & ♢¬E!a)›
 using "∃E"[rotated, OF "thm-cont-e:1"[THEN "BF♢"[THEN "→E"]]] by blast
 AOT_hence ‹♢♢¬E!a›
 using "KBasic2:3"[THEN "→E", THEN "&E"(2)] by blast
 AOT_hence ‹♢¬E!a›
 using "4♢"[THEN "→E"] by blast
 AOT_thus ‹∃x ♢¬E!x› using "∃I" by fast
 

  "thm-cont-e:5": ‹Contingent([E!])›
  -
 AOT_have ‹∀F (Contingent([F]) ≡ ♢∃x [F]x & ♢∃ *() ha "v'x + t =c" byauto
 using "thm-cont-prop:2" GEN by fast
 AOT_hence ‹Contingent([E!]) ≡ ♢∃1) 🚫
 using "∀E"(2) by blast
 thus ?thesis
 using "thm-cont-e:3" "thm-cont-e:4" "≡
 

  "thm-cont-e:6": ‹Contingent([E!]^-)›
  -
 AOT_have ‹∀F (Contingent([«F::<\<kappa>>¬]) ≡with I' hav *: "c < v
 using "thm-cont-prop:3" GEN by fast
 AOT_hence ‹Contingent([E!]) ≡ Contingent([E!]^-)›
 using "∀E"(2) by fast
 thus ?thesis using "thm-cont-e:5" "≡E" by blast
 

  "thm-cont-e:7":
 ‹∃F∃G (Contingent([«F::<\<kappa>>¬]) & Contingent([G]) & F ≠ G)›
  (rule "∃I")+
  with 🚫
 using "thm-relation-negation:5" GEN by fast
 AOT_hence ‹[E!] ≠ [E!]^-›
 using "∀E" by fast
 AOT_thus ‹Contingent([E!]) & Contingent([E!]⁼X. \exists I'' x = In= Int d}"
 using "thm-cont-e:5" "thm-cont-e:6" "&I" by metis
 
 AOT_show ‹E!^-↓›
 by (fact AOT)
 ("cqt:2")

  "property-facts:1":
 ‹NonContingent([F]) → ¬∃G (Contingent([G]) & G = F)›> X" ay:y <n 
  (rule "→I"; rule "raa-cor:2")
 AOT_assume ‹NonContingent([F])›
 AOT_hence 1: ‹Necessary([F]) ∨ Impossible([F])›
 using "contingent-properties:3"[THEN "≡_dfE"] by blast
 AOT_assume ‹∃G (Contingent([G]) & G = F)›
 then AOT_obtain G where ‹
 using "∃E"[rotated] by blast
 AOT_hence ‹Contingent([F])› using "rule=E" "&E" by blast
 AOT_hence ‹¬(Necessary([F]) ∨ Impossible([F]))›
 using "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1),
 OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
 AOT_thus ‹(Necessary([F]) ∨ Impossible([F])) &
 ¬(Necessary([F]) ∨ Impossible([F]))›
 using 1 "&I" by blast
 

  "property-facts:2":
 ‹Contingent([F]) → ¬∃G (NonContingent([G]) & G = F)›
  (rule "→I"; rule "raa-cor:2")
 AOT_assume ‹
 AOT_hence 1: ‹¬(Necessary([F]) ∨ Impossible([F]))›
 using "contingent-properties:4"[THEN "≡ t)) I'x)" "vali(k x) I'x) b au
 OF "cqt:2[const_var]"[axiom_inst], THEN "≡E"(1)] by blast
 AOT_assume ‹∃G (NonContingent([G]) & G = F)›
 then AOT_obtain G where ‹NonContingent([G]) & G = F›
 using "∃E"[rotated] by blast
 AOT_hence ‹NonContingent([F])›
 using "rule=E" "&E" by blast
 AOT_hence ‹Necessary([F]) ∨ Impossible([F])›
 using "contingent-properties:3"[THEN "≡_dfE"] by blast
 AOT_thus ‹(Necessary([F]) ∨
 ¬
 using 1 "&I" by blast
 

  "property-facts:3":
 ‹L ≠ [L]^- & L ≠ E! & L ≠ E!^- & [L]^- ≠ [E!]^- & E! ≠ [E!]^-›
  -
 AOT_have noneqI: ‹Π ≠ Π'› if ‹φ{Π}›X k I r}"}"
 apply (rule "=-infix"[THEN "≡ "vlid_rX k " "v \in r v' <>region
 using "rule=E"[where φ=φ and τ=Π and σ = Π'] that "&I" by blast
 AOT_have contingent_denotes: ‹
 using that "contingent-properties:4"[THEN "≡_dreal( + 1)≤
 AOT_have not_noncontingent_if_contingent:
 ‹
 proof(rule RAA(2))
 AOT_show ‹¬(Necessary([Π]) ∨ Impossible([Π]))›
 using that "contingent-properties:4"[THEN "≡Df", THEN "≡S"(1),
 OF contingent_denotes[OF that], THEN "≡E"(1)]
 by blast
 next
 AOT_assume ‹NonContingent([Π])›
 AOT_thus ‹Necessary([Π]) ∨ Impossible([Π])›
 using "contingent-properties:3"[THEN "≡_dfE"] by blast
 qed

 show ?thesis
 proof (safe intro!: "&I")
 AOT_show ‹L ≠ [L]^-›
 apply with ‹
 apply "cqt:2[lambda]"
 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]^-¬"{ ∈<>.
 apply (rule GEN) apply (fact AOT)
 by "cqt:2[lambda]"
 next
 AOT_show ‹L ≠ E!›
 apply (rule noneqI)
 using "thm-noncont-e-e:3"
 not_noncontingent_if_contingent[OF "thm-cont-e:5"]
 by auto
 next
java.lang.NullPointerException
 apply (rule noneqI)
 using "thm-noncont-e-e:3" apply fast
 apply (rule not_noncontingent_if_contingent)
 apply (rule "∀E"(1)[
 where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]^-)¬",
 rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
 using
 using "thm-cont-e:5" by fast+
 next
 AOT_show ‹[L]^- ≠ E!^-›
 apply (rule noneqI)
 noncont--e:4 apply fast
 apply (rule not_noncontingent_if_contingent)
 apply (rule "∀E"(1)[
 where φ="λ Π . «Contingent([Π]) ≡ Contingent([Π]^-)¬",
 rotated, OF contingent_denotes, THEN "≡E"(1), rotated])
 using "thm-cont-prop:3" GEN apply fast
 using "thm-cont-e:5" by fast+
 next
 AOT_show ‹
 apply (rule "=_dfI"(2)[OF L_def])
 apply "cqt:2[lambda]"
 apply (rule "∀E"(1)[where φ="λ Π . «Π ≠ [Π]^-¬"])
 apply (rule GEN) apply (fact AOT)
 by "cqt:2"
 qed
 

  "thm-cont-propos:1":
 ‹≤
 (rule "≡I"; rule "→I")
 AOT_assume ‹NonContingent0(p)›
 AOT_hence ‹Necessary0(p) ∨ Impossible0(p)›
 using "contingent-properties:3[zero]"[THEN "≡_dfEle> v x + t}
 moreover {
 AOT_assume ‹Necessary0(p)›
 AOT_hence 1: ‹◻p›
 using "contingent-properties:1[zero]"[THEN "≡_dfE"] by blast
java.lang.NullPointerException
 by (AOT_subst ‹
 (auto simp add: 1 "thm-relation-negation:4")
 AOT_hence ‹)
 by (rule "contingent-properties:2[zero]"[THEN "≡_dfI"])
 }
 moreover {
 AOT_assume ‹Impossible0(p)›
 AOT_hence 1: ‹◻¬p›
 by (rule "contingent-properties:2[zero]"[THEN "≡_dfE"])
 AOT_have ‹◻((p)^-)›
 by (AOT_subst ‹((p)^-)› ‹¬p›)
 (auto simp: 1 "thm-relation-negation:3")
 AOT_hence ‹Necessary0(((p)^-))›
 by (rule "contingent-properties:1[zero]"[THEN "≡_dfI"])
 }
 ultimately AOT_have ‹Necessary0(((p)^-)) ∨ Impossible0(((p)^-))›
 using "∨E"(1) "∨I" "→I" by metis
 AOT_thus ‹NonContingent0(((p)^-))›
 using "contingent-properties:3[zero]"[THEN "≡_dfI"] by blast
 
 AOT_assume ‹NonContingent0(((p)^-))›
 AOT_hence ‹∧ + t}
 using "contingent-properties:3[zero]"[THEN "≡_dfE"] by blast
 moreover {
 AOT_assume ‹Impossible0(((p)^-))›
 AOT_hence 1: ‹◻¬((p)^-)›t ha "🚫
 by (rule "contingent-properties:2[zero]"[THEN "≡_dfE"])
 AOT_have ‹◻p›
 by (AOT_subst (reverse) ‹p› ‹ ?R"
 assume F2: "\forall>X. \<notisConst
 AOT_hence ‹Necessary0(p)›
 using "contingent-properties:1[zero]"[THEN "≡_dfI"] by blast
 }
 moreover {
 AOT_assume ‹Necessary0(((p)^-))›
 AOT_hence 1: ‹◻((p)^-)›
 by (rule "contingent-properties:1[zero]"[THEN "≡_dfE"])
 AOT_have ‹◻¬p›
 by (AOT_subst (reverse) ‹¬p› ‹((p)^-)›)
 (auto simp: 1 "thm-relation-negation:3")
 AOT_hence ‹Impossible0(p)›
 by (rule "contingent-properties:2[zero]"[THEN "≡su0 \<orally
 }
 ultimately AOT_have ‹Necessary0(p) ∨ Impossible0(p)›
 using "∨E"(1) "∨I" "→I" by metis
 AOT_thus ‹NonContingent0(p)›
 using "contingent-properties:3[zero]"[THEN "≡_dfI"] by blast
 

  "thm-cont-propos:2": ‹Contingent0(φ) ≡ ♢φ & ♢¬φ›
  -
 AOT_have ‹Contingent0(φ) ≡ ¬(Necessary0(φ) ∨ Impossible0(φ))›
 using "contingent-properties:4[zero]"[THEN "≡Df"] by simp
 also AOT_have ‹… ≡ ¬Necessary0(φ) & ¬Impossible0(φ)›
 by (fact AOT)
 also AOT_have ‹… ≡ ¬Impossible0(φ) & ¬Necessary0(φ)›
 by (fact AOT)
 also AOT_have ‹… ≡ ♢φ & ♢¬φ›
 apply (AOT_subst ‹♢φ› ‹¬◻¬φ›)
 apply (simp add: "conventions:5" "≡Df")
 apply (AOT_subst ‹Impossible0(φ)› ‹◻¬φ›)
 apply (simp add: "contingent-properties:2[zero]" "≡Df")
  and const_ex: "\<xists\
 apply (simp add: "KBasic:11")
 apply (AOT_subst ‹
 apply (simp add: "contingent-properties:1[zero]" "≡Df")
 by (simp add: "oth-class-taut:3:a")
 finally show ?thesis.
 

  "thm-cont-propos:3": ‹Contingent0(p) ≡ Contingent0(((p)^-))›
  -
 AOT_have ‹R"
 also AOT_have ‹… ≡ ♢¬p & ♢p› by (fact AOT)
 also AOT_have ‹… ≡ ♢& \diamond🚫
 by (AOT_subst ‹((p)^-)› ‹¬with t11,3) show ?thesby
 (auto simp: "thm-relation-negation:3" "oth-class-taut:3:a")
 also AOT_have ‹… ≡ ♢((p)^-) & ♢¬((p)^-)›
 by (AOT_subst ‹¬((p)^-)› ‹p›)
 (auto simp: "thm-relation-negation:4" "oth-class-taut:3:a")
 also AOT_have ‹
 using "thm-cont-propos:2"[symmetric] by blast
 finally show ?thesis.
 

  noncontingent_prop :: ‹φ› (‹p₀›)
 p₀_def: "(p₀) =_df (∀x (E!x → E!x))"

  "thm-noncont-propos:1": ‹Necessary0((p₀))›
 (rule "contingent-properties:1[zero]"[THEN "≡_dfI"])
 AOT_show ‹◻(p₀)›
 apply (rule "=_dfI"(2)[OF p₀_def])
 using "log-prop-prop:2" apply simp
 using "if-p-then-p" RN GEN by fast
 

  "thm-noncont-propos:2": ‹c" y linarith
 (rule "contingent-properties:2[zero]"[THEN "≡_dfI"])
 AOT_show ‹◻¬((p₀)^-)›
 apply (AOT_subst ‹((p₀)^-)› ‹¬p₀› y ob c' wh c':"y<>X
 using "thm-relation-negation:3" GEN "∀E"(1)[rotated, OF "log-prop-prop:2"]
 apply fast
 apply (AOT_subst (reverse) ‹
 apply (simp add: "oth-class-taut:3:b")
 apply (rule "=_dfI"(2)[OF p₀_def])
 using "log-prop-prop:2" apply simp
 using "if-p-then-p" RN GEN by fast
 

  "thm-noncont-propos:3": ‹NonContingent0((p₀))›
 apply(rule "contingent-properties:3[zero]"[THEN "≡_dfI"])
 using "thm-noncont-propos:1" "∨I" by blast

  "thm-noncont-propos:4": ‹NonContingent0(((p₀)^-))›
 apply(rule "contingent-properties:3[zero]"[THEN "≡_dfI"])
 using "thm-noncont-propos:2" "∨I" by blast

  "thm-noncont-propos:5":
 ‹∃p∃q (NonContingent0((p)) & NonContingent0((q)) & p ≠ q)›
 (rule "∃I")+
 AOT_have 0: ‹φ ≠ (φ)^-› for φ
 using "thm-relation-negation:6" "∀I"
 "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
 AOT_thus ‹
 using "thm-noncont-propos:3" "thm-noncont-propos:4" "&I" by auto
 (auto simp: "log-prop-prop:2")

  "no-cnac": ‹¬∃x(E!x & ¬\AE!x)›
 (rule "raa-cor:2")
 AOT_assume ‹∃x(E!x & ¬\AE!x)›
 then AOT_obtain a where a: ‹E!a & ¬\AE!a›
 using "∃E"[rotated] by blast
 AOT_hence ‹\A¬E!a›
 using "&E" "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
 AOT_hence ‹¬E!a›
 using "logic-actual"[act_axiom_inst, THEN "→E"] by blast
 AOT_hence ‹E!a & ¬E!a›
 using a "&E" "&I" by blast
 AOT_thus ‹p & ¬p› for p using "raa-cor:1" by blast
 

  "pos-not-pna:1": ‹¬\A∃x (E!x & ¬\AE!x)›
 (rule "raa-cor:2")
 AOT_assume ‹\A∃x (E!x & ¬\AE!x)›
 AOT_hence ‹∃x \A(E!x & ¬\AE!x)›
 using "Act-Basic:10"[THEN "≡E"(1)] by blast
 then AOT_obtain a where ‹opent > 0<>]
 using "∃E"[rotated] by blast
 AOT_hence 1: ‹\AE!a & \A¬\AE!a›
 using "Act-Basic:2"[THEN "≡E"(1)] by blast
 AOT_hence ‹¬\A\AE!a›
 using "&E"(2) "logic-actual-nec:1"[axiom_inst, THEN "≡E"(1)] by blast
 AOT_hence ‹¬\AE!a›
 using "logic-actual-nec:4"[axiom_inst, THEN "≡E"(1)] RAA by blast
 AOT_thus ‹p & ¬p› for p using 1[THEN "&E"(1)] "&I" "raa-cor:1" by blast
 

  "pos-not-pna:2": ‹♢¬∃x(E!x & ¬\A> " "(( ⊕
  (rule RAA(1))
 AOT_show ‹¬\A∃x (E!x & ¬\AE!x)›
 using "pos-not-pna:1" by blast
 
 AOT_assume ‹¬ have "(v \\> t1) ⊕
 AOT_hence ‹◻∃x (E!x & ¬\AE!x)›
 using "KBasic:12"[THEN "≡E"(2)] by blast
 AOT_thus ‹\A∃x (E!x & ¬\AE!x)›
 using "nec-imp-act"[THEN "→E"] by blast
 

  "pos-not-pna:3": ‹∃x (♢E!x & ¬\AE!x)›
  -
 AOT_obtain a where ‹
 using "qml:4"[axiom_inst] "BF♢"[THEN "→E"] "∃E"[rotated] by blast
java.lang.NullPointerException
 using "KBasic2:3"[THEN "→E"] "&E" by blast+
 AOT_have ‹¬◻\AE!a›
 using ξ "KBasic:11"[THEN "≡E"(2)] by blast
 AOT_hence ‹¬\AE!a›
 using "Act-Basic:6"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(2)] by blast
java.lang.NullPointerException
 thus ?thesis using "∃I" by fast
 

  contingent_prop :: φ (‹q₀›R
 q₀_def: ‹(q₀) =_df (∃x (E!x & ¬\AE!x))›

  q₀_prop: ‹♢q₍1 t2)) (v ⊕⊕
java.lang.NullPointerException
 apply (fact "log-prop-prop:2")
 apply (rule "&I")
 apply (fact "qml:4"[axiom_inst])
 by (fact "pos-not-pna:2")

  "basic-prop:1": ‹Contingent0((q₀))›
 (rule "contingent-properties:4[zero]"[THEN "≡_dfI"])
 AOT_have ‹¬Necessary0((q₀)) & ¬Impossible0((q₀))›
 proof (rule "&I";
 rule "=_dfI"(2)[OF q₀_def];
 (rule "log-prop-prop:2" | rule "raa-cor:2"))
 AOT_assume ‹Necessary0(∃x (E!x & ¬\AE!x))›
 AOT_hence ‹◻∃x (E!x & ¬
 using "contingent-properties:1[zero]"[THEN "≡_dfE"] by blast
  \open^bold>AE!x)›
 using "Act-Basic:8"[THEN "→E"] "qml:2"[axiom_inst, THEN "→E"] by blast
java.lang.NullPointerException
 using "pos-not-pna:1" "&I" by blast
 next
 AOT_assume ‹
 AOT_hence ‹◻¬(∃x (E!x & ¬\AE!x))›
 using "contingent-properties:2[zero]"[THEN "≡_dfE"] by blast
 AOT_hence ‹¬♢(∃x (E!x & ¬\AE!x))›
 using "KBasic2:1"[THEN "≡E"(1)] by blast
 AOT_thus ‹♢(∃x (E!x & ¬\AE!x)) & ¬♢(∃x (E!x & ¬\AE!x))›
 using "qml:4"[axiom_inst] "&I" by blast
 qed
 AOT_thus ‹¬(Necessary0((q₀)) ∨ Impossible0((q₀)))›
 using "oth-class-taut:5:d" "≡E"(2) by blast
 

  "basic-prop:2": ‹∃p Contingent0((p))›
 using "∃I"(1)[rotated, OF "log-prop-prop:2"] "basic-prop:1" by blast

  "basic-prop:3": ‹Contingent0(((q₀)^-))›
 apply (AOT_subst ‹((q₀)^-)› ‹¬q₀›)
 apply (insert "thm-relation-negation:3" "∀I"
 "∀E"(1)[rotated, OF "log-prop-prop:2"]; fast)
 apply (rule "contingent-properties:4[zero]"[THEN "\<    case
 apply (rule "oth-class-taut:5:d"[THEN "≡E"(2)])
 apply (rule "&I")
 apply (rule "contingent-properties:1[zero]"[THEN "df-rules-formulas[3]",
 THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
 apply (rule "conventions:5"[THEN "≡
 apply (rule "=_dfE"(2)[OF q₀_def])
 apply (rule "log-prop-prop:2")
 apply (rule q₀_prop[THEN "&E"(1)])
 apply (rule "contingent-properties:2[zero]"[THEN "df-rules-formulas[3]",
 THEN "useful-tautologies:5"[THEN "→E"], THEN "→E"])
 apply (rule "conventions:5"[THEN "≡_dfE"])
 by (rule q_{goal_cases}

  "basic-prop:4":
 ‹∃p∃q (p ≠ q & Contingent0(p) & Contingent0(q))›
 (rule "∃I")+
 AOT_have 0: ‹φ ≠ (φ)^-› for φ
 using "thm-relation-negation:6" "∀I"
 "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
 AOT_show ‹(q₀) ≠ (q₀)^- & Contingent0(q₀) & Contingent0(((q₀)^-))›
 using "basic-prop:1" "basic-prop:3" "&I" 0 by presburger
 (auto simp: "log-prop-prop:2")

  "proposition-facts:1":
 ‹
 (rule "→I"; rule "raa-cor:2")
 AOT_assume ‹NonContingent0(p)›
 AOT_hence 1: ‹
 using "contingent-properties:3[zero]"[THEN "≡_dfE"] by blast
 AOT_assume ‹∃q (Contingent0(q) & q = p)›
 then AOT_obtain q where ‹Contingent0(q) & q = p›
 using "∃E"[rotated] by blast
 AOT_hence ‹Contingent0(p)›
 using "rule=E" "&E" by fast
 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) &
 with A have u: "u x > c" "u x > d" unfolding cccval_de by auto
 using "contingent-properties:4[zero]"[THEN "≡_dfE"] 1 "&I" by blast
 

  "proposition-facts:2":
 ‹Contingent0(p) → ¬∃q (NonContingent0(q) & q = p)›
 (rule "→I"; rule "raa-cor:2")
 AOT_assume ‹ with d d u have "c' ≥
 AOT_hence 1: ‹¬(Necessary0(p) ∨ Impossible0(p))›
java.lang.NullPointerException
 AOT_assume ‹∃q (NonContingent0(q) & q = p)›
 then AOT_obtain q where ‹NonContingent0(q) & q = p›
 using "∃E"[rotated] by blast
 AOT_hence ‹
 using "rule=E" "&E" by fast
 AOT_thus ‹(Necessary0(p) ∨ Impossible0(p)) &
 ¬(Necessary0(p) ∨ Impossible0(p))›
 using "contingent-properties:3[zero]"[THEN "≡_dfE"] 1 "&I" by blast
 

  "proposition-facts:3":
 ‹ u)
  -
 {
 fix χ φ ψ
 AOT_assume ‹χcase (1 c') wA u show ?case by fastforce
 moreover AOT_assume ‹¬χ{ψ}›
 ultimately AOT_have ‹¬(χ B: "c' u x" "u x x <c 
 using RAA "≡E" by metis
 moreover {
 AOT_have ‹∀p∀q ((¬(χ{p} ≡ χ{q})) → (3 c')
 by (rule "∀I"; rule "∀I"; rule "pos-not-equiv-ne:4[zero]")
 AOT_hence ‹((¬(χ{φ} ≡ χ{ψ})) → φ ≠ ψ)›
 using "∀E" "log-prop-prop:2" by blast
 }
 ultimately AOT_have ‹φ ≠ ψ›
 using "→E" by blast
 } note 0 = this
 AOT_have contingent_neg: ‹Contingent0(φ) ≡ Contingent0(((φ)^-))› for φ
 using "thm-cont-propos:3" "∀I"
 "∀E"(1)[rotated, OF "log-prop-prop:2"] by fast
 AOT_have not_noncontingent_if_contingent:
 ‹¬NonContingent0(φ)› if ‹Contingent0(φ)› for φ
 apply (rule "contingent-properties:3[zero]"[THEN "≡Df",
 THEN "oth-class-taut:4:b"[THEN "≡E"(1)], THEN "≡
 using that "contingent-properties:4[zero]"[THEN "≡_dfE"] by blast
 show ?thesis
 apply (rule "&I")+
 using "thm-relation-negation:6" "∀I"
 "∀E"(1)[rotated, OF "log-prop-prop:2"]
 apply fast
 apply (rule 0)
 using "thm-noncont-propos:3" apply fast
 apply (rule not_noncontingent_if_contingent)
 apply (fact AOT)
 apply (rule 0)
 apply (rule "thm-noncont-propos:3")
 apply (rule not_noncontingent_if_contingent)
 apply (rule contingent_neg[THEN "≡E"(1)])
 apply (fact AOT)
 apply (rule 0)
 apply (rule "thm-noncont-propos:4")
 apply (rule not_noncontingent_if_contingent)
 apply (rule contingent_neg[THEN "≡E"(1)])
 apply (fact AOT)
 using "thm-relation-negation:6" "∀I"
 "∀
 

  ContingentlyTrue :: ‹t: auesiein> gb ds" and eq: "level p = level b" shows "p = b"
 tf o>Con(p) ≡_f ♢p›

 ve \dots2ix +1)*2(v ' d - lv p d)" using lv b at
 "cont-tf:2": ‹p {d yato

 
  r: "r \<>?
  ?- ) ^l d- lv? d)"
 fix x
  lv x d"
 AOT_have ‹
 roH "\equivDf",
 THEN "oth-class-taut:4:b"[THEN "≡ d" and "length p ≤rd byauo
 refls "rerefl_on (?X\^0 - {x}) ?r" unf refl_on by auto
 ¬Impossible0((p))›
 apply (rule "contingent-properties:2[zero]"[THEN "≡
 -t:4:
 apply (rule "conventions:5"[THEN "≡_p ∈ grid y d'close> by auto
 using "T♢"[THEN "→ }"
 ultimately AOT_have \open¬(Necessary0((p)) ∨
 using DeMorgan(2)[THEN "≡E"(2)] "&I" by blast
 AOT_thus <Contingent0( grid (start dm) ds"
java.lang.NullPointerException
 

  "cont-true-cont:2":
 ‹ = lgrid p ds (level p + Suc l)"
 (rule "→I")
 AOT_assume ‹
 AOT_hence 1: ‹not thesis"
 AOT_have ‹
 apply (rule "contingent-properties:1[zero]"[THEN "≡
 THEN "oth-class-taut:4:b"[THEN "≡E"(2)])
 11<E"(2)] "T♢"[THEN "→E", OF 1] by blast
 moreover AOT_have ‹ a
 apply cinenrprs2r]"[TH"\>f",
 THEN fr iraeOxr a_igiFprd,wee s''0dm} ∪
 apply (rule "conventions:5"[TN eq>\^sub>d_fE"])
 using 2.
 ultimately AOT_have ‹
 using DeMorgan(2)[THEN "≡
 AOT_thus ‹
 using "contingent-propertie(uen_qulI
 

  "cont-true-cont:3":
 ‹ContingentlyTrue((p)) ≡ ContingentlyFalse(((p)^-))›
 (rule "≡I")
 AOT_assume ‹'}" using False by auto
 AOT_hence 0: ‹
  ‹
 equiv>_fI)
 apply (AOT_subst (reverse) ‹ p)
 
 AOT_show ‹
 apply (AOT_subst ‹((p)^-)› ‹
 by (auto simp: "thm-relation-negation:3" 1)
 
 AOT_assume 1: ‹ L:: "f (v y) \<in 
 AOT_have ‹ContingentlyFalse(¬p)›
 by (AOT_subst (reverse) ‹¬p› ‹have "fr v y)\<le 
 (auto simp: "thm-relation-negation:3" 1)
 AOT_hence ‹¬¬p & ♢¬p› using "cont-tf:2"[THEN "≡_dfE"] by blast
 AOT_hence ‹p & ♢¬p›L_bound = this
 using "&I" "&E" "useful-tautologies:1"[THEN "→E"] by metis
 AOT_thus ‹ContingentlyTrue((p))›
 using "cont-tf:1"[THEN "≡_dfI"] by blast
 

  "cont-true-cont:4":
 ‹ContingentlyFalse((p)) ≡ ContingentlyTrue(((p)^-))›
 (rule "≡I"; rule "→I")
 AOT_assume ‹ContingentlyFalse(p)›
 AOT_hence 0: ‹
 using " ith nat_intv_frac_gt0[OF this] frac_ l(1)) hav "0 <Max?
 &\diamond>¬
 by (AOT_subst (reverse) ‹
 (auto simp: "oth-class-taut:3:b" 0)
 AOT_hence 1: ‹ContingentlyTrue(¬p)›
 by (rule "cont-tf:1"[THEN "≡_dfI"])
 AOT_show ‹ContingentlyTrue(((p)^-))›
 by (AOT_subst ‹((p)^-)› ‹¬p›)
 (auto simp: "thm-relation-negation:3" 1)
 
 AOT_assume 1: ‹ContingentlyTrue(((p)^-))›
 AOT_have ‹ContingentlyTrue(¬p)›
 by (AOT_subst (reverse) ‹¬p› ‹((p)^-)›)
 (auto simp add: "thm-relation-negation:3" 1)
 AOT_hence 2: ‹¬p & ♢next
 AOT_have ‹♢p›
 by (AOT_subst p ‹¬¬p›
 (auto simp add: "oth-clas ave l_bound': "c < ?
 AOT_hence ‹¬p & ♢p› using 2[THEN "&E"(1)] "&I" by blast
 AOT_thus ‹ContingentlyFalse(p)›
 by (rule "cont-tf:2"[THEN "≡_dfI"])
 

  "cont-true-cont:5":
 ‹(ContingentlyTrue((p)) & Necessary0((q))) → p ≠ q›
  (rule "→I"; frule "&E"(1); drule "&E"(2); rule "raa-cor:1")
 AOT_assume ‹ContingentlyTrue((p))›
 AOT_hence ‹♢¬p›
 using "cont-tf:1"[THEN "≡_dfE"] "&E" by blast
 AOT_hence 0: ‹¬◻p› using "KB
 AOT_assume ‹Necessary0((q))›
 moreover AOT_assume ‹¬(p ≠ q)›
 AOT_hence ‹p = q›
 using "=-infix"[THEN "≡Df",
 THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(1)]
 "useful-tautologies:1"[THEN "→E"] by blast
 ultimately AOT_have ‹Necessary0((p))› using "rule=E" id_sym by blast
 AOT_hence ‹◻p›
 using "contingent-properties:1[zero]"[THEN "≡_dfE"] by blast
 AOT_thus ‹◻p & ¬◻p› using 0 "&I" by blast
 

  "cont-true-cont:6":
 ‹(ContingentlyFalse((p)) & Impossible0((q))) → p ≠ q›
  (rule "→I"; frule "&E pproof (cases "?L = {}")
 AOT_assume ‹ContingentlyFalse((p))›
 AOT_hence ‹♢p›
 using "cont-tf:2"[THEN "≡_dfE"] "&E" by blast
 AOT_hence 1: ‹¬◻¬p›
 using "conventions:5"[THEN "≡_dfE"] by blast
 AOT_assume ‹
 moreover AOT_assume ‹¬(p ≠ q)›
 AOT_hence ‹p = q›
 using "=-infix"[THEN "≡Df",
 :<>"
 THEN "≡E"(1)]
 "useful-tautologies:1"[THEN "→E"] by blast
 ultimately AOT_have ‹Impossible0((p))› using "rule=E" id_sym by blast
 AOT_hence ‹◻¬p›
 using "contingent-properties:2[zero]"[THEN "≡_dfE"] by blast
 AOT_thus ‹◻¬p & ¬◻¬p› using 1 "&I" by blast
 

  "q0cf:1": ‹ContingentlyFalse(q₀)›
 apply (rule "cont-tf:2"[THEN "≡_dfI"])
 apply (rule "=_dfI"(2)[OF q₀_def])
 apply (fact "log-prop-prop:2")
 apply (rule "&I")
 apply (fact "no-cnac")
 by (fact "qml:4"[axiom_inst])

  "q0cf:2": ‹
 apply (rule "cont-tf:1"[THEN "≡_dfI"])
 apply (rule "=_dfI"(2)[OF q₀_def])
 apply (fact "log-prop-prop:2")
 apply (rule "&I")
 apply (rule "thm-relation-negation:3"
 [unvarify p, OF "log-prop-prop:2", THEN "≡E"(2)])
 apply (fact "no-cnac")
 apply (rule "rule=E"[rotated,
 OF "thm-relation-negation:7"
 [unvarify p, OF "log-prop-prop:2", THEN id_sym]])
 apply (AOT_subst (reverse) ‹¬¬(∃x (E!x & ¬
 by (auto simp: "oth-class-taut:3:b" "qml:4"[axiom_inst])

  "cont-tf-thm:1": ‹∃p ContingentlyTrue((p))›
 (rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
 AOT_assume ‹q₀›
 AOT_hence ‹
java.lang.NullPointerException
 by (rule "cont-tf:1"[THEN "≡_dfI"])
 
 AOT_assume ‹¬q₀›
java.lang.NullPointerException
 AOT_hence ‹ContingentlyFalse(q₀)›
 by (rule "cont-tf:2"[THEN "≡_dfI"])
 AOT_thus ‹ContingentlyTrue(((q₀)^-))›
 by (rule "cont-true-cont:4"[unvarify p,
 OF "log-prop-prop:2", THEN "≡by simp
 (auto simp: "log-prop-prop:2")


  "cont-tf-thm:2": ‹
 (rule "∨E"(1)[OF "exc-mid"]; rule "→I"; rule "∃I")
 AOT_assume ‹q₀›
java.lang.NullPointerException
 AOT_hence ‹ContingentlyTrue(q₀)›
 
 proof c "?L = {}")
 by (rule "cont-true-cont:3"[unvarify p,
 OF "log-prop-prop:2", THEN "≡E"(1)])
 
 AOT_assume ‹ from L_iL_intv[OF False] have "0 ≤ Max ?L" "Max ?L < 1
 AOT_hence ‹¬q₀ & ♢q₀› using q₀_prop[THEN "&E"(1)] "&I" by blast
 AOT_thus ‹ContingentlyFalse(q₀)›
 by (rule "cont-tf:2"[THEN "≡_dfI"])
 (auto simp: "log-prop-prop:2")

  "property-facts1:1": ‹
  -
 fix x
java.lang.NullPointerException
 using "cont-tf-thm:1" "∃E"[rotated] by blast
 AOT_hence 1: ‹p₁ & ♢¬p₁› using "cont-tf:1"[THEN "≡_dfE"] by blast
 AOT_modally_strict {
 AOT_have ‹for arbitrary p: \⊨_\◻ ([λz p]x ≡ p)›
 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
 AOT_hence ‹for arbitrary p: \⊨_\◻ ◻ ([λz p]x ≡ p)›
 by (rule RN)
 AOT_hence ‹∀p ◻([λz p]x ≡ p)› using GEN by fast
 AOT_hence ‹◻([λz p₁]x ≡ p₁)› using "∀E" by fast
 } note 2 = this
 AOT_hence ‹◻([λz p₁]x ≡ p₁)› using "∀E" by blast
 AOT_hence ‹[λz p₁]x›
 using 1[THEN "&E"(1)] "qml:2"[axiom_inst, THEN "→E"] "≡E"(2) by blast
 moreover AOT_have ‹♢¬[λz p₁]x›
 using 2[THEN "qml:2"[axiom_inst, THEN "→E"]]
 apply (AOT_subst ‹[λz p₁]x› ‹
 using 1[THEN "&E"(2)] by blast
 ultimately AOT_have ‹[λz p₁]x & ♢¬[λz p₁]x› using "&I" by blast
 AOT_hence ‹:y,) \notinr"
 moreover AOT_have ‹[λz p₁]↓› by "cqt:2[lambda]"
 ultimately AOT_show ‹∃F∃x ([F]x & ♢¬[F]x)› by (rule "∃I"(1))
 

  "property-facts1:2": ‹
  -
 fix x
 AOT_obtain p₁ where ‹ContingentlyFalse((p₁))›
 using "cont-tf-thm:2" "∃E"[rotated] by blast
 AOT_hence 1: ‹¬p₁ & ♢p₁› using "cont-tf:2"[THEN "≡_dfE"] by blast
 AOT_modally_strict {
 AOT_have ‹for arbitrary p: \⊨_\◻ ([λz p]x ≡ p)›
 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
 AOT_hence ‹for arbitrary p: \⊨_\◻ (¬[λz p]x ≡ ¬p)›
 using "oth-class-taut:4:b" "≡ fr ?u" using u bby auto
 AOT_hence ‹for arbitrary p: \⊨_\◻ ◻(¬[λz p]x ≡ ¬p)›
 by (rule RN)
 AOT_hence ‹∀p ◻(¬[λz p]x ≡ ¬p)› using GEN by fast
 AOT_hence ‹◻(¬[λz p₁]x ≡ ¬p₁)› using "∀E" by fast
 } note 2 = this
 AOT_hence ‹◻(¬[λz p₁]x ≡ ¬p₁)› using "∀E" by blast
 AOT_hence
 )q"[axiom_inst, THEN \rightarrowE"] "≡
 AOT_modally_strict {
 AOT_have ‹for arbitrary p: \⊨_\◻ ([λz p]x ≡ p)›
 by (rule "beta-C-cor:3"[THEN "∀E"(2)]) cqt_2_lambda_inst_prover
 AOT_hence ‹for arbitrary p: \⊨_\◻ ◻([λz p]x ≡ p)›
 by (rule RN)
 AOT_hence ‹∀p ◻([λz p]x ≡ p)› using GEN by fast
  ‹1]x ≡1)›E" by fast
 } note 4 = this
  ‹♢[λz p₁]x›
 THEN<>E
 pplyTsubt <>[1]x\\›p₁›
 using 1[THEN "&E"(2)] by blast
 AOT_hence ‹1]x & ♢[λ1]x›
 AOT_hence ‹z p_z p using "∃
 moreover AOT_have ‹[λz p_var{mg}$t ascedg onfiguati var{c'}.
 ultimately AOT_show ‹"ini c
 

 
 

 ζ‹L\equiv (E!x → E!x)›
 d_fI"2[OF _ef]
 apply "cqt:2[lambda]"
 apply (rule "beta-C-meta"[THEN "→L!
 by "cqt:2[lambda]"

  (0::nat) < msgs
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lamb using InitRInitC xIs0OrN C0otC WogesExigtsbyauo

 : ‹ [λ]x)›
 (rule "≡I"; rule "→I"; (rule "∀I")?)
 fix
 AOT_assume 1: ‹
 [L]x ≡ (E!x → E!x)› using eqnotnec_123_Aux_ζ
 also AOT_have ‹… ≡ φ›
 \equiv>" "→I" b smp
 also AOT_have ‹…f1=(s1
 using "Commutativity of ≡by simp
 how<pen[.
 

  > [\lambdax)🚫
 AOT_hence ‹E" by blast
 … ≡ using eqnotnec_123_Aux_ψ.
 finally AOT_have \<          also
 using "Commutativity of ≡"[
 proofc
 <\<phi>› using "≡E" "if-p-then-p" by fast
 
  lemmas eqnotnec_123_Aux_ξ =
 eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
 THEN "RM♢"]
  lemmas eqnotnec_123_Aux_ξ' =
 eqnotnec_123_Aux_θ[
 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
 THEN "RM♢"]

  "eqnotnec:1": ‹∃F∃G(∀x([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›
 -
 AOT_obtain p₁ where ‹ContingentlyTrue(p₁)›
 using "cont-tf-thm:1" "∃E"[rotated] by blast
 AOT_hence ‹p₁ & ♢¬p₁› using "cont-tf:1"[THEN "≡_dfE"] by blast
 AOT_hence ‹∀x ([L]x ≡ [λz p
 apply - apply (rule "&I")
 using "&E" eqnotnec_123_Aux_θ[THEN "≡E"(1)]
 eqnotnec_123_Aux_ξ "→E" by fast+
 AOT_hence ‹∃G (∀x([L]x ≡ [G]x) & ♢¬∀x([L]x ≡ [G]x))›
 by (rule "∃I") "cqt:2[lambda]"
 AOT_thus ‹∃F∃G (∀x([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›
 apply (rule "∃I")
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 

  "eqnotnec:2": ‹∃F∃G(¬∀x([F]x ≡ [G]x) & ♢∀hav "d = ?u" b auto
 -
java.lang.NullPointerException
 using "cont-tf-thm:2" "∃E"[rotated] by blast
 AOT_hence ‹¬p₁ & ♢p₁› using "cont-tf:2"[THEN "≡_dfE"] by blast
 AOT_hence ‹¬∀x ([L]x ≡ [λz p₁]x) & ♢∀x([L]x ≡ [λz p next
 apply - apply (rule "&I")
 using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(1)]
 "&E" eqnotnec_123_Aux_ξ' "→E" by fast+
 AOT_hence ‹∃G (¬∀x([L]x ≡ [G]x) & ♢∀x([L]x ≡ [G]x))›
 by (rule "∃I") "cqt:2[lambda]"
 AOT_thus ‹∃F∃G (¬∀x([F]x ≡ [G]x) & ♢∀
 apply (rule "∃I")
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 

  "eqnotnec:3": ‹∃F∃G(\A¬∀x([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 -
 AOT_have ‹¬\Aq₀›
 apply (rule "=_d also have "…?\closea
 apply (fact "log-prop-prop:2")
 by (fact AOT)
 AOT_hence ‹\A¬q₀›
 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
 AOT_hence ‹\A¬∀x ([L]x ≡ [λz q₀]x)›
 using eqnotnec_123_Aux_θ[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "conventions:3"[THEN "≡Df", THEN "≡E"(1), THEN "&E"(1)],
 THEN "RA[2]", THEN "act-cond"[THEN "→E"], THEN "→E"] by blast
 moreover AOT_have ‹♢∀x ([L]x ≡ [λz q₀]x)›
 using eqnotnec_123_Aux_ξ'[THEN "→E"] q₀_prop[THEN "&E"(1)] by blast
 ultimately AOT_have ‹\A¬∀x ([L]x ≡ [λz q₀]x) & ♢∀x ([L]x ≡ [λz q₀]x)›
 using "&I" by blast
 AOT_hence ‹∃G (\A¬∀x([L]x ≡ [G]x) & ♢∀x([L]x ≡ [G]x))›
 by (rule "∃I") "cqt:2[lambda]"
 AOT_thus ‹∃F∃G (\A¬∀x([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 apply (rule "∃I")
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 

 

  "eqnotnec:4": ‹∀F∃G(∀x([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›
 (rule GEN)
 fix F
 AOT_have Aux_A: ‹\⊨_\◻ ψ → ∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
 proof(rule "→I"; rule GEN)
 AOT_modally_strict {
 fix x
 AOT_assume 0: ‹ψ›
 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E"
 apply blast
 using 0 "&I" by blast
 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_B: ‹\⊨_\◻ ψ → ∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
 proof (rule "→I"; rule GEN)
 AOT_modally_strict {
 fix x
 AOT_assume 0: ‹ψ›
 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0]
 "&E"
 apply blast
 apply (rule "∨I"(1)) using 0 "&I" by blast
 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_C:
 ‹\⊨_\◻ ♢¬ψ → ♢¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
 proof(rule "RM♢"; rule "→I"; rule "raa-cor:2")
 AOT_modally_strict {
 AOT_assume 0: ‹¬ψ›
 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "∀E" by blast
 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 ultimately AOT_have ‹[F]z & ψ ≡ (([F]z & ψ) ∨ ¬ψ)› for z
 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z using 0 "∨I" by blast
 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
 }
 qed

 AOT_have Aux_D: ‹◻∀z ([F]z ≡ [λz [F]z & ψ]z) →
 (♢¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡
 ♢¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
 proof (rule "→I")
 AOT_assume A: ‹◻∀z([F]z ≡ [λz [F]z & ψ]z)›
 AOT_show ‹♢¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡
 ♢¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 proof(rule "≡I"; rule "KBasic:13"[THEN "→E"];
 rule "RN[prem]"[where Γ="{«∀z([F]z ≡ [λz [F]z & ψ]z)¬}", simplified];
 (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
 AOT_modally_strict {
 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
 AOT_hence 1: ‹[F]z ≡ [λshow ?c?cas
 using "∀E" by blast
 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ d(1,) h "ntv y ? (I )" y (ases" , a) (c " y" auto)
 using "∀E" by blast
 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "≡E" 1 2 by meson
 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 by (rule GEN)
 }
 next
 AOT_modally_strict {
 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z
 using "∀E" by blast
 AOT_assume ‹
 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "∀E" by blast
 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z›then obtd wh "v(x:=d)<> 
 using 1 2 "≡E" by meson
 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 by (rule GEN)
 }
 qed(auto simp: A)
 qed

 AOT_obtain p₁ where p₁_prop: ‹p₁ & ♢¬p₁›
 using "cont-tf-thm:1" "∃E"[rotated]
 "cont-tf:1"[THEN "≡_dfE"] by blast
 {
 AOT_assume 1: ‹
 AOT_have 2: ‹∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using Aux_B[THEN "→E", OF p₁_prop[THEN "&E"(1)]].
 AOT_have ‹♢¬∀x([λz [F]z & p₁]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using Aux_C[THEN "→E", OF p₁_prop[THEN "&E"(2)]].
 AOT_hence 3: ‹♢¬∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
 AOT_hence ‹∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x) &
 ♢¬∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using 2 "&I" by blast
 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›
 by (rule "∃I"(1)) "cqt:2[lambda]"
 }
 moreover {
 AOT_assume 2: ‹¬◻'_d cv by s
 AOT_hence ‹♢¬∀x([F]x ≡ [λz [F]z & p₁]x)›
 using "KBasic:11"[THEN "≡E"(1)] by blast
 AOT_hence ‹∀x ([F]x ≡ [λz [F]z & p₁]x) & ♢¬∀x([F]x ≡ [λz [F]z & p₁]x)›
 using Aux_A[THEN "→E", OF p₁_prop[THEN "&E"(1)]] "&I" by blast
 AOT_hence ‹∃G (∀x ([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›"
 (\<>"
 }
 ultimately AOT_show ‹∃G (∀x ([F]x ≡ [G]x) & ♢¬∀x([F]x ≡ [G]x))›
 using "∨E"(1)[OF "exc-mid"] "→I" by blast
 

  "eqnotnec:5": ‹∀F∃G(¬∀x([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 (rule GEN)
 fix F
 AOT_have Aux_A: ‹\⊨_\◻ ♢ψ → ♢∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
 proof(rule "RM♢"; rule "→I"; rule GEN)
 AOT_modally_strict {
 fix x
 AOT_assume 0: ‹ψ›
 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E"
 apply blast
 using 0 "&I" by blast
 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_B: ‹\⊨_\◻ ♢ψ → ♢∀x([F]x ≡ [λz [F]z & ψ ∨ ¬-
 proof (rule "RM♢"; rule "→I"; rule GEN)
 
 fix x
 AOT_assume 0: ‹
 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡ (([F]x & ψ) ∨ ¬ψ)›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E"
 apply blast
 apply (rule "∨r: r 🚫
 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_C: ‹\⊨_\◻ ¬ψ → ¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
 proof(rule "→I"; rule "raa-cor:2")
 AOT_modally_strict {
 AOT_assume 0: ‹
 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
 AOT_hence ‹[λz [F]z & ψ]z ≡
 using "∀E" by blast
 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 moreover AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]z ≡ (([F]z & ψ) ∨ ¬ψ)› for z
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 ultimately AOT_have ‹[F]z & ψ
 using "Commutativity of ≡"[THEN "≡E"(1)] "≡E"(5) by meson
 moreover AOT_have ‹(([F]z & ψ) ∨ ¬ψ)› for z
 using 0 "∨I" by blast
 ultimately AOT_have ‹ψ› using "≡E" "&E" by metis
 AOT_thus ‹ψ & ¬ψ› using 0 "&I" by blast
 
 qed

 AOT_have Aux_D: ‹∀z ([F]z ≡ [λz [F]z & ψ]z) →
 (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x) ≡
 ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x))› for ψ
 proof (rule "→I"; rule "≡I";
 (rule "useful-tautologies:5"[THEN "→E"]; rule "→I")?)
 AOT_modally_strict {
 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z
 using "∀E" by blast
 AOT_assume ‹∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 AOT_hence 2: ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "∀E" by blast
 AOT_have ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "≡E" 1 2 by meson
 AOT_thus ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 by (rule GEN)
 }
 next
  dertthe existence of a countable
 AOT_assume ‹∀z ([F]z ≡ [λz [F]z & ψ]z)›
 AOT_hence 1: ‹[F]z ≡ [λz [F]z & ψ]z› for z
 using "∀E" by blast
 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 AOT_hence 2: ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "∀E" by blast
 AOT_have ‹[F]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using 1 2 "≡E" by meson
 AOT_thus ‹ ∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)›
 by (rule GEN)
 }
 qed

 AOT_obtain p₁ where p₁_prop: ‹
 using "cont-tf-thm:2" "∃E"[rotated] "cont-tf:2"[THEN "≡_dfE"] by blast
 {
 AOT_assume 1: ‹∀x([F]x ≡ [λz [F]z & p₁]x)›
 AOT_have 2: ‹♢∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using Aux_B[THEN "→E", OF p₁_prop[THEN "&E"(2)]].
 AOT_have ‹¬∀x([λz [F]z & p₁]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using Aux_C[THEN "→E", OF p₁_prop[THEN "&E"(1)]].
 AOT_hence 3: ‹¬∀
 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x) &
 ♢∀x([F]x ≡ [λz [F]z & p₁ ∨ ¬p₁]x)›
 using 2 "&I" by blast
 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 by (rule "∃I"(1)) "cqt:2[lambda]"
 }
 moreover {
 AOT_assume 2: ‹¬∀x([F]x ≡ [λz [F]z & p₁]x)›
 AOT_hence ‹¬∀x([F]x ≡ [λz [F]z & p₁]x)›
 using "KBasic:11"[THEN "≡E"(1)] by blast
 AOT_hence ‹¬∀x ([F]x ≡ [λz [F]z & p₁]x) &
 ♢∀x([F]x ≡ [λz [F]z & p₁]x)›
 using Aux_A[THEN "→E", OF p₁_prop[THEN "&E"(2)]] "&I" by blast
 AOT_hence ‹∃G (¬∀x ([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 by (rule "∃I"(1)) "cqt:2[lambda]"
 }
 ultimately AOT_show ‹∃G (¬∀x ([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 using "∨E"(1)[OF "exc-mid"] "→I" by blast
 

  "eqnotnec:6": ‹∀F∃G(\A¬∀x([F]x ≡ [G]x) & ♢∀x([F]x ≡ [G]x))›
 (rule GEN)
 fix F
 AOT_have Aux_A: ‹\⊨_\◻ ♢ψ → ♢∀x([F]x ≡ [λz [F]z & ψ]x)› for ψ
 proof(rule "RM♢"; rule "→I"; rule GEN)
 AOT_modally_strict {
 fix x
 AOT_assume 0: ‹ψ›
 AOT_have ‹[λz [F]z & ψ]x ≡ [F]x & ψ›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0]
 "&E"
 apply blast
 using 0 "&I" by blast
 finally AOT_show ‹[F]x ≡ [λz [F]z & ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_B: ‹\⊨_\◻ ♢ψ → ♢∀x([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)› for ψ
 proof (rule "RM♢"; rule "→I"; rule GEN)
 AOT_modally_strict {
 fix x
 AOT_assume 0: ‹ψ›
 AOT_have ‹[λz ([F]z & ψ) ∨ ¬ψ]x ≡"
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹... ≡ [F]x›
 apply (rule "≡I"; rule "→I")
 using "∨E"(3)[rotated, OF "useful-tautologies:2"[THEN "→E"], OF 0] "&E"
 apply blast
 apply (rule "∨I"(1)) using 0 "&I" by blast
 finally AOT_show ‹[F]x ≡ [λz ([F]z & ψ) ∨ ¬ψ]x›
 using "Commutativity of ≡"[THEN "≡E"(1)] by blast
 }
 qed

 AOT_have Aux_C:
 ‹\⊨_\◻ \A¬ψ → \A¬∀z([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)› for ψ
 proof(rule "act-cond"[THEN "→E"]; rule "RA[2]"; rule "→I"; rule "raa-cor:2")
 AOT_modally_strict {
 AOT_assume 0: ‹¬ψ›
 AOT_assume ‹∀z ([λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z)›
 AOT_hence ‹[λz [F]z & ψ]z ≡ [λz [F]z & ψ ∨ ¬ψ]z› for z
 using "∀E" by blast
 moreover AOT_have ‹[λz [F]z & ψ]z ≡ [F]z & ψ› for z
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 _have ‹ψ (([F]z & ψ ¬)›
  x z \<> 
 ultimately AOT_have ‹z =\close
 " ≡"() by mes
 moreover AOT_have ‹)\<close x y. if (x, y) ∈ r then Eq else Lt else Gt)"
 using 0 "∨
 E" "&E" by metis
 AOT_thus ‹
 }
 qed

  [consumes 1, case_names Nil Cons:
 ∀ [λ [Fz & ψ ¬]x ≡
 ¬∀x ([F]x ≡ [λz [F]z & ψ ∨ ¬ψ]x)))› for ψ
 proof (rule RN; rule "→I")
 
 AOT_assume ‹ 'c ==> (('a × 'b) list ==> 'd) list) ==>
 AOT_thus ‹Lt ==>f kx vx 0; aux = map2_val_pair f g xs ((k, vy) # ys) in if v = 0 then aux else (kx, v) # au
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 apply - assu "oalist_inv_raw xs"
 proof(rule "≡
 AOT_assume ‹∀z ([F]z ≡ [λ
 AOT_hence 1: ‹
 using "∀E" by blast
 AOT_assume ‹
 AOT_hence 2: ‹" and "olist_inv_ra ys and lookup_pair x = lookup_pair ys"
 using "∀E" by blast
 withCons4) have "k \<notin 
  k"by (im on:eq)
 AOT_thus ‹x ([λ]x ≡ ∨ψ

 next
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 AOT_hence 1: ‹
  "∀bst
 AOT_assume ‹∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψ ∨as k =ft d x"
 AOT_hence 2: ‹[λz & ψ ¬]z›
 forall>E" by blast
 AOT_have ‹
 using 1 2 "≡E" by meson
 AOT_thus ‹
 by (rule GEN)
 qed
 }
 qed
 AOT_hence ‹s"
 (¬∀x ([λz [F]z & ψ]x ≡ [λz [F]z & ψt s
 pairk u) )"fo ub (u on())
 using "nec-imp-act"[THEN "→E"] by blast
 AOT_hence ‹t_conv)
 \A
 ¬∀x ([F]x ≡ [λz [F]z & ψ
 using "act-cond"[THEN "→m spi: oderspit, simp add: eq)
 AOT_hence Aux_D: ‹k1 = k2›
 (\¬x ([λ>]x≡[Fz& 🚫
 \A¬∀x ([F]x ≡k1 ≠ k2›ti eq Gt_lt_conv Ltt_ov)
 by (auto intro!: "→I" "Act-Basic:5"[THEN "≡a_y_aiOass(1)])

 AOT_have ‹
 apply (rule "=_dir
 apply (fact "log-prop-prop:2")
 by (fact AOT)
java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
 using "logic-actualhave q:"pte_ynpikf = x @i =0ten] else[k 0])"
 {
 AOT_assume 1: ‹
 AOT_have 2: ‹♢∀
 using Aux_B[THEN "→E", OF q₀_prop[THEN "&E"(1)]].
 AOT_have ‹\A¬ca by simp
 using Aux_C[THEN "→E", OF q₀_prop_1].
java.lang.NullPointerException
 using Aux_D[THEN "→E", OF 1, THEN "≡E"(1)] by blast
java.lang.NullPointerException
 ♢
 using 2 "&I" by blast
 AOT_hence ‹A∀ [G]x) & ♢∀x ≡G]x))\)>
 by (rule "∃I"(1)) "cqt:2[lambda]"
 }
 moreover {
 AOT_assume 2: ‹
java.lang.NullPointerException
 using "logic-actual-nec:1"[axiom_inst, THEN "≡E"(2)] by blast
 AOT_hence ‹\A¬∀x ([F]x ≡ [λz [F]z & q₀]x) this(1) ave 1: "di (map fst (map_pair f xs))" by (rule Cons(1))
 Aux_A[THEN "→0_prop[THEN "&E"(1)]] "&I" by blast
 AOT_hence ‹∃G (\
 by (rule "∃
 
 ultimately AOT_show ‹) k = lo (map_val_pair f xs) (fst (?f (k, 0)))"
 using "∨E"(1)[OF "exc-mid"] "→I" by blast
 

  "oa-contingent:1": ‹
 rule"<>\")
 fix x
 AOT_assume 1: ‹O! = A!› have "fs ` set ( [])\subseteq]:' ×
 AOT_hence ‹
 by (rule "=_dfE"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
  next
 by (rule "=_dfE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
 moreover AOT_have ‹[λx ♢E!x]x ≡ ♢E!x›
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 ultimately AOT_have ‹[λx ¬♢E!x]x ≡ ♢E!x›
 using "rule=E" by fast
 moreover AOT_have ‹
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 ultimately AOT_have ‹
 using "≡E"(6) "Commutativity of ≡"[THEN "≡E"(1)] by blast
 AOT_thus "(♢
 using "oth-class-taut:3:c" "&I" by blast


  "oa-contingent:2": ‹
  -
 AOT_have ‹O!x ≡ [λx ♢assume "map2_vl_pai f g h xs (k', v') # ys) ≠
 apply (rule "≡)Cons(1(1) as(3, 4) have "... \subseteq` et ∪
 apply (rule "=_dfE"(2)[OF AOT_ordinary])
 apply "cqt:2[lambda]"
 apply argo
 subd<>(
 apply "cqt:2[lambda]"
 by argo
 also AOT_ (simp add: Let_def, intro conjI impI)
 by (rule "beta-C-meta"[THEN "→E"]) "cqt:2[lambda]"
 also AOT_have ‹… ≡ ¬¬♢E!x›
 using "oth-class-taut:3:b".
 also AOT_have ‹… ≡ ¬[λx ¬♢E!x]x›
 by (rule "beta-C-meta"[THEN "→E",
 THEN "oth-class-taut:4:b"[THEN "≡
 "cqt:2"
 also AOT_have ‹… ≡ ¬A!x›
 apply(rul "\<>I
 apply (rule "=_dfI"(2)[OF AOT_abstract])
 apply "cqt:2[lambda]"
 apply argo
 apply (rule "=_dfE"(2)[OF AOT_abstract])
 apply "cqt:2[lambda]"
 by argo
 finally show ?thesis.
 

  "oa-contingent:3": \<AOT_theorem 
 by (AOT_subst ‹A!x›
 (auto simp add: "oth-class-taut:3:b" "oa-contingent:2"[THEN
 "oth-class-taut:4:b"[THEN "≡E"(1)], symmetric])

 >)›
  (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:1", THEN "≡E"(2)];
 rule "&I")
 AOT_have ‹
 AOT_hence ‹ys)
 then AOT_obtain a where ‹♢E!a› using "∃E"[rotated] by blast
 AOT_hence ‹[λx ♢E!x]a›
 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2"
 AOT_hence ‹O!a›
 by (rule "=_dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2"
 AOT_hence ‹ ys k)" by (rule lookup_pair_map_val_pair fact Nil, fa as(7))
 AOT_thus ‹♢∃x O!x› using "T♢"[THEN "→E"] by blast
 
 AOT_obtain a where ‹k v xs)
 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
 AOT_hence ‹¬O!a› using "oa-contingent:3"[THEN "≡E"(1)] by blast
 AOT_hence ‹∃x ¬O!x› using "∃lookup_pai, ffa*(1, faas(7))
 AOT_thus ‹v' ys)
 

  "oa-contingent:5": ‹Contingent(A!)›
  (rule "thm-cont-prop:2"[unvarify F, OF "oa-exist:2", THEN "≡E"(2)];
 rule "&I")
 AOT_obtain a where ‹A!a›
 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
 AOT_hence ‹∃x A!x› using "\          
 AOT_thus ‹♢∃
 
  \open<diamond\
 AOT_hence ‹∃x ♢E!x›
 then AOT_obtain a where ‹
 AOT_hence ‹[λx ♢E!x]a›
 by (rule "beta-C-meta"[THEN "→E", THEN "≡E"(2), rotated]) "cqt:2[lambda]"
 AOT_hence ‹O!a›
 by (rule "=_dfI"(2)[OF AOT_ordinary, rotated]) "cqt:2[lambda]"
 AOT_hence ‹¬A!a› using "oa-contingent:2"[THEN "≡E"(1)] by blast
 AOT_hence ‹∃x ¬A!x› using "∃I" by fast
 AOT_thus ‹
 

java.lang.NullPointerException
  -
 AOT_have ‹O!x ≡ ¬A!x›
 using "oa-contingent:2" by blast
 also AOT_have ‹
 using"threlation-negation:1"[sy, unv F, OF "oa-exist:2"].
 finally AOT_have 1: ‹ ?the ..

 AOT_have ‹A!x ≡ ¬O!x›
 using "oa-contingent:3" by blast
 also AOT_have ‹… ≡ O!^have fs ` set x ∪
 using "thm-relatio p
 finally AOT_have 2: ‹A!x ≡ O!^-x›.

 AOT_show ‹O!^-x ≡ ¬A!^-x›
 using 1[THEN "oth-class-taut:4:b"[THEN "≡E"(1)]]
 "oa-contingent:3"[of _ x] 2[symmetric]
 "≡E"(5) by blast
 

  "oa-contingent:6": ‹O!^- ≠ A!^-›
  (rule "=-infix"[THEN "≡_#xs k0 (lookup_pair ((k', v') # ys) k0)"
 AOT_assume 1: ‹O!^- = A!^-›
 fix x
 AOT_have ‹A!^-x ≡ O!^-x›
 apply (rule "rule=E"[rotated, OF 1])
 by (fact "oth-class-taut:3:a")
 AOT_hence ‹A!^-x ≡ ¬A!^-x›
 using "oa-contingent:7" "≡E" by fast
 AOT_thus ‹(A!^-x ≡ ¬A!^-x) & ¬(A!^-x ≡ ¬A!^-x)›
 using "oth-class-taut:3:c" "&I" by blast
 

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:1", THEN "≡' k = Lt›
 OF "oa-contingent:4"].

  "oa-contingent:9": ‹Contingent(A!^-)›
 using "thm-cont-prop:3"[unvarify F, OF "oa-exist:2", THEN "≡E"(1),
 OF "oa-contingent:5"].

  WeaklyContingent :: ‹Π ==> φ› (‹WeaklyContingent'(_')›)
 "df-cont-nec":
 ‹WeaklyContingent([F]) ≡_df Contingent([F]) & ∀x (♢[F]x → ◻[F]x)›

  "cont-nec-fact1:1":
 ‹WeaklyContingent([F]) ≡ WeaklyContingent([F]^-)›
  -
 AOT_have ‹WeaklyContingent([F]) ≡ Contingent([F]) & ∀x (♢[F]x → ◻[F]x)›
 using "df-cont-nec"[THEN "≡
 (^u>-) \forall\diamondFx →
 apply (rule "oth-class-taut:8:f"[THEN "≡E"(2)]; rule "→I")
 simp
 also AOT_have ‹
 proof (rule "oth-class-taut:8:e"[THEN "≡E"(2)];
 rule "→ y k0) S Eq"" .
 fix x
  C k v xs)
 AOT_assume 1: ‹♢[F]^-x›
 AOT_have ‹♢¬[F]x›
 by next
 (auto simp add: "thm-relation-negation:1" 1)
 AOT_hence 2: ‹¬◻[F]x›f kv 0in if = ome E thelex_ f xs (k', v')#y) e aux)= So Eq"
 using "KBasic:11"[THEN "≡E"(2)] by blast
 " \<in 
 proof (rule "raa-cor:1")
 k
 AOT_have ‹¬◻¬[F]x›
 by (AOT_subst (reverse) ‹¬[F]x› ‹ :\<> 
 (auto simp add: "thm-relation-negation:1" 3)
 AOT_hence ‹
 
 AOT_hence ‹on Regio›
 AOT_thus ‹◻[F]x & ¬◻[F]x› using "&I" 2 by blast

 next
 fix x
 AOT_assume 0: ‹ lex_ord_pair_va:
 AOT_assume 1: ‹♢[F]x›
 AOT_have ‹♢¬[F]^-x› Ni
 by (AOT_subst ‹¬[F]^Cons ' \<>  step_r:
 (auto simp: "thm-relation-negation:2" 1)
 AOT_hence 2: ‹
 usingeq2: "f k' 00 v' = S E" by si
 proof(s add: Let_def eq, ru Cons(5)
 proof (rule "raa-cor:1")
 AOT_assume 3: ‹¬◻[F]x›
 AOT_have ‹¬◻¬[F]^assume f `set []\union`seys"
 by (AOT_subst ‹
 (auto simp add: "thm-relation-negation:2" 3)
 AOT_hence ‹
 using "conventions:5"[THEN "≡_dfI"] by simp
 AOT_hence ‹◻[F]^-x› using 0 "∀E" "→E" by fast
 AOT_thus ‹
 qed
 qed
 also AOT_have ‹… ≡ WeaklyContingent([F]^-)›
 using "df-cont-nec"[THEN "≡Df", symmetric] by blast
 finally show ?thesis.
 

  "cont-nec-fact1:2":
 ‹(WeaklyContingent([F]) & ¬WeaklyContingent([G])) → F ≠ G›
  (rule "→I"; rule "=-infix"[THEN "≡_dfI"]; rule "raa-cor:2")
 AOT_assume 1: ‹WeaklyContingent([F]) & ¬WeaklyContingent([G])›
 AOT_hence ‹WeaklyContingent([F])› using "&E" by blast
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 ultimately AOT_have ‹WeaklyContingent([G])›
 using "rule=E" by blast
 AOT_thus ‹WeaklyContingent([G]) & ¬WeaklyContingent([G])›
 using 1 "&I" "&E" by blast
 

  "cont-nec-fact2:1": ‹WeaklyContingent(O!)›
  (rule "df-cont-nec"[THEN "≡_dfI"]; rule "&I")
 AOT_show ‹Contingent(O!)›
 using "oa-contingent:4".
 
 AOT_show ‹∀x (♢[O!]x → ◻[O!]x)›
 apply (rule GEN; rule "→I")
 using "oa-facts:5"[THEN "≡E"(1)] by blast
 


  "cont-nec-fact2:2": ‹
  (rule "df-cont-nec"[THEN "≡_dfI"]; rule "&I")
 AOT_show ‹Contingent(A!)›
 using "oa-contingent:5".
 
 AOT_show ‹
 apply (rule GEN; rule "→I")
 using "oa-facts:6"[THEN "≡E"(1)] by blast
 

  "cont-nec-fact2:3": ‹¬WeaklyContingent(E!)›
  (rule "df-cont-nec"[THEN "≡Df",
 THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(2)];
 rule DeMorgan(1)[THEN "≡E"(2)]; rule "∨I"(2); rule "raa-cor:2")
 AOT_have ‹♢∃x (E!x & ¬\AE!x)› using "qml:4"[axiom_inst].
 AOT_hence ‹∃x ♢(E!x & ¬\AE!x)› using "BF♢"[THEN "→E"] by blast
 then AOT_obtain a where ‹♢(E!a & ¬\AE!a)› using "∃E"[rotated] by blast
 AOT_hence 1: ‹♢E!a & ♢¬\AE!a› using "KBasic2:3"[THEN "→E"] by simp
 moreover AOT_assume ‹∀x (♢[E!]x → ◻[E!]x)›
 ultimately AOT_have ‹◻E!a› using "&E" "∀E" "→E" by fast
 AOT_hence ‹\AE!a› using "nec-imp-act"[THEN "→E"] by blast
 AOT_hence ‹◻\AE!a› using "qml-act:1"[axiom_inst, THEN "→E"] by blast
 moreover AOT_have ‹¬◻\AE!a›
 E"(2)] 1[THEN "&E"(2)] by meson
 ultimately AOT_have ‹
 AOT_thus ‹"yblat
 

  "cont-nec-fact2:4": \<pen< s = p*v - e*q*u ∧e∣
 apply (rule "df-cont-nec"[THEN "≡
 THEN "oth-class-taut:4:b"[THEN "≡
 E"(2)];
  "<u\∣ = ∣p^2-N*q^2∣ ∧ ∣v∣ = ∣2*p*q∣" by (rule qfN_square_prime)
 apply (rule "contingent-properties:4"
 [THEN "≡Df",
 THEN "oth-class-taut:4:b"[THEN "\<equivv*2) by(sip d:ealna_urlfeld_ip)
 THEN "≡ R. prime R ∧ R dvd Q ∧ is_qfN R 3"
 apply (rule DeMorgan(1)[THEN "≡
 
 rule "useful-tautologies:2"[THEN "\<rightarrowE
 using "thm-noncont-e-e:3"[THEN "contingent-properties:3"[THEN "≡ x dvd b" by simp

  "cont-nec-fact2:5": \<open 
  -ssm "\ (c≠0)" hence "c=0 ∧
 AOT_have 1: ‹
 by (rule "=_g^2*0" by (simp only: zmult_zless_mono2)
 {
 fix φeconr)
 <>\¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹notφ{Π›
 proof (rule "raa- with have"b=3^2q- 3*^" im
 AOT_assume ‹
 open>φ{Π'}› using that(1) "≡
 AOT_thus ‹i∈#ps. i)^3
 qedaume"alp> = 0" with alphabeta have "3 dvd ?p" by auto
 AOT_have ‹ u v. a^2+3*b^2 = (u^2 + 3*v^2)*(c^2+3*d^2)
 using "pos-not-equiv-ne:4"[unvarify F have "corme c d"
 OF that(1,2), OF A[OF that(3, 4)]].
 
 show ?thesis
 apply(safe intro!: "&I"; rule 0)
 
 
 using "cont-nec-fact2:3" apply fast
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:1" apply fast
 using "rel-neg-T:3" apply fast
 using "oa-exist:1" apply blast
 using "cont-nec-fact1:1"[THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(1), rotated, OF "cont-nec-fact2:3"] apply fast
 -a2"THEN ""\<>]
 fact21 app bl
 apply (rule "=_dfI"(2)[OF L_def]; "cqt:2[lambda]")
 using "oa-exist:1" apply fast
 using "cont-nec-fact2:4" apply fast
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 using "cont-nec-fact2:1" apply fast
 using "rel-neg-T:3" apply fast
 using "oa-exist:1" apply fast
 apply (rule "cont-nec-fact1:1"[unvarify F,
 THEN "oth-class-taut:4:b"[THEN "≡: \open=k''\close
 THEN "≡
 apply (rule "=_dfI"(2)[OF L_def]; "cqt:2[lambda]")
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:1" by blast
 

  "cont-nec-fact2:6": ‹A! ≠ E! & A! ≠ E!^- & A! ≠ L & A! ≠ L^-› add: Con(7) <>k
  -
 AOT_have 1: ‹
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 {
 
 AOT_have A: ‹¬(φ{Π'} ≡ φ{Π})› if ‹φ{Π}› and ‹¬φ{Π'}›
 proof (rule "raa-cor:2")
 AOT_assume ‹φ{Π'} ≡ φ{Π}›
 AOT_hence ‹φ{Π'}› using that(1) "≡E" by blast
 AOT_thus ‹
 qed
 AOT_have ‹
 using "pos-not-equiv-ne:4"[unvarify F G, THEN "→E",
 OF that(1,2), OF A[OF that(3, 4)]].
 } note 0 = this
 show ?thesis
 apply(safe intro!: "&I"; rule 0)
 apply "cqt:2"
 using "oa-exist:2" apply blast
 using "cont-nec-fact2:3" apply fast
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:2" apply fast
 using "rel-neg-T:3" apply fast
 using "oa-exist:2" apply blast
 using "cont-nec-fact1:1"[THEN "oth-class-taut:4:b"[THEN "≡
 THEN "≡E"(1), rotated, OF "cont-nec-fact2:3"] apply fast
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:2" apply blast
 apply (rule "=_dfI"(2)[OF L_def]; "cqt:2[lambda]")
 using "oa-exist:2" apply fast
 using "cont-nec-fact2:4" apply fast
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:2" apply fast
 using "rel-neg-T:3" apply fast
 using "oa-exist:2" apply fast
 apply (rule "cont-nec-fact1:1"[unvarify F,
 THEN "oth-class-taut:4:b"[THEN "≡E"(1)],
 THEN "≡E"(1), rotated, OF "cont-nec-fact2:4"])
 apply (rule "=_dfI"(2)[OF L_def]; "cqt:2[lambda]")
 apply (rule "useful-tautologies:2"[THEN "→E"])
 using "cont-nec-fact2:2" by blast
 

  necessary_or_contingently_false :: ‹
 ‹) lookup_pair ys k)"

  sixteen:
 shows ‹∃F₁∃F₂∃F f xs ys = So Eq" (rulex_)
 «F₁::<\<kappa>>¬ ≠ F₂ & F f ` se xs ∪
java.lang.NullPointerException
 F₁ ≠ F₁₄ & F₁ ≠ F₁₅ & F₁ ≠ F₁₆ &
java.lang.NullPointerException
 F_hence*: "∧ \Longrightarrow
 F₂ ≠ F₁₅ & F₂ ≠ F₁₆ &
 F₃ ≠ F₄ & F₃ ≠ F₅ & F₃ ≠ F₆ & F (smp on: a(3))
java.lang.NullPointerException
 F₄ ≠ F₅ & F₄ ≠>@{constprod_}›
 F_:
 F₅ ≠ F₆ & F₅ ≠
java.lang.NullPointerException
 F₆ ≠ F₇ & F (4P kx vx xs ky vy ys)
 F₆ ≠ F₁₄ & F₆ ≠ F₁₅ & F₆ ≠ F₁₆ &
java.lang.NullPointerException
 F₇ ≠ F₁₅ & F₇ ≠ F₁₆ &
java.lang.NullPointerException
 
java.lang.NullPointerException
java.lang.NullPointerException
 F₁₁ ≠ F₁₂ & F₁₁ ≠ F₁₃ & F₁ prby (rule lex_or)
 F₁₂ ≠ F₁₃ & F₁₂ ≠ F₁₄ & F₁₂ ≠ F₁₅ & F₁₂ ≠ F₁₆ &
 F₁₃ ≠ F₁₄ & F₁₃ ≠ F₁₅ & F₁₃ ≠ F₁₆ &
 F₁₄ ≠ F₁₅ & F₁₄ ≠ F₁₆ &
 F₁₅ ≠ F₁₆)› "(pr P xs ys) ⟷ x ∪
  -
 AOT_have Delta_pos: ‹( A)⊆assumes "oalst_inv_raw ys"
 proof(rule "→I")
 AOT_assume ‹\Δφ›
 AOT_hence ‹
 using "≡_dfE"[OF necessary_or_contingently_false] by blast
 moreover {
 AOT_assume ‹◻: oali)
 AOT_hence ‹♢φ›
 by (metis "B♢" "T♢" "vdash-properties:10")
 }
 moreover {
java.lang.NullPointerException
 AOT_hence ‹♢φ›
 using "&E" by blast
 }
 ultimately AOT_show ‹♢φ›
 by (metis "∨E"(2) "raa-cor:1")
 qed

 AOT_have act_and_not_nec_not_delta: ‹¬‹
 using "≡_d have "set (update_by_pa (k, v) (sort_oalist xs)) = insert (k v) set (sort xs) - range (Pair k k))"
 raa:3 that(1,2) by bla
 AOT_have act_and_pos_not_not_delta: ‹¬\Δφ› if ‹\Aφ
 using "KBasic:11" act_and_not_nec_not_delta "≡E"(2) that(1,2) by blast
 AOT_have impossible_delta: ‹¬\Δφ› if ‹¬♢φ›
 using Delta_pos "modus-tollens:1" that by blast
 AOT_have not_act_and_pos_delta: ‹\Δφ› if ‹¬\Aφ› and ‹♢φ› : "s (x # x) update_by_p (, v) ((ortxs)" by (s ad: xsort_oalist_def)
 by (meson "≡_dfI" "&I" "∨= k' th v else lookup_ xs k')"
 AOT_have nec_delta: ‹\Δφ› sh "l (sort_o (x # xs)) ' = lookup_ (x # xs) k'" .
 using "≡_dfI" "∨I"(1) necessary_or_contingently_false that by blast

 AOT_obtain a where a_prop: ‹A!a›
 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
 AOT_obtain b where b_prop: ‹♢[E!]b & ¬\A[E!]b›
 

 
 proof(rule "=_dfI"(2)[OF AOT_ordinary])
 AOT_show ‹
 next
 AOT_show ‹
 proof (rule "β←C"(1); ("cqt:2[lambda]")?)
 AOT_show ‹
 AOT_show ‹
 qed
 qed

  AOT_have nec_not_L_neg: \<open>\<box>\<not>[L\<^sup>-]x\<close> for x
    using "thm-noncont-e-e:2" "contingent-properties:2"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&an>E"
          CBF[THEN "\<rightarrow>E"] "\<forall>E" by blast
  AOT_have nec_L: \<open>\<box>[L]x\<close> for x
    using "thm-noncont-e-efun tl_raw : "('a, ', ') oalist_raw <> (a, ':,')oalist_raw"
      CBF[THEN "\<rightarrow>E"] "\<forall>E" by blast

  AOT_have act_ord_b: \<open>\<^bold>\<A>[O!]b\<close>
    using b_ord "\<equiv>E"(1) "oa-facts:7" by blast
  AOT_have delta_ord_b: \<open>\<^bold>\<Delta>[O!]b\<close>
    by (meson "\<equiv>\<^sub>d\<^sub>fI" b_ord "\<or>I"(1) necessary_or_contingently_false
              "oa-facts:1" "\<rightarrow>E")
  AOT_have not_act_ord_a: \<open>\<not>\<^bold>\<A>[O!]a\<close>
    by (meson a_prop "\<equiv>E"(1) "\<equiv>E"(3) "oa-contingent:3" "oa-facts:7")
  AOT_have not_delta_ord_a: \<open>\<not>\<^bold>\<Delta>[O!]a\<close>
    by (metis Delta_pos "\<equiv>E"(4) not_act_ord_a "oa-facts:3" "oa-facts:7"
              "reductio-aa:1" "\<rightarrow>E")

  AOT_have not_act_abs_b: \<open>\<not>\<^bold>\<A>[A!]b\<close>
    by (meson b_ord "\<equiv>E"(1) "\<equiv>E"(3) "oa-contingent:2" "oa-facts:8")
AOT_have:\open\not\^><>[A]b<>
  proof(rule "raa-cor:2")
    AOT_assume \<open>\<^bold>\<Delta>[A!]b\<close>
    AOT_hence \<open>\<diamond>[A!]b\<close>
      by (metis Delta_pos "vdash-properties:10")
    AOT_thus \<open>[A!]b & \<not>[A!]b\<close>
      by (metis b_ord "&I" "\<equiv>E"(1) "oa-contingent:2"
                "oa-facts:4" "\<rightarrow>E")
  qed
  AOT_have act_abs_a: \<open>\<^bold>\<A>[A!]a\<close>
    using a_prop "\<equiv>E"(1) "oa-facts:8" by blast
  AOT_have delta_abs_a: \<open>\<^bold>\<Delta>[A!]a\<close>
    by (metis "\<equiv>\<^sub>d\<^sub>fI" a_prop "oa-facts:2" "\<rightarrow>E" "\<or>I"(1)
              necessary_or_contingently_false)

  AOT_have not_act_concrete_b: \<open>\<not>\<^bold>\<A>[E!]b\<close>
    using b_prop "&E"(2) by blast

  proof (rule "\<equiv>\<^sub>d\<^sub>fI"[OF necessary_or_contingently_false];
rule\or>"2  &I)
    AOT_show \<open>\<not>\<^bold>\<A>[E!]b\<close> using b_prop "&E"(2) by blast
  next
    AOT_show \<open>\<diamond>[E!]b\<close
  qed
  AOT_have not_act_concrete_a: \<open>\<not>\<^bold>\<A>[E!]a\<close>
  proof (rule "raa-cor:2")
    AOT_assume \<open>\<^bold>\<A>[E!]a\<close>
    AOT_hence 1: \<open>\<diamond>[E!]a\<close> by     ubsection \<open>@{const tl_raw}\<close>
    AOT_have \<open>[A!]a\<close> by (simp add: a_prop)
    AOT_hence \<open>[\<lambda>x \<not>\<diamond>[E!]x]a\<close>
      by (rule "=\<^sub>d\<^sub>fE"(2)[OF AOT_abstract, rotated]) "cqt:2"
    AOT_hence \<open>\<not>\<diamond>[E!]a\<close> using "\<beta>\<rightarrow>C"(1) by blast
    AOT_thus \<open>\<diamond>[E!]a & \<not>\<diamond>[E!]a\<close> using 1 "proof-
  qed
  AOT_have not_delta_concrete_a: \<open>\<not>\<^bold>\<Delta>[E!]a\<close>
  proof (rule "raa-cor:2")
    AOT_assume \<open>\<^bold>\<Delta>[E!]a\<close>
    AOT_hence 1: \<open>\<diamond>[E!]a\<close> by (metis Delta_pos "vdash-properties:10")
    AOT_have \<open>[A!]a\<close> by (simp add: a_prop)
    AOT_hence \<open>[\<lambda>x \<not>\<diamond>[E!]x]a\<close>
      by (rule "=\<^sub>d\<^sub>fE"(2)[OF AOT_abstract, rotated]) "cqt:2[lambda]"
    AOT_hence \<open>\<not>\<diamond>[E!]a\<close> using "\<beta>\<rightarrow>C"(1) by blast
    AOT_thus \<open>\<diamond>[E!]a & \<not>\<diamond>[E!]a\<close> using 1 "&I" by blast
  

  AOT_have not_act_q_zero: \<open>\<not>\<^bold>\<A>q\<^sub>0\<close>
    by (meson "log-prop-prop:2" "pos-not-pna:1"
              q\<^sub>0_def "reductio-aa:1" "rule-id-df:2:a[zero]")
  AOT_have delta_q_zero: \<open>\<^bold>\<Delta>q\<^sub>0\<close>
  proof(rule "\<equiv>\<^sub>d\<^sub>fI"[OF necessary_or_contingently_false];
        rule "\<or>I"(2); rule "&I")
    AOT_show \<open>\<not>\<^bold>\<A>q\<^sub>0\<close> using not_act_q_zero.
   ?thesisunfoldingxs
  qed
  AOT_have act_not_q_zero: \<open>\<^bold>\<A>\<not>q\<^sub>0\<close>
    using "Act-Basic:1" "\<or>E"(2) not_act_q_zero by blast
  AOT_have not_delta_not_q_zero: \<open>\<not>\<^bold>\<Delta>\<not>q\<^sub>0\<close>
    using "\<equiv>\<^sub>d\<^sub>fE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta
          "&E"(1) "\<or>E"(2) not_act_q_zero q\<^sub>0_prop by blast

  AOT_have \<open>[L\<^sup>-]\<down>\<close> by (simp add: "rel-neg-T:3")
moreover\open><ot><bold\A[\^>-b&\not>\<bold\Delta[\<>-b&\<>\<bold>\<A>\^>-a&\not\^old\>[^>-a<>
  proof (safe intro!: "&I")
    AOT_show \<open>\<not>\<^bold>\<A>[L\<^sup>-]b\<close>
      by (meson "\<equiv>E"(1) "logic-actual-nec:1"[axiom_inst] "nec-imp-act"
nec_not_L_neg<>E)
    AOT_show \<open>\<not>\<^bold>\<Delta>[L\<^sup>-]b\<close>
      by (meson Delta_pos "KBasic2:1" "\<equiv>E"(1)
                "modus-tollens:1" nec_not_L_neg)
    AOT_show \<open>\<not>\<^bold>\<A>[L\<^sup>-]a\<close>
      by (meson "\<equiv>E"(1) "logic-actual-nec:1"[axiom_inst]
                "nec-imp-act" nec_not_L_neg "\<rightarrow>E")
AOT_show><ot><bold\<>[L\^>-\close
      using Delta_pos "KBasic2:1" "\<equiv>E"(1) "modus-tollens:1"
            nec_not_L_neg by blast
  qed
  ultimately AOT_obtain F\<^sub>0
    using "\<exists>I"(1)[rotated, THEN "\<exists>E"[rotated]] by fastforce
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>0]b\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>0]b\<close> and \<open>\<not>\<^bold>\<A>[F\<^sub>0]a\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>0]a\<close>
    using "&E" by blast+
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  let ?\<Pi> = "\<guillemotleft>[\<lambda>y [A!]y & q\<^sub>0]\<guillemotright>"
  AOT_modally_strict {
      assumes "oalist_inv xs"
  } note 1 = this
  moreover AOT_have \<open>\<not>\<^bold>\<A>[\<guillemotleft>?\<Pi>\<guillemotright
  proof (safe intro!: "&I"; AOT_subst \<open>[\<lambda>y A!y & q\<^sub>0]x\<close> \<open>A!x & q\<^sub>0\<close> for: x)
    AOT_show \<open>\<not>\<^bold>\<A>([A!]b & q\<^sub>0)\<close>
      using "Act-Basic:2" "&E"(1) "\<equiv>E"(1) not_act_abs_b "raa-cor:3" by blast
  next AOT_show \<open>\<not>\<^bold>\<Delta>([A!]b & q\<^sub>0)\<close>
      by (metis Delta_pos "KBasic2:3" "&E"(1) "\<equiv>E"(4) not_act_abs_b
                "oa-facts:4" "oa-facts:8" "raa-cor:3" "\<rightarrow>E")
  next AOT_show \<open>\<not>\<^bold>\<A>([A!]a & q\<^sub>0)\<close>
      using "Act-Basic:2" "&E"(2) "\<equiv>E"(1) not_act_q_zero
            "raa-cor:3" by blast
  next AOT_show \<open>\<^bold>\<Delta>([A!]a & q\<^sub>0)\<close>
    proof (rule not_act_and_pos_delta)
      AOT_show \<open>\<not>\<^bold>\<A>([A!]a proof -
        using "Act-Basic:2" "&E"(2) "\<equiv>E"(4) not_act_q_zero
              "raa-cor:3" by blast
    next AOT_show \<open>\<diamond>([A!]a & q\<^sub>0)\<close>
        by (metis "&I" "\<rightarrow>E" Delta_pos "KBasic:16" "&E"(1) delta_abs_a
                  "\<equiv>E"(1) "oa-facts:6" q\<^sub>0_prop)
    qed
  qed(auto simp: "beta-C-meta"[THEN "\<rightarrow>E", OF 1])
  ultimately AOT_obtain F\<^sub>1 where \<open>\<not>\<^bold>\<A>[F\<^sub>1]b & \<not>\<^bold>\<Delta>[F\<^sub>1]b & \<not>\<^bold>\<A>[F\<^sub>1]a & \<^bold>\<Delta>[F\<^sub>1]a\<close>
    using "\<existsw fxs = map_val_raw gxs"
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>1]b\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>1]b\<close> and \<open>\<not>\<^bold>\<A>[F\<^sub>1]a\<close> and \<open>\<^bold>\<Delta>[F\<^sub>1]a\<close>
    using "&E" by blast+
  note props = props this

  let ?\<Pi> = "\<guillemotleft>[\<lambda>y [A!]y & \<not>q\<^sub>0]\<guillemotright>"
  AOT_modally_strict {
    AOT_have \<open>[\<guillemotleft>?\<Pi>\<guillemotright>]\<down>\<close> by "cqt:2[lambda]"
  } note 1 = this
  moreover AOT_have \<open>\<not>\<^bold>\<A>[\<guillemotleft>?\<Pi>\<guillemotright>]b & \<not>\<^bold>\<Delta>[\<guillemotleft  unfolding map2_val_compat
  proof (safe intro!: "&I"; AOT_subst \<open>[\<lambda>y A!y & \<not>q\<^sub>0]x\<close> \<open>A!x & \<not>q\<^sub>0\<close> for: x)
    AOT_show \<open>\<not>\<^  assumes "map2_val_compat' f" and"oalist_inv zs"
      using "Act-Basic:2" "&E"(1) "\<equiv>E"(1) not_act_abs_b "raa-cor:3" by blast
  next AOT_show \<open>\<not>\<^bold>\<Delta>([A!]b & \<not>q
      by (meson "RM\<diamond>" Delta_pos "Conjunction Simplification"(1) "\<equiv>E"(4)
                "modus-tollens:1" not_act_abs_b "oa-facts:4" "oa-facts:8")
  next AOT_show \<open>\<^bold>\<A>([A!]a & \<not>q\<^sub>0)\<close>
      by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "\<or>E"(2)
                "\<equiv>E"(3) not_act_q_zero "raa-cor:3")
  next AOT_show \<open>\<not>\<^bold>\<Delta>([A!]a & \<not>q\<^sub>0)\<close>
    proof (rule act_and_not_nec_not_delta)
      AOT_show \<open>\<^bold>\<A>([A!]a & \<not>q\<^sub>0)\<close>
        by (metis "Act-Basic:1" "Act-Basic:2" act_abs_a "&I" "\<or>E"(2)
                  "\<equiv>E"(3) not_act_q_zero "raa-cor:3")
    next
      AOT_show \<open>\<not>\<box>([A!]a & \<not>q\<^sub>0)\<close>
        by (metis "KBasic2:1" "KBasic:3" "&E"(1) "&E"(2) "\<equiv>E"(4)
                  q\<^sub>0_prop "raa-cor:3")
    qed
       (rule map2_val_compat'imp_map2_val_compat
  ultimately AOT_obtain F\<^sub>2 where \<open>\<not>\<^bold>\<A>[F\<^sub>2]b & \<not>\<^bold>  thus ?hesis by (simp add:  oalist_inv_alt)
    using "\<exists>I"(1)[rotated, THEN "\<exists>E"[rotated]] by fastforce
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>2]b\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>2]b\<close> and \<open>\<^bold>\<A>[F\<^sub>2]a\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>2]a\<close>
    using "&E" by blast+
  note props = props this

  AOT_have abstract_prop: \<open>\<not>\<^bold>\<A>[A!]b & \<not>\<^bold>\<Delta>[A!]b & \<^bold>\<A>[A!]a & \<^bold>\<Delta>[A!]a\<close>
    using act_abs_a "&I" delta_abs_a not_act_abs_b not_delta_abs_b
    by presburger
  then AOT_obtain F\<^sub>3 where \<open>\<not>\<^bold>\<A>[F\<^sub>3]b & \<not>\<^bold>\<Delta>[F\<^sub>3]b & \<^bold>\<A>[F\<^sub>3]a & \<^bold>\<Delta>[F\<^sub>3]a\<close>
using\exists")rotated  \<>E"[rotated]] "-exist2  java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>3]b\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>3]b\<close> and \<open>\<^bold>\<A>[F\<^sub>3]a\<close> and \<open>\<^bold>\<Delta>[F\<^sub>3]a\<close>
    using "&E" by blast+
 

  AOT_have \<open>\<not>\<^bold>\<A>[E!]b & \<^bold>\<Delta>[E!]b & \<not>\<^bold>\<A>[E!]a &n> \<not>\<^bold>\<Delta>[E!]a\<close>
    by (meson "&I" delta_concrete_b not_act_concrete_a
              not_act_concrete_b not_delta_concrete_a)
  then AOT_obtain F\<^sub>4 where \<open>\<not>\<^bold>\<A>[F\<^sub>4]b & \<^bold>\<Delta>[F\<^sub>4]b & \<not>\<^bold>\<A>[F\<^sub>4]a & \<not>\<^bold>\<Delta>[F\<^sub>4]a\<close>
 I1rotated  \existsE[]]
    by fastforce
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>4]b\<close> and \<open>\<^bold>\<Delta>[F       ?thesis unfolding equsinginv_raw assms2)
    using "&E" by blast+
  note props = props this

  AOT_modally_strict {
    AOT_have \<open>[\<lambda>y q\<^sub>0]\<down>\<close> by "cqt:2[lambda]"
  } note 1 = this
  moreover AOT_have \<open>\<not>\<^bold>\<A>[\<lambda>y q\<^sub>0]b & \<^bold>\<Delta>[\<lambda>y q\<^sub>0]b & \<not>\<^bold>\<A>[\<lambda>y q\<^sub>0]a & \<^bold>\<Delta>[\<lambda>y q\<^sub>0]a\<close>
    by (    "\And>k. '\in    fst)<>fst  fstys <>ltko 'k\<ongrightarrow
       (auto simp: not_act_q_zero delta_q_zero  "beta-C-meta"[THEN "\<rightarrow>E", OF 1])
  ultimately AOT_obtain F\<^sub>5 where \<open>\<not>\<^bold>\<A>[F\<^sub>5]b & \<^bold>\<Delta>[F\<^sub>5]b & \<not>\<^bold>\<A>[F\<^sub>5]a & \<^bold>\<Delta>[F\<^sub>5]a\<close>
    using "\<exists>I"(1)[rotated, THEN "\<exists>E"[rotated]]
   assume"t ko 'k
  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>5]b\<close> and \<open>\<^bold>\<Delta>[F\<^sub>5]b\<close> and \<open>\<not>\<^bold>\<A>[F\<^sub>5]a\<    by (imp :)
    using "&E" by blast+
   props = props this

  let ?\<Pi> = "\<guillemotleft>[\<lambda>y [E!]y \<or> ([A!]y & \<not>q\<^sub>0)]\<guillemotright>"
  AOT_modally_strict {
AOT_haveopen[<>?<Pi>\<guillemotright>]\<>\<lose by "cqt:[lambda]"
  } note 1 = this
  <>\not\^>\A[<>?<Pi\<>b& <bold\Delta\guillemotleft<><>]  <bold<[guillemotleft>?<><guillemotrighta&span>not>\^bold>\Delta[<>?<>\guillemotrightaclose>
  proof(safe intro!: "&I";
        AOT_subst \<open>[\<lambda>y E!y \<or> (A!y & \<not>q\<^sub>0)]x\<close> \<open>E!x \<or> (A!x & \<not>q\<^sub>0)\<close> for: x)
    AOT_have \<open>\<^bold>\<A>\<not>([A!]b & \<not>q\<^sub>0)\<close>
      by (metis "Act-Basic:1" "Act-Basic:2" abstract_prop "&E"(1) "\<or>E"(2)
                "\<equiv>E"(1) "raa-cor:3")
    moreover AOT_have \<open>\<not>\<^bold>\<A>[E!]b\<close>
      using b_prop "&E"(2) by blast
    ultimately AOT_have 2: \<open>\<^bold>\<A>(\<not>[E!]b & \<not>([A!]b & \<not>q\<^sub>0))\<close>
      by (metis "Act-Basic:2" "Act-Sub:1" "&I" "\<equiv>E"(3) "raa-cor:1")
    AOT_have \<open>\<^bold>\<A>\<not>([E!]b \<or> ([A!]b & \<not>q\<^sub>0))\<close>
      by (AOT_subst \<open>\<not>([E!]b \<or> ([A!]b & \<not>q\<^sub>0))\<close> \<open>\<not>[E!]b &> \<not>([A!]b & \<not>q\<^sub>0)\<close>)
         (auto simp: "oth-class-taut:5:d" 2)
    AOT_thus \<open>\<not>\<^bold>\<A>([E!]b \<or> ([A!]b & \<not>q\<^sub>0))\<close>
      by (metis "\<not>\<not>I" "Act-Sub:1" "\<equiv>E"(4))
  next
    AOT_show \<open>\<^bold>\<Delta>([E!]b \<or> ([A!]b & \<not>q\<^sub>0))\<close>
    proof (rule not_act_and_pos_delta)
      AOT_show \<open>\<not>\<^bold>\<A>([E!]b \<or> ([A!]b & \<not>q\<^sub>0))\<close>
        by (metis "Act-Basic:2" "Act-Basic:9" "\<or>E"(2) "raa-cor:3"
                  "Conjunction Simplification"(1) "\<equiv>E"(4)
                  "modus-tollens:1" not_act_abs_b not_act_concrete_b)
    next
      AOT_show \<open>\<diamond>([    thus "P  (ookup_pairoxxs')(lookup_pair  (  ys k"
        using "KBasic2:2" b_prop "&E"(1) "\<or>I"(1) "\<equiv>E"(3) "raa-cor:3" by blast
    qed
  next AOT_show \<open>\<^bold>\<A>([E!]a \proof-
   from assms(3) have"prod_ord_pairoxP xs' (ort_oalist_aux ox ys)  simp :xsjava.lang.StringIndexOutOfBoundsException: Index 96 out of bounds for length 96
                "\<  thus ?thesisby( add xslookup_pair_sort_oalist_aux (2))
  next AOT_show \<open>\  rollary prod_ord_raw_alt:
    proof (rule act_and_not_nec_not_delta)
      AOT_show \<open>\<^bold>\<A>([E!]a \<or> ([A!]a & \<not>q\<^sub>0))\<close>
  assumes"oalist_inv xs" and"oalist_invys"
                  "\<or>I"(2) "\<or>E"(2) "\<equiv>E"(3) not_act_q_zero "raa-cor:1")
    
      AOT_have \<open>\<box>\<not>[E!]a\<close>
        by (metis "\<equiv>\<^sub>d\<^sub>fI" "conventions:5" "&I" "\<or>I"(2)
necessary_or_contingently_false
                  not_act_concrete_a not_delta_concrete_a "raa-cor:3")
      moreover AOT_have \<open>\<diamond>\<not>([A!]a & \<not>q\<^sub>0)\<close>
        by (metis "KBasic2:1" "KBasic:11" "KBasic:3"
                  "&E"(1,2) "\<equiv>E"(1) q\<^sub>0_prop "raa-cor:3")
      ultimately AOT_have\<>\<diamond>(\<not>[E!]a & \<not>([A!]a & \<not>q\<^sub>0))\close>
        by (metis "KBasic:16" "&I" "vdash-properties:10")
      AOT_hence \<open>\<diamond>\<not>([E!]a \<or> ([A!]a & \<not>q\<^sub>0))\<close>
        by (metis "RE\<diamond>" "\<equiv>E"(2) "oth-class-taut:5:d")
      AOT_thus \<open>\<not>\<box>([E!]a \<or> ([A!]a & \<not>q\<^sub>0))\<close>
        by (metis "KBasic:12" "\<equiv>E"(1) "raa-cor:3")
    qed
  qed(auto simp: "beta-C-meta"[THEN "\<rightarrow>E", OF 1])
   AOT_obtain F\<^sub>6 \<open>\<not\<^bold>\<A>[F\<^sub>6]b & \<^bold\<>[<>6]b & \<^bold>\<A>[F\<^sub>6]a & \<ot><bold>\<><>6]\close

  AOT_hence \<open>\<not>\<^bold>\<A>[F\<^sub>6]b\<close> and \<open>\<^bold>\<Delta>[F\<^sub>6]b\<close> and \<open>\<^   assmshave distinct(  )  (simpadd: )
    using "&E" by blast+
  note props = props this

  aoalist_inv_list_of_oalist: "oalist_inv (list_of_oalist x)"
 AOT_modally_strict {
 AOT_have ‹[«as "oalist_inv xs"
 } note 1 = this
 moreover AOT_have ‹¬\A
 proof(safe intro!: "&I"; AOT_subst ‹[λy A!y ∨ E!y]x›
 AOT_show ‹¬\A
 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b
 not_act_concrete_b "raa-cor:3" by blast
 next AOT_show ‹\Δ([A!]b ∨ [E!]b)›
 proof (rule not_act_and_pos_delta)
 AOT_show ‹ 'x"
 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_abs_b
 not_act_concrete_b "raa-cor:3" by blast
 >
 using "KBasic2:2" b_prop "&E"(1) "∨I"(2) "≡E"(2) by blast
 qed
 next AOT_show ‹\A([A!]a ∨ [E!]a)›
 by (meson "Act-Basic:9" act_abs_a "∨I"(1) "≡E"(2))
  AO‹Δ([A!]a ∨[E!]a) ›
 proof (rule nec_delta)
  ‹
 by (metis "KBasic:15" act_abs_a act_and_not_nec_not_delta
 "Disjunction Addition"(1) delta_abs_a "raa-cor:3" "→E")
 qed
 qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₇ where
 using "∃I"(1)[rotated, THEN roof - 
 AOT_hence ‹
 using "&E" by blast+
 note props = props this

 
 AOT_modally_strict by (sim add: list_ lookup_def look)
 AOT_have ‹
 } note 1 = this
 moreover AOT_have ‹\A
 proof(safe intro!: "&I"; AOT_subst ‹
 \<>[
 by (metis "Act-Basic:1" "Act-Basic:2" act_ord_b "&I" "∨E"(2)
 "≡E"(3) not_act_concrete_b "raa-cor:3")
 next AOT_show ‹¬\Δ([O!]b & ¬[E!]b)›
 by (metis (no_types, opaque_lifting) "conventions:5" "Act-Sub:1" "RM:1"
 act_and_not_nec_not_delta "act-conj-act:3"
 act_ord_b b_prop "&I" "&E"(1) "Conjunction Simplification"(2)
 "df-rules-formulas[3]"
 "≡E"(3) "raa-cor:1" "→E")
java.lang.NullPointerException
 using "Act-Basic:2" "&E"(1) "≡E"(1) not_act_ord_a "raa-cor:3" by blast
 next AOT_have ‹¬♢([O!]a & ¬[E!]a)›
 by (metis "KBasic2:3" "&E"(_id, rule oaoalist_inv_tl_raw, facfact oalist)
 "oa-facts:7" "raa-cor:3" "vdash-properties:10")
 AOT_thus ‹¬\Δ([O!]a & ¬[E!]a)›
 by (rule impossible_delta)
 qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₈ where ‹\A[F₈]b & ¬\Δ[F₈]b & an style='font-size: 18px;'>¬\A[F₈]a & ¬\Δ[F₈]a›
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
 AOT_hence ‹\A[F₈]b› and ‹¬\Δ[F₈]b› and ‹¬\A[F₈]a› and ‹¬\Δ[F₈]a›
 using "&E" by blast+
 note props = props this

 let ?Π = "«[λy ¬[E!]y & ([O!]y ∨ q₀)]¬"
 AOT_modally_strict {
 AOT_have ‹[«?Π¬]↓› by "cqt:2[lambda]"
 } note 1 = this
 moreover AOT_have ‹\A[«?Π¬]b & ¬\Δ[«?Π¬]b & ¬\A[«?Π¬]a & \Δ[«?Π¬]a›
 proof(safe intro!: "&I";
 AOT_subst ‹[λy ¬E!y & (O!y ∨ q₀)]x› ‹¬E!x & (O!x ∨ q₀)› for: x)
java.lang.NullPointerException
 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I"
 "∨I"(1) "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 proof (rule act_and_pos_not_not_delta)
 AOT_show ‹\A(¬[E!]b & ([O!]b ∨ q₀))›
 by (metis "Act-Basic:1" "Act-Basic:2" "Act-Basic:9" act_ord_b "&I"
 "∨I"(1) "∨E"(2) "≡E"(3) not_act_concrete_b "raa-cor:1")
 next
 AOT_show ‹♢¬(¬[E!]b & ([O!]b ∨ q₀))›
 proof (AOT_subst ‹¬(¬[E!]b & ([O!]b ∨ q₀))› ‹[E!]b ∨ ¬([O!]b ∨ q₀)›)
 AOT_modally_strict {
 AOT_show ‹¬(¬[E!]b & ([O!]b ∨ q₀)) ≡ [E!]b ∨ ¬([O!]b ∨ q₀)›
 by (metis "&I" "&E"(1,2) "∨I"(1,2) "∨E"(2)
 "→I" "≡I" "reductio-aa:1")
 }
 next
 AOT_show ‹♢([E!]b ∨ ¬([O!]b ∨ q₀))›
 using "KBasic2:2" b_prop "&E"(1) "∨I"(1) "≡E"(3)
 "raa-cor:3" by blast
 qed
 qed
 next
 AOT_show ‹¬\A(¬[E!]a & ([O!]a ∨ q₀))›
 using "Act-Basic:2" "Act-Basic:9" "&E"(2) "∨E"(3) "≡E"(1)
 not_act_ord_a not_act_q_zero "reductio-aa:2" by blast
   next
     AOT_show \<open>\<^bold>\<Delta>(\<not>[E!]a & ([O!]a \<or> q\<^sub>0))\<close>
     proof (rule not_act_and_pos_delta)
       AOT_show \<open>\<not>\<^bold>\<A>(\<not>[E!]a & ([O!]a \<or> q\<^sub>0))\<close>
         by (metis "Act-Basic:2" "Act-Basic:9" "&E"(2) "\<or>E"(3) "\<equiv>E"(1)
                   not_act_ord_a not_act_q_zero "reductio-aa:2")
     next
       AOT_have \<open>\<box>\<not>[E!]a\<close>
         using "KBasic2:1" "\<equiv>E"(2) not_act_and_pos_delta not_act_concrete_a
               not_delta_concrete_a "raa-cor:5" by blast
       moreover AOT_have \<open>\<diamond>([O!]a \<or> q\<^sub>0)\<close>
         by (metis "KBasic2:2" "&E"(1) "\<or>I"(2) "\<equiv>E"(3) q\<^sub>0_prop "raa-cor:3")
       ultimately AOT_show \<open>\<diamond>(\<not>[E!]a & ([O!]a \<or> q\<^sub>0))\<close>
         by (metis "KBasic:16" "&I" "vdash-properties:10")
     qed
   qed(auto simp:  "beta-C-meta"[THEN "\<rightarrow>E", OF 1])
  ultimately AOT_obtain F\<^sub>9 where \<open>\<^bold>\<A>[F\<^sub>9]b & \<not>\<^bold>\<Delta>[F\<^sub>9]b & \<not>\<^bold>\<A>[F\<^sub>9]a & \<^bold>\<Delta>[F\<^sub>9]a\<close>
    using "\<exists>I"(1)[rotated, THEN "\<exists>E"[rotated]] by fastforce
  AOT_hence \<open>\<^bold>\<A>[F\<^sub>9]b\<close> and \<open>\<not>\<^bold>\<Delta>[F\<^sub>9]b\<close> and \<open>\<not>\<^bold>\<A>[F\<^sub>9]a\<close> and \<open>\<^bold>\<Delta>[F\<^sub>9]a\<close>
    using "&E" by blast+
  note props = props this

  AOT_modally_strict {
    AOT_have \<open>[\<lambda>y \<not>q\<^sub>0]\<down>\<close> by "cqt:2[lambda]"
  } note 1 = this
  moreover AOT_have \<open>\<^bold>\<A>[\<lambda>y \<not>q\<^sub>0]b & \<not>\<^bold>\<Delta>[\<lambda>y \<not>q\<^sub>0]b & \<^bold>\<A>[\<lambda>y \<not>q\<^sub>0]a & \<not>\<^bold>\<Delta>[\<lambda>y \<not>q\<^sub>0]a\<close>
 by (safe intro!: "&I"; OT_sub \>y \<notq
 (auto simp: act_not_q_zero not_delta_not_q_zero
 "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₁₀ where ‹\<^ oalist_eq_alt
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
 AOT_hence ‹
 using "&E" by blast+
 note props = props this

 AOT_modally_strict {
 AOT_have ‹[λy ¬[E!]y]↓› by "cqt:2[lambda]"
 } note 1 = this
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 \A[λy ¬[E!]y]a & \Δ[λy ¬[E!]y]a›
 proof (safe intro!: "&I"; AOT_subst ‹[λy ¬[E!]y]x› ‹¬[E!]x› for: x)
 AOT_show ‹\A¬[E!]b›
 using "Act-Basic:1" "∨E"(2) not_act_concrete_b by blast
 next AOT_show ‹¬\Δ¬[E!]b›
 using "≡_dfE" "conventions:5" "Act-Basic:1" act_and_not_nec_not_delta
 
 next AOT_show ‹\A¬[E!]a›
 using "Act-Basic:1" "∨E"(2) not_act_concr f k 0 c)) (list_of_olistxxs) (list_of_oalisty ys)"
 next AOT_show ‹\Δ¬[E!]a›
 using "KBasic2:1" "≡E"(2) nec_delta not_act_and_pos_delta
 not_act_concrete_a not_delta_concrete_a "reductio-aa:1"
 by blast
 qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₁₁ where ‹\A[F₁₁]b & ¬\Δ[F₁₁]b & \A[F₁₁]a & \Δ[F₁₁]a›
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
java.lang.NullPointerException
 using "&E" by blast+
 note props = props this

 AOT_have ‹\A[O!]b & \Δ[O!]b & ¬\A[O!
 by (simp add: act_ord_b "&I" delta_ord_b not_act_ord_a not_delta_ord_a)
java.lang.NullPointerException
 using "oa-exist:1" "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
  \open>\^>A1_]cl> a ‹Δ1 and ‹¬A[F₂]a›¬>F₂]a›
 using "&E" by blast+
 note props = props this

 let ?Π = "«[λy [O!]y ∨ q₀]¬"
 AOT_modally_strict {
 AOT_have ‹[«?Π¬]↓› by "cqt:2[lambda]"
 } note 1 = this
  AOT_have ‹<^><>
 proof (safe intro!: "&I"; AOT_subst ‹[λy O!y ∨ q₀]x› ‹O!x ∨ q₀› for: x)
 AOT_show ‹\A([O!]b ∨ q₀)›
 by (meson "Act-Basic:9" act_ord_b "∨I"(1) "≡E"(2))
 next AOT_show ‹\Δ([O!]b ∨ q₀)›
 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "→E")
 next AOT_show ‹¬from2,3) a "ux= f (ook xsk) (oay.look y )"
 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a
 not_act_q_zero "raa-cor:3" by blast
 next OT_ ‹
 proof (rule not_act_and_pos_delta)
 AOT_show ‹¬\A([O!]a ∨ q₀)›
 using "Act-Basic:9" "∨E"(2) "≡E"(4) not_act_ord_a
 not_act_q_zero "raa-cor:3" by blast
 next AOT_show ‹♢([O!]a ∨ q₀)›
 using "KBasic2:2" "&E"(1) "∨I"(2) "≡E"(2) q₀_prop by blast
 qed
 qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₁₃ where ‹\A[F₁₃]b & \Δ[F₁₃]b & ¬\A[F₁₃]a & \Δ[F₁₃]a›
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 using "&E" by blast+
 note props = props this

 let ?Π = "«[λy [O!]y ∨ ¬q₀]¬"
 AOT_modally_strict {
 AOT_have ‹[«?Π¬]↓› by "cqt:2[lambda]"
 } note 1 = this
 moreover AOT_have ‹\A[«?Π¬]b & \Δ[«?Π¬]b & \A[«?Π>a & \not^bo\Delta[\guillemotleft?\<i\
 proof (safe intro!: "&I"; AOT_subst ‹[λy O!y ∨ ¬q₀]x› ‹O!x ∨ ¬q₀› for: x)
 AOT_show ‹\A([O!]b ∨ ¬q₀)›
 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
 next AOT_show ‹\Δ([O!]b ∨ ¬q₀)›
 by (meson "KBasic:15" b_ord "∨I"(1) nec_delta "oa-facts:1" "→E")
 next AOT_show ‹\A([O!]a ∨ ¬q₀)›
 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
 next AOT_show ‹¬\Δ([O!]a ∨ ¬q₀)›
 proof(rule act_and_pos_not_not_delta)
 AOT_show ‹\A([O!]a ∨ ¬q₀)›
 by (meson "Act-Basic:9" act_not_q_zero "∨I"(2) "≡E"(2))
 next
 AOT_have \<  and
 using "KBasic2:1" "≡E"(2) not_act_and_pos_delta
 not_act_ord_a not_delta_ord_a "raa-cor:6" by blast
 moreover AOT_have ‹♢q₀›
 by (meson "&E"(1) q₀_prop)
 ultimately AOT_have 2: ‹♢(¬[O!]a & q₀)›
 by (metis "KBasic:16" "&I" "vdash-properties:10")
 AOT_show ‹♢¬([O!]a ∨ ¬q₀)›
 proof (AOT_subst (reverse) ‹¬ update_by_fun_gr_ko = ko.update
 AOT_modally_strict {
 AOT_show ‹¬[O!]a & q₀ ≡ ¬([O!]a ∨ ¬q₀)›
 by (metis "&I" "&E"(1) "&E"(2) "∨I"(1) "∨I"(2)
 "∨E"(3) "deduction-theorem" "≡I" "raa-cor:3")
 }
 next
 AOT_show ‹♢(¬[O!]a & q₀)›
 2" by blast
 qed
 qed
 qed(auto simp: "beta-C-meta"[THEN "→E", OF 1])
 ultimately AOT_obtain F₁"OAlist xs = Ab_oalist (sor xs)"
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
 AOT_hence ‹\A[F₁₄]b› and ‹\Δ[F₁₄]b› and ‹\A[F₁₄]a› and ‹¬: "list_of_oalist (OAlist xs) = sort_oalist_ko xs"
 using "&E" by blast+
 note props = props this

 AOT_have ‹[L]↓›
 by (rule "=_dfI"(2)[OF L_def]) "cqt:2[lambda]"+
 moreover AOT_have ‹koolstin_a ox x(simp add: [symmet] ko.oalist_inv_symmetric]oali
 proof (safe intro!: "&I")
 AOT_show ‹\A[L]b›
 by (meson nec_L "nec-imp-act" "vdash-properties:10")
 next AOT_show ‹\Δ[L]b› using nec_L nec_delta by blast
 next AOT_show ‹\A[L]a› by (meson nec_L "nec-imp-act" "→E")
 next AOT_show ‹
 qed
 ultimately AOT_obtain F₁₅ where ‹\AOAl = oa.map2_val_n
 using "∃I"(1)[rotated, THEN "∃E"[rotated]] by fastforce
 AOT_hence ‹\A[F₁₅]b› and ‹\Δ by (fact OAlist
 using "&E" by blast+
 note props = props this

 show ?thesis
 by (rule "∃I"(2)[where β=F₀]; rule "∃I"(2)[where β=F₁];
 rule "∃I"(2)[where β=F₂]; rule "∃I"(2)[where β=F₃];
 rule "∃I"(2)[where β=F₄]; rule "∃I"(2)[where β=F₅];
 rule "∃I"global_interpretation tc: comparator "comparator_of"
 rule "∃I"(2)[where β=F₈]; rule "∃I"(2)[where β=F₉];
 rule "∃I"(2)[where β=F₁₀]; rule "∃I"(2)[where β=F₁₁];
 rule "∃I"(2)[where β=F₁₂]; rule "∃I"(2)[where β=F₁₃];
 rule "∃I"(2)[where β=F₁₄]; rule "∃I"(2)[where β=F₁₅];
 safe intro!: "&I")
 (match conclusion in "[?v ⊨> [G]G]]" for F G ==>
 match props in A: "[?v ⊨ ¬φ{F}]" for φ ==> ‹
 match (φ) in "λa . ?p" ==> ‹fail› ∣ "λa . a" ==> ‹
 match props in B: "[?v ⊨ φ{G}]" ==> ‹
 fact "ps-e:4"[where =F an GGad 🚫φ, THEN "→E",
 OF "oth-class-taut:4:h"[THEN "≡E"(2)],
 OF "Disjunction Addition"(2)[THEN "→E"],
 OF "&I", OF A, OF B]››››)+
 

 ‹
 ‹\label{PLM: 9.11}›

  "o-objects-exist:1": ‹◻∃x O!x›
 (rule RN)
 AOT_modally_strict {
 AOT_obtain a where ‹♢(E!a & ¬\A[E!]a)›
 using "∃E"[rotated, OF "qml:4"[axiom_inst, THEN "BF♢"[THEN "→E"]]]
 by blast
 AOT_hence 1: ‹t_tc"
 AOT_have ‹[λx ♢[E!]x]a›
 proof (rule "β←C"(1); "cqt:2[lambda]"?)
 
 next
 AOT_show ‹♢E!a› by (fact 1)
 qed
 AOT_hence ‹
 AOT_thus ‹∃x [O!]x› by (rule "∃I")
 }
 

  "o-objects-exist:2": ‹◻∃x A!x›
  (rule RN)
 AOT_modally_strict {
 AOT_obtain a where ‹[A!]a›
 using "A-objects"[axiom_inst] "∃E"[rotated] "&E" by blast
 AOT_thus ‹∃x A!x› using "∃I" by blast
 }
 

  "o-objects-exist:3": ‹
 by (rule RN)
 (metis (no_types, opaque_lifting) "∃
 "≡E"(4) "modus-tollens:1" "o-objects-exist:2" "oa-contingent:2"
 iom_insteductio-aa:2")

  "o-objects-exist:4": ‹◻¬∀x A!x›
 by (rule RN)
 (metis (mono_tags, opaque_lifting) "∃E" "cqt-orig:1[const_var]"
 "≡E"(1) "modus-tollens:1" "o-objects-exist:1" "oa-contingent:2"
 

 
  (rule RN; rule "raa-cor:2")
  {
 AOT_assume ‹∀x E!x›
 moreover AOT_obtain a where abs: ‹A!a›
 using "o-objects-exist:2"[THEN "qml:2"[axiom_inst, THEN "→E"]]
 "∃E"[rotated] by blast
 ultimately AOT_have ‹E!a› using "∀E" by blast
 AOT_hence 1: ‹♢E!a› by (metis "T♢" "→E")
 AOT_have ‹[λy ♢E!y]a›
 proof (rule "β←mbda]"?)
 AOT_show\>›
 next
 AOT_show ‹♢E!a› by (fact 1)
 qed
 AOT_hence ‹O!a›
java.lang.NullPointerException
 AOT_hence ‹¬A!a› by (metis "≡E"(1) "oa-contingent:2")
 AOT_thus ‹ = "list_of_oalist_tc xs"
 }
 

  partition: ‹¬.. = h" b smp d:q
  proof
 AOT_assume ‹∃x (O!x & A!x)›
 then AOT_obtain a where ‹O!a & A!a›
 using "∃E"[rotated] by blast
 AOT_thus ‹p & ¬p› for p
 by (metis "&E"(1) "Conjunction Simplification"(2) "≡E"(1)
 "modus-tollens:1"list_of_oalist(OAlst_tc_inet s) pa_ypai_cx(ls_f_aist_t s)"
 

  eq_E :: ‹Π› (‹'(=_E')›)
 "=E": ‹(=_E) =_df [λxy O!x & O!y & ◻∀F ([F]x ≡ [F]y)]›

  "_AOT_eq_E_infix" :: ‹τ ==> τ ==> φ› (infixl ‹=_E› 50)
 
 "_AOT_eq_E_infix κ κ'" == "CONST AOT_exe (CONST eq_E) (CONST Pair κ κ')"
 ‹
 
 (🍋‹AOT_exe›, fn ctxt => fn [
 Const (🍋‹eq_E›, _),
 Const (🍋‹Pair›, _) $ lhs $ rhs
  => Const (🍋‹_AOT_eq_E_infix›, dummyT) $ lhs $ rhs)]›

 ‹Note: Not explicitly mentioned as theorem in PLM.›
  "=E[denotes]": ‹[(=_E)]↓›
 by (rule "=_dfI"(2)[OF "=E"]) "cqt:2[lambda]"+

  "=E-simple:1": ‹x =_E y ≡ (O!x & O!y & ◻∀F ([F]x ≡ [F]y))›
  -
 AOT_have 1: ‹[λxy [O!]x & [O!]y & ◻∀F ([F]x ≡ [F]y)]↓› by "cqt:2"
 show ?thesis
 apply (rule "=_dfI"(2)[OF "=E"]; "cqt:2[lambda]"?)
 using "beta-C-meta"[THEN "→E", OF 1, unvarify ν₁ν_n, of "(_,_)",
 OF tuple_denotes[THEN "≡_dfI"], OF "&I",
 const_var[axiom_inst],
 OF "cqt:2[const_var]"[axiom_inst]]
 by fast
 

  "=E-simple:2": ‹
  (rule "→I")
 AOT_assume ‹x =_E y›
 AOT_hence ‹O!x & O!y & ◻∀F ([F]x ≡ [F]y)›
 using "=E-simple:1"[THEN "≡E"(1)] by blast
 AOT_thus ‹x = y›
 using "≡_dfI"[OF "identity:1"] "∨I" by blast
 

  "id-nec3:1": ‹x =_E y ≡ ◻(x =_E y)›
  (rule "≡I"; rule "→I")
 AOT_assume ‹x =_E y›
 AOT_hence ‹O!x & O!y & ◻∀F ([F]x ≡ [F]y)›
 using "=E-simple:1" "≡E" by blast
 AOT_hence ‹◻map2_val_compa, fact tc.map2_val_compat_map_val_pa, rule refl, simp onl: assms)
 by (metis "S5Basic:6" "&I" "&E"(1) "&E"(2) "≡E"(4)
 "oa-facts:1" "raa-cor:3" "vdash-properties:10")
 AOT_hence ‹◻(O!x & O!y & ◻∀F ([F]x ≡ [F]y))›
 by (metis "&E"(1) "&E"(2) "≡E"(2) "KBasic:3" "&I")
 AOT_thus ‹◻(x =_E y)›
 using "=E-simple:1"
 by (AOT_subst ‹
 
 AOT_assume ‹◻ ‹
 AOT_thus ‹x =_E y› using "qml:2"[axiom_inst, THEN "→E"] by blast
 

  "id-nec3:2": ‹♢(x =_E y) ≡ x =_E y›
 by (meson "RE♢" "S5Basic:2" "id-nec3:1" "≡E"(1,5) "Commutativity of ≡")

  "id-nec3:3": ‹♢(x =_E y) ≡ ◻(x =_E y)›
 by (meson "id-nec3:1" "id-nec3:2" "≡E"(5))

  "_AOT_non_eq_E" :: ‹Π› (‹'(≠_E')›)
 
 (Π) "(≠_E)" == (Π) "(=_E)^-"
  "_AOT_non_eq_E_infix" :: ‹list_tc)union> fst ` se is_f_it_ty)"
 
 OT_non_eq_E_infix🚫
 "CONST AOT_exe (CONST relation_negation (CONST eq_E)) (CONST Pair κ κ')"
 ‹
 
 (🍋‹AOT_exe›, fn ctxt => fn [
 Const (🍋‹ 1 show ?thesis ..
 Const (🍋‹Pair›, _) $ lhs $ rhs
  => Const (🍋‹_AOT_non_eq_E_infix›, dummyT) $ lhs $ rhs)]›
  "thm-neg=E": ‹x ≠_E y ≡ ¬(x =_E y)›
  -
 AOT_have θ: ‹[λx₁...x₂ ¬(=_E)x₁...x₂]↓› by "cqt:2"
 AOT_have ‹x ≠_E y ≡ [λx₁...x₂ ¬(=_E)x₁...x₂]xy›
 by (rule "=_dfI"(1)[OF "df-relation-negation", OF θ])
 (meson "oth-class-taut:3:a")
 also AOT_have ‹… ≡ ¬(=_E)xy›
 by (safe intro!: "beta-C-meta"[THEN "→E", unvarify ν₁ν_n] "cqt:2"
 tuple_denotes[THEN "≡_dfI"] "&I")
 finally show ?thesis.
 

 
  -
 AOT_have ‹x ≠_E y ≡ ¬(x =_E y)› using "thm-neg=E".
 also AOT_have ‹…
 by (meson "id-nec3:2" "≡E"(1) "Commutativity of ≡" "oth-class-taut:4:b")
 also AOT_have ‹… ≡ ◻¬(x =_E y)›
 by (meson "KBasic2:1" "≡E"(2) "Commutativity of ≡:nat, 4:nat), 1, 3), (0, 2), (3,1))
 also AOT_have ‹… ≡ ◻(x ≠_E y)›
 by (AOT_subst (reverse) ‹
 (auto simp: "thm-neg=E" "oth-class-taut:3:a")
 finally show ?thesis.
 

  "id-nec4:2": ‹♢(x ≠_E y) ≡ (x ≠_E y)›
 by (meson "RE♢" "S5Basic:2" "id-nec4:1" "≡E"(2,5) "Commutativity of ≡")

  "id-nec4:3": ‹♢(x ≠_E y) ≡ ◻(x ≠_E y)›
 by (meson "id-nec4:1" "id-nec4:2" "≡E"(5))

  "id-act2:1": ‹x =_E y ≡ \Ax =_E y›
 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec3:2" "≡E"(1,6))
  "id-act2:2": ‹x ≠_E y ≡ \Ax ≠_E y›
 by (meson "Act-Basic:5" "Act-Sub:2" "RA[2]" "id-nec4:2" "≡E"(1,6))

  "ord=Eequiv:1": ‹O!x → x =_E x›
  (rule "→I")
 AOT_assume 1: ‹O!x›
 AOT_show ‹x =_E x›
 apply (rule "=_dfI"(2)[OF "=E"]) apply "cqt:2[lambda]"
 apply (rule "β←C"(1))
 apply "cqt:2[lambda]"
 apply (simp add: "&I" "cqt:2[const_var]"[axiom_inst] prod_denotesI)
 by (simp add: "1" RN "&I" "oth-class-taut:3:a" "universal-cor")
 

  "ord=Eequiv:2": ‹x =_E y → y =_E x›
 (rule CP)
 AOT_assume 1: ‹x =_E y›
 AOT_hence 2: ‹x = y› by (metis "=E-simple:2" "vdash-properties:10")
 AOT_have ‹O!x› using 1 by (meson "&E"(1) "=E-simple:1" "≡E"(1))
 AOT_hence ‹x =_E x› using "ord=Eequiv:1" "→E" by blast
 AOT_thus ‹y =_E x› using "rule=E"[rotated, OF 2] by fast
 

  "ord=Eequiv:3": ‹(x =_E y & y =_E z) → x =_E z›
  (rule CP)
 AOT_assume 1: ‹x =_E y & y =_E z›
 AOT_hence ‹x = y & y = z›
 by (metis "&I" "&E"(1) "&E"(2) "=E-simple:2" "vdash-properties:6")
 AOT_hence ‹x = z› by (metis "id-eq:3" "vdash-properties:6")
 moreover AOT_have ‹x =_E x›
 using 1[THEN "&E"(1)] "&E"(1) "=E-simple:1" "≡E"(1)
 "ord=Eequiv:1" "→E" by blast
 ultimately AOT_show ‹x =_E z›
 using "rule=E" by fast
 

  "ord-=E=:1": ‹(O!x ∨ O!y) → ◻(x = y ≡ x =_E y)›
 (rule CP)
 AOT_assume ‹O!x ∨ O!y›
 moreover {
 AOT_assume ‹O!x›
 AOT_hence ‹◻O!x› by (metis "oa-facts:1" "vdash-properties:10")
 moreover {
 AOT_modally_strict {
 AOT_have ‹O!x → (x = y ≡ x =_E y)›
 proof (rule "→I"; rule "≡I"; rule "→I")
 AOT_assume ‹O!x›
 AOT_hence ‹x =_E x› by (metis "ord=Eequiv:1" "→E")
 moreover AOT_assume ‹x = y›
 ultimately AOT_show ‹x =_E y› using "rule=E" by fast
 next
 AOT_assume ‹x =_E y›
 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
 qed
 }
 AOT_hence ‹◻O!x → ◻(x = y ≡ x =_E y)› by (metis "RM:1")
 }
 ultimately AOT_have ‹◻(x = y ≡ x =_E y)› using "→E" by blast
 }
 moreover {
 AOT_assume ‹O!y›
 AOT_hence ‹◻O!y› by (metis "oa-facts:1" "vdash-properties:10")
 moreover {
 AOT_modally_strict {
 AOT_have ‹O!y → (x = y ≡ x =_E y)›
 proof (rule "→I"; rule "≡I"; rule "→I")
 AOT_assume ‹O!y›
 y =_E y›E")
 moreover AOT_assume ‹ True thus ?thesis using i id_k unfnfolding id_upt_k_def by presburger ―Easy due to the inductive hypothesis›
 ultimately AOT_show ‹ y›
 next
 AOT_assume ‹x =_E y›
 AOT_thus ‹x = y› by (metis "=E-simple:2" "→E")
 qed
 }
 AOT_hence ‹◻O!y → ◻(x = y ≡ x =_E y)› by (metis "RM:1")
 }
 ultimately AOT_have ‹◻(x = y ≡ x =_E y)› using "→E" by blast
 }
 ultimately AOT_show ‹◻(x = y ≡ x =_E y)› by (metis "∨E"(3) "raa-cor:1")
 

  "ord-=E=:2": ‹O!y → [λx x = y]↓›
  (rule "→I"; rule "safe-ext"[axiom_inst, THEN "→E"]; rule "&I")
 AOT_show ‹[λx x =_E y]↓› by "cqt:2[lambda]"
 
 AOT_assume ‹O!y›
 AOT_hence 1: ‹◻(x = y ≡ x =_E y)› for x
 using "ord-=E=:1" "→E" "∨I" by blast
 AOT_have ‹◻(x =_E y ≡ x = y)› for x
 by (AOT_subst ‹x =_E y ≡ x = y› ‹x = y ≡ x =_E y›)
 (auto simp add: "Commutativity of ≡" 1)
 AOT_hence ‹∀x ◻(x =_E y ≡ x = y)› by (rule GEN)
 AOT_thus ‹◻∀x (x =_E y ≡ x = y)› by (rule BF[THEN "→E"])
 


 [λxy O!x & O!y & x = y]↓› "register_tensor F G = register (\lambdaa. tensor_update (F a) id_update) (\<lambdabrgste_ensor_irer:
  (rule "safe-ext[2]"[axiom_inst, THEN "→
 AOT_show ‹('a×'b) update ==>
 
 AOT_show ‹◻∀x∀y ([O!]x & [O!]y & x =_E y ≡ [O!]x & [O!]y & x = y)›
 proof (rule RN; rule GEN; rule GEN; rule "≡I"; rule "→qer A= HN uco, whea])
 
 AOT_show ‹_nonliy:
  (metis "&I" "&E"(1) "Conjunction Simplification"(2) "=E-simple:2"
 
 }
 
 AOT_modally_strict {
 AOT_show \<pen[
 apply(safe intro!: "&I")
 apply (metis ththat[HEN&E(1 E"&E"(1)
 apply (metis that[THEN "&E"(1), THEN "&E"(2)])
 using "rule=E"[rotated, OF that[THEN "&E"(2)]]
 "ord=Eequiv:1"[THEN "→
  at
 }
 qed
 

  "ind-nec": ‹
 (rule "→I")
 AOT_assume ‹∀ [F]y)›
  ‹F ([F]x ≡›q:2[lmbda]
 ultimately AOT_have ‹x ◻F ([F]x ≡ F]y)] \<quiv 
 E" by blast
 moreover AOT_have ‹[λ∀(x ≡
 apply (ule "\beta←C"(1))
 apply "cqt:2[lambda]"
 apply (fact "cqt:2[const_var]"[axiom_inst])
 by -
 ultimately AOT_have ‹e F H"
 \>◻∀F ([F]x ≡ [F]y)›
 →1) by blaast
 

java.lang.NullPointerException
 (rule "\ "→→
 AOT_assume \open>🚫
 AOT_hence ‹◻ apply ( (simp add:: reorerd eise_pai_appy)
 using "ind-nec"[THEN "→E"] by blast
 moreover AOT_assume ‹_def by (auto simp: register_pair_apply tensor_update_mult)
 ultimately AOT_have ‹Φ c d (a ⊗_\') c ⊗u d) *_\\ b)›
 using "&I" by blast
 AOT_thussim: sso_e)
 

  "ord=E:2": ‹
  (rule "→I"; rule "→I")
java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 moreoverAT_sme\openfor>F ([F]x ≡ [F]y)›
 ultimately AOT_have ‹
 using "ord=E:1" "→E" by blast
 AOT_thus ‹iso_register F ==> iso_register (inv F)›
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

  "ord=E2:1":
 ‹
  (rule "→I"; rule "≡pr -
 rule "≡d_[OF "=-infi"; rue"raa-cor:2")
 AOT_assume 0: ‹inv (F ⊗s>r G) = inv F ⊗_r inv G›
 AOT_assume ‹
 
java.lang.NullPointerException
 
 apply (rule "β←C"(1))
 apply "cqt:2[lambda]"
 apply (fact "cqt:2[const_var]"[axiom_inst])
 "v1TEN"\rightarrowE" F0HE &E"(1]]
 ultimately AOT_have ‹[λz z =_E y]x› using "rule=E" by fast
  quvaetrgistrscm:
 AOT_hence ‹ by (metis "=E-simple:2 "vdash-properties:6:6")
 AOT_thus ‹x = y & ¬
 
java.lang.NullPointerException
 AOT_hence 0: ‹
 using "≡register F›^e>‹iso_register F›, term‹compatible F G› or a conjunction of these,
 AOT_have ‹
  ‹E x] = [λz z =🚫
 by (metis "rule=I:1")
 moreover AOT_assume ‹x = y›
 ultimately AOT_have ‹[λz z =_E x] = [λz z =_E y]›
 using "rule=E" by fast
 AOT_thus ‹[λz z =_E x] = [λz z =_E y] & ¬([λz z =_E x] = [λz z =_E y])›
 using 0 "&I" by blast
 

  "ord=E2:2":
 ‹(O!x & O!y) → (x ≠ y ≡ [λz z = x] ≠ [λz z = y])›
  (rule "→I"; rule "≡I"; rule "→I";
 rule "≡_dfI"[OF "=-infix"]; rule "raa-cor:2")
 AOT_assume 0: ‹O!x & O!y›
 AOT_assume ‹x ≠ y›
 AOT_hence 1: ‹¬(x = y)› using "≡_dfE"[OF "=-infix"] by blast
 AOT_assume ‹[λz z = x] = [λz z = y]›
 moreover AOT_have ‹[λz z = x]x›
 apply (rule "β←C"(1))
 apply (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
 apply (fact "cqt:2[const_var]"[axiom_inst])
 by (simp add: "id-eq:1")
 ultimately AOT_have ‹[λz z = y]x› using "rule=E" by fast
 AOT_hence ‹x = y› using "β→C"(1) by blast
 AOT_thus ‹x = y & ¬(x = y)› using 1 "&I" by blast
 
 AOT_assume 0: ‹O!x & O!y›
 AOT_assume ‹[λz z = x] ≠ [λz z = y]›
 AOT_hence 1: ‹¬([λz z = x] = [λz z = y])›
 using "≡_dfE"[OF "=-infix"] by blast
 AOT_have ‹[λz z = x]↓›
 by (fact "ord-=E=:2"[THEN "→E", OF 0[THEN "&E"(1)]])
 AOT_hence ‹[λz z = x] = [λz z = x]›
 by (metis "rule=I:1")
 moreover AOT_assume ‹x = y›
 ultimately AOT_have ‹[λz z = x] = [λz z = y]›
 using "rule=E" by fast
 AOT_thus ‹[λz z = x] = [λz z = y] & ¬([λz z = x] = [λz z = y])›
 using 1 "&I" by blast
 

  ordnecfail: ‹O!x → ◻¬∃F x[F]›
 by (meson "RM:1" "→I" nocoder[axiom_inst] "oa-facts:1" "→E")

  "ab-obey:1": ‹(A!x & A!y) → (∀F (x[F] ≡ y[F]) → x = y)›
  (rule "→I"; rule "→I")
 AOT_assume 1: ‹A!x & A!y›
 AOT_assume ‹∀F (x[F] ≡ y[F])›
 AOT_hence ‹x[F] ≡ y[F]› for F using "∀E" by blast
 AOT_hence ‹◻(x[F] ≡ y[F])› for F by (metis "en-eq:6[1]" "≡E"(1))
 AOT_hence ‹∀F ◻(x[F] ≡ y[F])› by (rule GEN)
 AOT_hence ‹◻∀F (x[F] ≡ y[F])› by (rule BF[THEN "→E"])
 AOT_thus ‹x = y›
 using "≡_dfI"[OF "identity:1", OF "∨I"(2)] 1 "&I" by blast
 

  "ab-obey:2":
 ‹(∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & ¬x[F])) → x ≠ y›
  (rule "→I"; rule "≡_dfI"[OF "=-infix"]; rule "raa-cor:2")
 AOT_assume 1: ‹x = y›
 AOT_assume ‹∃F (x[F] & ¬y[F]) ∨ ∃F (y[F] & 'font-size: 18px;'>¬x[F])›
 moreover {
 AOT_assume ‹∃F (x[F] & ¬y[F])›
 then AOT_obtain F where ‹x[F] & ¬y[F]›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹y[F]›
 using calculation[THEN "&E"(1)] 1 "rule=E" by fast
 ultimately AOT_have ‹p & ¬p› for p
 by (metis "Conjunction Simplification"(2) "modus-tollens:2" "raa-cor:3")
 }
 moreover {
 AOT_assume ‹∃F (y[F] & ¬x[F])›
 then AOT_obtain F where ‹y[F] & ¬x[F]›
 using "∃E"[rotated] by blast
 moreover AOT_have ‹¬y[F]›
 using calculation[THEN "&E"(2)] 1 "rule=E" by fast
 ultimately AOT_have ‹p & ¬p› for p
 by (metis "Conjunction Simplification"(1) "modus-tollens:1" "raa-cor:3")
 }
 ultimately AOT_show ‹p & ¬p› for p
 by (metis "∨E"(3) "raa-cor:1")
 

  "encoders-are-abstract": ‹∃F x[F] → A!x›
 by (meson "deduction-theorem" "≡E"(2) "modus-tollens:2" nocoder
 "oa-contingent:3" "vdash-properties:1[2]")

  "denote=:1": ‹∀H∃x x[H]›
 by (rule GEN; rule "existence:2[1]"[THEN "≡_dfE"]; "cqt:2")

  "denote=:2": ‹∀G∃x₁...∃x_n x₁...x_n[H]›
 by (rule GEN; rule "existence:2"[THEN "≡_dfE"]; "cqt:2")

  "denote=:2[2]": ‹∀G∃x₁∃x₂ x₁x₂[H]›
 by (rule GEN; rule "existence:2[2]"[THEN "≡_dfE"]; "cqt:2")

  "denote=:2[3]": ‹∀G∃x₁∃x₂∃x₃ x₁x₂x₃[H]›
 by (rule GEN; rule "existence:2[3]"[THEN "≡_dfE"]; "cqt:2")

  "denote=:2[4]": ‹∀G∃x₁∃x₂∃x₃∃x₄ x₁x₂x₃x₄[H]›
 by (rule GEN; rule "existence:2[4]"[THEN "≡_dfE"]; "cqt:2")

  "denote=:3": ‹∃x x[Π] ≡ ∃H (H = Π)›
 using "existence:2[1]" "free-thms:1" "≡E"(2,5)
 "Commutativity of ≡" "≡Df" by blast

  "denote=:4": ‹(∃x₁...∃x_n x₁...x_n[Π]) ≡ ∃H (H = Π)›
 using "existence:2" "free-thms:1" "≡E"(6) "≡Df" by blast

  "denote=:4[2]": ‹(∃x₁∃x₂ x₁x₂[Π]) ≡ ∃H (H = Π)›
 using "existence:2[2]" "free-thms:1" "≡E"(6) "≡Df" by blast

  "denote=:4[3]": ‹(∃x₁∃x₂∃x₃ x₁x₂x₃[Π]) ≡ ∃H (H = Π)›
 using "existence:2[3]" "free-thms:1" "≡E"(6) "≡Df" by blast

  "denote=:4[4]": ‹(∃x₁∃x₂∃x₃∃x₄ x₁x₂x₃x₄[Π]) ≡ ∃H (H = Π)›
 using "existence:2[4]" "free-thms:1" "≡E"(6) "≡Df" by blast

  "A-objects!": ‹∃!x (A!x & ∀F (x[F] ≡ φ{F}))›
  (rule "uniqueness:1"[THEN "≡_dfI"])
 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ φ{F})›
 using "A-objects"[axiom_inst] "∃E"[rotated] by blast
 AOT_have ‹(A!β & ∀F (β[F] ≡ φ{F})) → β = a› for β
 proof (rule "→I")
 AOT_assume β_prop: ‹[A!]β & ∀F (β[F] ≡ φ{F})›
 AOT_hence ‹β[F] ≡ φ{F}› for F
 using "∀E" "&E" by blast
 AOT_hence ‹β[F] ≡ a[F]› for F
 using a_prop[THEN "&E"(2)] "∀E" "≡E"(2,5)
 "Commutativity of ≡" by fast
 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
 AOT_thus ‹β = a›
 using "ab-obey:1"[THEN "→E",
 OF "&I"[OF β_prop[THEN "&E"(1)], OF a_prop[THEN "&E"(1)]],
 THEN "→E"] by blast
 qed
 AOT_hence ‹∀β ((A!β & ∀F (β[F] ≡ φ{F})) → β = a)› by (rule GEN)
 AOT_thus ‹∃α ([A!]α & ∀F (α[F] ≡ φ{F}) &
 ∀β ([A!]β & ∀F (β[F] ≡ φ{F}) → β = α))›
 using "∃I" using a_prop "&I" by fast
 

  "obj-oth:1": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y))›
 using "A-objects!" by fast

  "obj-oth:2": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y & [F]z))›
 using "A-objects!" by fast

  "obj-oth:3": ‹∃!x (A!x & ∀F (x[F] ≡ [F]y ∨ [F]z))›
 using "A-objects!" by fast

  "obj-oth:4": ‹∃!x (A!x & ∀F (x[F] ≡ ◻[F]y))›
 using "A-objects!" by fast

  "obj-oth:5": ‹∃!x (A!x & ∀F (x[F] ≡ F = G))›
 using "A-objects!" by fast

  "obj-oth:6": ‹∃!x (A!x & ∀F (x[F] ≡ ◻∀y([G]y → [F]y)))›
 using "A-objects!" by fast

  "A-descriptions": ‹\ιx (A!x & ∀F (x[F] ≡ φ{F}))↓›
 by (rule "A-Exists:2"[THEN "≡E"(2)]; rule "RA[2]"; rule "A-objects!")

  "thm-can-terms2":
 ‹y = \ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & ∀F (y[F] ≡ φ{F}))›
 using "y-in:2" by blast

  "can-ab2": ‹y = \ιx(A!x & ∀F (x[F] ≡ φ{F})) → A!y›
 (rule "→I")
 AOT_assume ‹y = \ιx(A!x & ∀F (x[F] ≡ φ{F}))›
 AOT_hence ‹\A(A!y & ∀F (y[F] ≡ φ{F}))›
 using "actual-desc:2"[THEN "→E"] by blast
 AOT_hence ‹\AA!y› by (metis "Act-Basic:2" "&E"(1) "≡E"(1))
 AOT_thus ‹A!y› by (metis "≡E"(2) "oa-facts:8")
 

  "desc-encode:1": ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
  -
 AOT_have ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
 by (simp add: "A-descriptions")
 AOT_hence ‹A!\ιx(A!x & ∀F (x[F] ≡ φ{F})) &
 ∀ιF (x[F] ≡ φ{F})›
 
 AOT_thus ‹
 using "&E" "∀E" by blast
 

  "desc-encode:2": ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))[G] ≡ φ{G}›
 using "desc-encode:1".

  "desc-nec-encode:1":
 ‹\ιx (A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ \Aφ{F}›
  -
 AOT_have 0: ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))↓›
 by (simp add: "A-descriptions")
 AOT_hence ‹\A(A!\ιx(A!x & ∀F (x[F] ≡ φ{F})) &
 ∀F(\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}))›
 using "actual-desc:4"[THEN "→E"] by blast
 AOT_hence ‹\A∀F (\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
 using "Act-Basic:2" "&E"(2) "≡E"(1) by blast
 AOT_hence ‹∀F \A(\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
 using "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]" by blast
 AOT_hence ‹\A(\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F})›
 using "∀E" by blast
 AOT_hence ‹\A\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ \Aφ{F}›
 using "Act-Basic:5" "≡E"(1) by blast
 AOT_thus ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ \Aφ{F}›
 using "en-eq:10[1]"[unvarify x₁, OF 0] "≡E"(6) by blast
 

  "desc-nec-encode:2":
 ‹{F}))[G] ≡A
 using "desc-nec-encode:1".

  "Box-desc-encode:1": ‹
  \rightarrow>I"; rul "es-ecenod:2[THEN "\<>( swap_swapsimp]: \open>swap (swap x) = x›(simpadd: pointfree_idE)
  nec-imp-act" "vdash-properties:10)

  "Box-desc-encode:2":
 ‹φ→(x(A!x & ∀ φ{G}))[G] ≡{G})›
 P
 ◻φ{G}›
 ◻◻{G}›E"())
java.lang.NullPointerException
 proof (rule RM; rule "→
 AOT_modally_strict {
 ◻φ{G}›
 AOT_hence ‹G o G' = id› ‹
 Boxeo1 \rightarrow
 
 using 1 by (meson "qml:2"[axiom_inst] "→E")
 ultimately AOT_show ‹\ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G}›
 using "→I" "≡I" by simp
 }
 qed
 ultimately AOT_show ‹◻(\ιx(A!x & ∀F (x[F] ≡ φ{G}))[G] ≡ φ{G})›
 using "→E" by blast
 

  rigid_condition where
 ‹rigid_condition φ ≡ ∀v . [v ⊨ ∀α (φ{α} → ◻φ{α})]›
  rigid_condition :: ‹id_position ==> AOT_prop› (‹RIGID'_CONDITION'(_')›)

  "strict-can:1[E]":
 assumes ‹RIGID_CONDITION(φ)›
 shows ‹∀α (φ{α} → ◻φ{α})›
 using assms[unfolded rigid_condition_def] by auto

  "strict-can:1[I]":
 assumes ‹\⊨_\◻ ∀α (φ{α} → ◻φ{α})›
 shows ‹RIGID_CONDITION(φ)›
 using assms rigid_condition_def by auto

  "box-phi-a:1":
 assumes ‹RIGID_CONDITION(φ)›
 shows ‹(A!x & ∀F (x[F] ≡ φ{F})) → ◻(A!x & ∀F (x[F] ≡ φ{F}))›
  (rule "→I")
 AOT_assume a: ‹A!x & ∀F (x[F] ≡ φ{F})›
 AOT_hence b: ‹◻A!x›
 by (metis "Conjunction Simplification"(1) "oa-facts:2" "→E")
 AOT_have ‹x[F] ≡ φ{F}› for F
 using a[THEN "&E"(2)] "∀E" by blast
 moreover AOT_have ‹◻(x[F] → ◻x[F])› for F
 by (meson "pre-en-eq:1[1]" RN)
 moreover AOT_have ‹◻(φ{F} → ◻φ{F})› for F
 using RN "strict-can:1[E]"[OF assms] "∀E" by blast
 ultimately AOT_have ‹◻(x[F] ≡ φ{F})› for F
 using "sc-eq-box-box:5" "qml:2"[axiom_inst, THEN "→E"] "→E" "&I" by metis
 AOT_hence ‹∀F ◻(x[F] ≡ φ{F})› by (rule GEN)
 AOT_hence ‹◻∀F (x[F] ≡ φ{F})› by (rule BF[THEN "→E"])
 AOT_thus ‹◻([A!]x & ∀F (x[F] ≡ φ{F}))›
 using b "KBasic:3" "≡S"(1) "≡E"(2) by blast
 

  "box-phi-a:2":
 assumes ‹RIGID_CONDITION(φ)›
 shows ‹y = \ιx(A!x & ∀F (x[F] ≡ φ{F})) → (A!y & e='font-size: 18px;'>∀F (y[F] ≡ φ{F}))›
 (rule "→I")
 AOT_assume ‹y = \ιx(A!x & ∀F (x[F] ≡ φ{F}))›
 AOT_hence ‹\A(A!y & ∀F (y[F] ≡ φ{F}))›
 using "actual-desc:2"[THEN "→E"] by fast
 AOT_hence abs: ‹\AA!y› and ‹\A∀F (y[F] ≡ φ{F})›
 using "Act-Basic:2" "&E" "≡E"(1) by blast+
 AOT_hence ‹∀F \A(y[F] ≡ φ{F})›
 by (metis "≡E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
 AOT_hence ‹\A(y[F] ≡ φ{F})› for F
 using "∀E" by blast
 AOT_hence ‹\Ay[F] ≡ \Aφ{F}› for F
 by (metis "Act-Basic:5" "≡E"(1))
 AOT_hence ‹y[F] ≡ φ{F}› for F
 using "sc-eq-fur:2"[THEN "→E",
 OF "strict-can:1[E]"[OF assms,
 THEN "∀E"(2)[where β=F], THEN RN]]
 by (metis "en-eq:10[1]" "≡E"(6))
 AOT_hence ‹∀F (y[F] ≡ φ{F})› by (rule GEN)
 AOT_thus ‹[A!]y & ∀F (y[F] ≡ φ{F})›
 using abs "&I" "≡E"(2) "oa-facts:8" by blast
 

  "box-phi-a:3":
 assumes ‹RIGID_CONDITION(φ)›
 shows ‹\ιx(A!x & ∀F (x[F] ≡ φ{F}))[F] ≡ φ{F}›
 using "desc-nec-encode:2"
 "sc-eq-fur:2"[THEN "→E",
 OF "strict-can:1[E]"[OF assms,
 THEN "∀E"(2)[where β=F], THEN RN]]
 "≡E"(5) by blast

  Null :: ‹τ ==> φ› (‹Null'(_')›)
 "df-null-uni:1": ‹Null(x) ≡_df A!x & ¬∃F x[F]›

  Universal :: ‹τ ==> φ› (‹Universal'(_')›)
 "df-null-uni:2": ‹Universal(x) ≡_df A!x & ∀F x[F]›

  "null-uni-uniq:1": ‹∃!x Null(x)›
  (rule "uniqueness:1"[THEN "≡_dfI"])
 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ ¬(F = F))›
 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
 AOT_have a_null: ‹¬a[F]› for F
 proof (rule "raa-cor:2")
 AOT_assume ‹a[F]›
 AOT_hence ‹¬(F = F)› using a_prop[THEN "&E"(2)] "∀E" "≡E" by blast
 AOT_hence ‹F = F & ¬(F = F)› by (metis "id-eq:1" "raa-cor:3")
 AOT_thus ‹p & ¬p› for p by (metis "raa-cor:1")
 qed
 AOT_have ‹Null(a) & ∀β (Null(β) → β = a)›
 proof (rule "&I")
 AOT_have ‹¬∃F a[F]›
 using a_null by (metis "instantiation" "reductio-aa:1")
 AOT_thus ‹Null(a)›
 using "df-null-uni:1"[THEN "≡_dfI"] a_prop[THEN "&E"(1)] "&I" by metis
 next
 AOT_show ‹∀β (Null(β) → β = a)›
 proof (rule GEN; rule "→I")
 fix β
 AOT_assume a: ‹Null(β)›
 AOT_hence ‹¬∃F β[F]›
 using "df-null-uni:1"[THEN "≡_dfE"] "&E" by blast
 AOT_hence β_null: ‹¬β[F]› for F
 by (metis "existential:2[const_var]" "reductio-aa:1")
 AOT_have ‹∀F (β[F] ≡ a[F])›
 apply (rule GEN; rule "≡I"; rule CP)
 using "raa-cor:3" β_null a_null by blast+
 moreover AOT_have ‹A!β›
 using a "df-null-uni:1"[THEN "≡_dfE"] "&E" by blast
 ultimately AOT_show ‹β = a›
 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"]
 "&I" by blast
 qed
 qed
 AOT_thus ‹∃α (Null(α) & ∀β (Null(β) → β = α))›
 using "∃I"(2) by fast
 

  "null-uni-uniq:2": ‹∃!x Universal(x)›
  (rule "uniqueness:1"[THEN "≡_dfI"])
 AOT_obtain a where a_prop: ‹A!a & ∀F (a[F] ≡ F = F)›
 using "A-objects"[axiom_inst] "∃E"[rotated] by fast
 AOT_hence aF: ‹a[F]› for F using "&E" "∀E" "≡E" "id-eq:1" by fast
 AOT_hence ‹Universal(a)›
 using "df-null-uni:2"[THEN "≡_dfI"] "&I" a_prop[THEN "&E"(1)] GEN by blast
 moreover AOT_have ‹∀β (Universal(β) → β = a)›
 proof (rule GEN; rule "→I")
 fix β
 AOT_assume ‹Universal(β)›
 AOT_hence abs_β: ‹A!β› and ‹β[F]› for F
 using "df-null-uni:2"[THEN "≡_dfE"] "&E" "∀E" by blast+
 AOT_hence ‹β[F] ≡ a[F]› for F
 using aF by (metis "deduction-theorem" "≡I")
 AOT_hence ‹∀F (β[F] ≡ a[F])› by (rule GEN)
 AOT_thus ‹β = a›
 using a_prop[THEN "&E"(1)] "ab-obey:1"[THEN "→E", THEN "→E"]
 "&I" abs_β by blast
 qed
 ultimately AOT_show ‹∃α (Universal(α) & ∀β (Universal(β) → β = α))›
 using "&I" "∃I" by fast
 

  "null-uni-uniq:3": ‹\ιx Null(x)↓›
 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:1" by blast

  "null-uni-uniq:4": ‹\ιx Universal(x)↓›
 using "A-Exists:2" "RA[2]" "≡E"(2) "null-uni-uniq:2" by blast

  Null_object :: ‹κ_s› (‹a_\∅›)
 "df-null-uni-terms:1": ‹a_\∅ =_df \ιx Null(x)›

  Universal_object :: ‹κ_s› (‹a_V›)
 "df-null-uni-terms:2": ‹a_V =_df \ιx Universal(x)›

  "null-uni-facts:1": ‹Null(x) → ◻Null(x)›
  (rule "→I")
 AOT_assume ‹Null(x)›
 AOT_hence x_abs: ‹A!x› and x_null: ‹¬∃F x[F]›
 using "df-null-uni:1"[THEN "≡_dfE"] "&E" by blast+
 AOT_have ‹¬x[F]› for F using x_null
 using "existential:2[const_var]" "reductio-aa:1"
 by metis
 AOT_hence ‹◻¬x[F]› for F by (metis "en-eq:7[1]" "≡E"(1))
 AOT_hence ‹∀F ◻¬x[F]› by (rule GEN)
 AOT_hence ‹◻∀F ¬x[F]› by (rule BF[THEN "→E"])
 moreover AOT_have ‹◻∀F ¬x[F] → ◻¬∃F x[F]›
 apply (rule RM)
 by (metis (full_types) "instantiation" "cqt:2[const_var]"[axiom_inst]
 "→I" "reductio-aa:1" "rule-ui:1")
 ultimately AOT_have ‹◻¬∃F x[F]›
 by (metis "→E")
 moreover AOT_have ‹◻A!x› using x_abs
 using "oa-facts:2" "vdash-properties:10" by blast
 ultimately AOT_have r: ‹◻(A!x & ¬∃F x[F])›
 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
 AOT_show ‹◻Null(x)›
 by (AOT_subst ‹Null(x)› ‹A!x & ¬∃F x[F]›)
 (auto simp: "df-null-uni:1" "≡Df" r)
 

  "null-uni-facts:2": ‹Universal(x) → ◻Universal(x)›
  (rule "→I")
 AOT_assume ‹Universal(x)›
 AOT_hence x_abs: ‹A!x› and x_univ: ‹∀F x[F]›
 using "df-null-uni:2"[THEN "≡_dfE"] "&E" by blast+
 AOT_have ‹x[F]› for F using x_univ "∀E" by blast
 AOT_hence ‹◻x[F]› for F by (metis "en-eq:2[1]" "≡E"(1))
 AOT_hence ‹∀F ◻x[F]› by (rule GEN)
 AOT_hence ‹◻∀F x[F]› by (rule BF[THEN "→E"])
 moreover AOT_have ‹◻A!x› using x_abs
 using "oa-facts:2" "vdash-properties:10" by blast
 ultimately AOT_have r: ‹◻(A!x & ∀F x[F])›
 by (metis "KBasic:3" "&I" "≡E"(3) "raa-cor:3")
 AOT_show ‹◻Universal(x)›
 by (AOT_subst ‹Universal(x)› ‹A!x & ∀F x[F]›)
 (auto simp add: "df-null-uni:2" "≡Df" r)
 

  "null-uni-facts:3": ‹Null(a_\∅)›
 apply (rule "=_dfI"(2)[OF "df-null-uni-terms:1"])
 apply (simp add: "null-uni-uniq:3")
 using "actual-desc:4"[THEN "→E", OF "null-uni-uniq:3"]
 "sc-eq-fur:2"[THEN "→E",
 OF "null-uni-facts:1"[unvarify x, THEN RN, OF "null-uni-uniq:3"],
        THEN "\<equiv>E"(1)]
  by blast

AOT_theorem "null-uni-facts:4": \<open>Universal(a\<^sub>V)\<close>
  apply (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:2"])
   apply (simp add: "null-uni-uniq:4")
  using "actual-desc:4"[THEN "\<rightarrow>E", OF "null-uni-uniq:4"]
    "sc-eq-fur:2"[THEN "\<rightarrow>E",
        OF "null-uni-facts:2"[unvarify x, THEN RN, OF "null-uni-uniq:4"],
        THEN "\<equiv>E"(1)]
  by blast

AOT_theorem "null-uni-facts:5": \<open>a\<^sub>\<emptyset> \<noteq> a\<^sub>V\<close>
proof (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
    rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
    rule "\<equiv>\<^sub>d\<^sub>fI"[OF "=-infix"];
    rule "raa-cor:2")
  AOT_obtain x where nullx: \<open>Null(x)\<close>
    by (metis "instantiation" "df-null-uni-terms:1" "existential:1"
              "null-uni-facts:3" "null-uni-uniq:3" "rule-id-df:2:b[zero]")
  AOT_hence act_null: \<open>\<^bold>\<A>Null(x)\<close>
    by (metis "nec-imp-act" "null-uni-facts:1" "\<rightarrow>E")
  AOT_assume \<open>\<^bold>\<iota>x Null(x) = \<^bold>\<iota>x Universal(x)\<close>
  AOT_hence \<open>\<^bold>\<A>\<forall>x(Null(x) \<equiv> Universal(x))\<close>
    using "actual-desc:5"[THEN "\<rightarrow>E"] by blast
  AOT_hence \<open>\<forall>x \<^bold>\<A>(Null(x) \<equiv> Universal(x))\<close>
    by (metis "\<equiv>E"(1) "logic-actual-nec:3" "vdash-properties:1[2]")
  AOT_hence \<open>\<^bold>\<A>Null(x) \<equiv> \<^bold>\<A>Universal(x)\<close>
    using "Act-Basic:5" "\<equiv>E"(1) "rule-ui:3" by blast
  AOT_hence \<open>\<^bold>\<A>Universal(x)\<close> using act_null "\<equiv>E" by blast
  AOT_hence \<open>Universal(x)\<close>
    by (metis RN "\<equiv>E"(1) "null-uni-facts:2" "sc-eq-fur:2" "\<rightarrow>E")
  AOT_hence \<open>\<forall>F x[F]\<close> using "\<equiv>\<^sub>d\<^sub>fE"[OF "df-null-uni:2"] "&pan>E" by metis
  moreover AOT_have \<open>\<not>\<exists>F x[F]\<close>
    using nullx "\<equiv>\<^sub>d\<^sub>fE"[OF "df-null-uni:1"] "&E" by metis
  ultimately AOT_show \<open>p & \<not>p\<close> for p
    by (metis "cqt-further:1" "raa-cor:3" "\<rightarrow>E")
qed

AOT_theorem "null-uni-facts:6": \<open>a\<^sub>\<emptyset> = \<^bold>\<iota>x(A!x & \<forall>F (x[F] \<equiv> F \<noteq> F))\<close>
proof (rule "ab-obey:1"[unvarify x y, THEN "\<rightarrow>E", THEN "\<rightarrow>E"])
  AOT_show \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))\<down>\<close>
    by (simp add: "A-descriptions")
next
  AOT_show \<open>a\<^sub>\<emptyset>\<down>\<close>
    by (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
       (simp add: "null-uni-uniq:3")
next
  AOT_have \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))\<down>\<close>
    by (simp add: "A-descriptions")
  AOT_hence 1: \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F)) = \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))\<close>
    using "rule=I:1" by blast
  AOT_show \<open>[A!]a\<^sub>\<emptyset> & [A!]\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))\<close>
    apply (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"];
           rule "&I")
     apply (meson "\<equiv>\<^sub>d\<^sub>fE" "Conjunction Simplification"(1)
                  "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3"
                  "null-uni-uniq:3" "rule-id-df:2:a[zero]" "\<rightarrow>E")
    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "\<rightarrow>E", OF 1].
next
  AOT_show \<open>\<forall>F (a\<^sub>\<emptyset>[F] \<equiv> \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))[F])\<close>
  proof (rule GEN)
    fix F
    AOT_have \<open>\<not>a\<^sub>\<emptyset>[F]\<close>
      by (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:1", OF "null-uni-uniq:3"])
         (metis (no_types, lifting) "\<equiv>\<^sub>d\<^sub>fE" "&E"(2) "\<or>I"(2) "\<or>E"(3) "\<exists>I"(2)
                "df-null-uni:1" "df-null-uni-terms:1" "null-uni-facts:3"
                "raa-cor:2" "rule-id-df:2:a[zero]"
                "russell-axiom[enc,1].\<psi>_denotes_asm")
    moreover AOT_have \<open>\<not>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))[F]\<close>
    proof(rule "raa-cor:2")
      AOT_assume 0: \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))[F]\<close>
      AOT_hence \<open>\<^bold>\<A>(F \<noteq> F)\<close>
        using "desc-nec-encode:2"[THEN "\<equiv>E"(1), OF 0] by blast
      moreover AOT_have \<open>\<not>\<^bold>\<A>(F \<noteq> F)\<close>
        using "\<equiv>\<^sub>d\<^sub>fE" "id-act:2" "id-eq:1" "\<equiv>E"(2)
              "=-infix" "raa-cor:3" by blast
      ultimately AOT_show \<open>\<^bold>\<A>(F \<noteq> F) & \<not>\<^bold>\<A>(F \<noteq> F)\<close> by (rule "&I")
    qed
    ultimately AOT_show \<open>a\<^sub>\<emptyset>[F] \<equiv> \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F \<noteq> F))[F]\<close>
      using "deduction-theorem" "\<equiv>I" "raa-cor:4" by blast
  qed
qed

AOT_theorem "null-uni-facts:7": \<open>a\<^sub>V = \<^bold>\<iota>x(A!x & \<forall>F (x[F] \<equiv> F = F))\<close>
proof (rule "ab-obey:1"[unvarify x y, THEN "\<rightarrow>E", THEN "\<rightarrow>E"])
  AOT_show \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))\<down>\<close>
    by (simp add: "A-descriptions")
next
  AOT_show \<open>a\<^sub>V\<down>\<close>
    by (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
       (simp add: "null-uni-uniq:4")
next
  AOT_have \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))\<down>\<close>
    by (simp add: "A-descriptions")
  AOT_hence 1: \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F)) = \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))\<close>
    using "rule=I:1" by blast
  AOT_show \<open>[A!]a\<^sub>V & [A!]\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))\<close>
    apply (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"];
           rule "&I")
     apply (meson "\<equiv>\<^sub>d\<^sub>fE" "Conjunction Simplification"(1) "df-null-uni:2"
                  "df-null-uni-terms:2" "null-uni-facts:4" "null-uni-uniq:4"
                  "rule-id-df:2:a[zero]" "\<rightarrow>E")
    using "can-ab2"[unvarify y, OF "A-descriptions", THEN "\<rightarrow>E", OF 1].
next
  AOT_show \<open>\<forall>F (a\<^sub>V[F] \<equiv> \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))[F])\<close>
  proof (rule GEN)
    fix F
    AOT_have \<open>a\<^sub>V[F]\<close>
      apply (rule "=\<^sub>d\<^sub>fI"(2)[OF "df-null-uni-terms:2", OF "null-uni-uniq:4"])
      using "\<equiv>\<^sub>d\<^sub>fE" "&E"(2) "df-null-uni:2" "df-null-uni-terms:2"
            "null-uni-facts:4" "null-uni-uniq:4" "rule-id-df:2:a[zero]"
            "rule-ui:3" by blast
    moreover AOT_have \<open>\<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))[F]\<close>
      using "RA[2]" "desc-nec-encode:2" "id-eq:1" "\<equiv>E"(2) by fastforce
    ultimately AOT_show \<open>a\<^sub>V[F] \<equiv> \<^bold>\<iota>x([A!]x & \<forall>F (x[F] \<equiv> F = F))[F]\<close>
      using "deduction-theorem" "\<equiv>I" by simp
  qed
qed

AOT_theorem "aclassical:1":
\<open>\<forall>R\<exists>x\<exists>y(A!x & A!y & x \<noteq> y & [\<lambda>z [R]zx] = [\<lambda>z [R]zy])\<close>
proof(rule GEN)
  fix R
  AOT_obtain a where a_prop:
    \<open>A!a & \<forall>F (a[F] \<equiv> \<exists>y(A!y & F = [\<lambda>z [R]zy] & \<not>y[F]))\<close>
    using "A-objects"[axiom_inst] "\<exists>E"[rotated] by fast
  AOT_have a_enc: \<open>a[\<lambda>z [R]za]\<close>
  proof (rule "raa-cor:1")
    AOT_assume 0: \<open>\<not>a[\<lambda>z [R]za]\<close>
    AOT_hence \<open>\<not>\<exists>y(A!y & [\<lambda>z [R]za] = [\<lambda>z [R]zy] & \<not>y[\<lambda>z [R]za])\<close>
      by (rule a_prop[THEN "&E"(2), THEN "\<forall>E"(1)[where \<tau>="\<guillemotleft>[\<lambda>z [R]za]\<guillemotright>"],
                THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)],
                THEN "\<equiv>E"(1), rotated])
         "cqt:2[lambda]"
    AOT_hence \<open>\<forall>y \<not>(A!y & [\<lambda>z [R]za] = [\<lambda>z [R]zy] & \<not>y[\<lambda>z [R]za])\<close>
      using "cqt-further:4" "vdash-properties:10" by blast
    AOT_hence \<open>\<not>(A!a & [\<lambda>z [R]za] = [\<lambda>z [R]za] & \<not>a[\<lambda>z [R]za])\<close>
      using "\<forall>E" by blast
    AOT_hence \<open>(A!a & [\<lambda>z [R]za] = [\<lambda>z [R]za]) \<rightarrow> a[\<lambda>z [R]za]\<close>
      by (metis "&I" "deduction-theorem" "raa-cor:3")
    moreover AOT_have \<open>[\<lambda>z [R]za] = [\<lambda>z [R]za]\<close>
      by (rule "=I") "cqt:2[lambda]"
    ultimately AOT_have \<open>a[\<lambda>z [R]za]\<close>
      using a_prop[THEN "&E"(1)] "\<rightarrow>E" "&I" by blast
    AOT_thus \<open>a[\<lambda>z [R]za] & \<not>a[\<lambda>z [R]za]\<close>
      using 0 "&I" by blast
  qed
  AOT_hence \<open>\<exists>y(A!y & [\<lambda>z [R]za] = [\<lambda>z [R]zy] & \<not>y[\<lambda>z [R]za])\<close>
    by (rule a_prop[THEN "&E"(2), THEN "\<forall>E"(1), THEN "\<equiv>E"(1), rotated])
       "cqt:2"
  then AOT_obtain b where b_prop:
    \<open>A!b & [\<lambda>z [R]za] = [\<lambda>z [R]zb] & \<not>b[\<lambda>z [R]za]\<close>
    using "\<exists>E"[rotated] by blast
  AOT_have \<open>a \<noteq> b\<close>
    apply (rule "\<equiv>\<^sub>d\<^sub>fI"[OF "=-infix"])
    using a_enc b_prop[THEN "&E"(2)]
    using "\<not>\<not>I" "rule=E" id_sym "\<equiv>E"(4) "oth-class-taut:3:a"
          "raa-cor:3" "reductio-aa:1" by fast
  AOT_hence \<open>A!a & A!b & a \<noteq> b & [\<lambda>z [R]za] = [\<lambda>z [R]zb]\<close>
    using b_prop "&E" a_prop "&I" by meson
  AOT_hence \<open>\<exists>y (A!a & A!y & a \<noteq> y & [\<lambda>z [R]za] = [\<lambda>z [R]zy])\<close> by (rule "\<exists>I")
  AOT_thus \<open>\<exists>x\<exists>y (A!x & A!y & x \<noteq> y & [\<lambda>z [R]zx] = [\<lambda>z [R]zy])\<close> by (rule "\<exists>I")
qed

AOT_theorem "aclassical:2":
  \<open>\<forall>R\<exists>x\<exists>y(A!x & A!y & x \<noteq> y & [\<lambda>z [R]xz] = [\<lambda>z [R]yz])\<close>
proof(rule GEN)
  fix R
  AOT_obtain a where a_prop:
    \<open>A!a & \<forall>F (a[F] \<equiv> \<exists>y(A!y & F = [\<lambda>z [R]yz] & \<not>y[F]))\<close>
    using "A-objects"[axiom_inst] "\<exists>E"[rotated] by fast
  AOT_have a_enc: \<open>a[\<lambda>z [R]az]\<close>
  proof (rule "raa-cor:1")
    AOT_assume 0: \<open>\<not>a[\<lambda>z [R]az]\<close>
    AOT_hence \<open>\<not>\<exists>y(A!y & [\<lambda>z [R]az] = [\<lambda>z [R]yz] & \<not>y[\<lambda>z [R]az])\<close>
      by (rule a_prop[THEN "&E"(2), THEN "\<forall>E"(1)[where \<tau>="\<guillemotleft>[\<lambda>z [R]az]\<guillemotright>"],
                THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)],
                THEN "\<equiv>E"(1), rotated])
         "cqt:2[lambda]"
    AOT_hence \<open>\<forall>y \<not>(A!y & [\<lambda>z [R]az] = [\<lambda>z [R]yz] & \<not>y[\<lambda>z [R]az])\<close>
      using "cqt-further:4" "vdash-properties:10" by blast
    AOT_hence \<open>\<not>(A!a & [\<lambda>z [R]az] = [\<lambda>z [R]az] & \<not>a[\<lambda>z [R]az])\<close>
      using "\<forall>E" by blast
    AOT_hence \<open>(A!a & [\<lambda>z [R]az] = [\<lambda>z [R]az]) \<rightarrow> a[\<lambda>z [R]az]\<close>
      by (metis "&I" "deduction-theorem" "raa-cor:3")
    moreover AOT_have \<open>[\<lambda>z [R]az] = [\<lambda>z [R]az]\<close>
      by (rule "=I") "cqt:2[lambda]"
    ultimately AOT_have \<open>a[\<lambda>z [R]az]\<close>
      using a_prop[THEN "&E"(1)] "\<rightarrow>E" "&I" by blast
    AOT_thus \<open>a[\<lambda>z [R]az] & \<not>a[\<lambda>z [R]az]\<close>
      using 0 "&I" by blast
  qed
  AOT_hence \<open>\<exists>y(A!y & [\<lambda>z [R]az] = [\<lambda>z [R]yz] & \<not>y[\<lambda>z [R]az])\<close>
    by (rule a_prop[THEN "&E"(2), THEN "\<forall>E"(1), THEN "\<equiv>E"(1), rotated])
       "cqt:2"
  then AOT_obtain b where b_prop:
    \<open>A!b & [\<lambda>z [R]az] = [\<lambda>z [R]bz] & \<not>b[\<lambda>z [R]az]\<close>
    using "\<exists>E"[rotated] by blast
  AOT_have \<open>a \<noteq> b\<close>
    apply (rule "\<equiv>\<^sub>d\<^sub>fI"[OF "=-infix"])
    using a_enc b_prop[THEN "&E"(2)]
    using "\<not>\<not>I" "rule=E" id_sym "\<equiv>E"(4) "oth-class-taut:3:a"
          "raa-cor:3" "reductio-aa:1" by fast
  AOT_hence \<open>A!a & A!b & a \<noteq> b & [\<lambda>z [R]az] = [\<lambda>z [R]bz]\<close>
    using b_prop "&E" a_prop "&I" by meson
  AOT_hence \<open>\<exists>y (A!a & A!y & a \<noteq> y & [\<lambda>z [R]az] = [\<lambda>z [R]yz])\<close> by (rule "\<exists>I")
  AOT_thus \<open>\<exists>x\<exists>y (A!x & A!y & x \<noteq> y & [\<lambda>z [R]xz] = [\<lambda>z [R]yz])\<close> by (rule "\<exists>I")
qed

AOT_theorem "aclassical:3":
  \<open>\<forall>F\<exists>x\<exists>y(A!x & A!y & x \<noteq> y & [\<lambda> [F]x] = [\<lambda> [F]y])\<close>
proof(rule GEN)
  fix R
  AOT_obtain a where a_prop:
    \<open>A!a & \<forall>F (a[F] \<equiv> \<exists>y(A!y & F = [\<lambda>z [R]y] & \<not>y[F]))\<close>
    using "A-objects"[axiom_inst] "\<exists>E"[rotated] by fast
  AOT_have den: \<open>[\<lambda>z [R]a]\<down>\<close> by "cqt:2[lambda]"
  AOT_have a_enc: \<open>a[\<lambda>z [R]a]\<close>
  proof (rule "raa-cor:1")
    AOT_assume 0: \<open>\<not>a[\<lambda>z [R]a]\<close>
    AOT_hence \<open>\<not>\<exists>y(A!y & [\<lambda>z [R]a] = [\<lambda>z [R]y] & \<not>y[\<lambda>z [R]a])\<close>
      by (safe intro!: a_prop[THEN "&E"(2), THEN "\<forall>E"(1)[where \<tau>=\<open>\<guillemotleft>[\<lambda>z [R]a]\<guillemotright>\<close>],
                THEN "oth-class-taut:4:b"[THEN "\<equiv>E"(1)],
                THEN "\<equiv>E"(1), rotated] "cqt:2") 
    AOT_hence \<open>\<forall>y \<not>(A!y & [\<lambda>z [R]a] = [\<lambda>z [R]y] & \<not>y[\<lambda>z [R]a])\<close>
      using "cqt-further:4" "\<rightarrow>E" by blast
    AOT_hence \<open>\<not>(A!a & [\<lambda>z [R]a] = [\<lambda>z [R]a] & \<not>a[\<lambda>z [R]a])\<close> using "\<forall>E" by blast
    AOT_hence \<open>(A!a & [\<lambda>z [R]a] = [\<lambda>z [R]a]) \<rightarrow> a[\<lambda>z [R]a]\<close>
      by (metis "&I" "deduction-theorem" "raa-cor:3")
    AOT_hence \<open>a[\<lambda>z [R]a]\<close>
      using a_prop[THEN "&E"(1)] "\<rightarrow>E" "&I"
      by (metis "rule=I:1" den)
    AOT_thus \<open>a[\<lambda>z [R]a] & \<not>a[\<lambda>z [R]a]\<close>  by (metis "0" "raa-cor:3") 
  qed
  AOT_hence \<open>\<exists>y(A!y & [\<lambda>z [R]a] = [\<lambda>z [R]y] & \<not>y[\<lambda>z [R]a])\<close>
    by (rule a_prop[THEN "&E"(2), THEN "\<forall>E"(1), OF den, THEN "\<equiv>E"(1), rotated])
  then AOT_obtain b where b_prop: \<open>A!b & [\<lambda>z [R]a] = [\<lambda>z [R]b] & \<not>b[\<lambda>z [R]a]\<close>
    using "\<exists>E"[rotated] by blast
  AOT_have 1: \<open>a \<noteq> b\<close>
    apply (rule "\<equiv>\<^sub>d\<^sub>fI"[OF "=-infix"])
    using a_enc b_prop[THEN "&E"(2)]
    using "\<not>\<not>I" "rule=E" id_sym "\<equiv>E"(4) "oth-class-taut:3:a"
          "raa-cor:3" "reductio-aa:1" by fast
  AOT_have a: \<open>[\<lambda> [R]a] = ([R]a)\<close>
    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
    by (meson "log-prop-prop:2")
  AOT_have b: \<open>[\<lambda> [R]b] = ([R]b)\<close>
    apply (rule "lambda-predicates:3[zero]"[axiom_inst, unvarify p])
    by (meson "log-prop-prop:2")
  AOT_have \<open>[\<lambda> [R]a] = [\<lambda> [R]b]\<close>
    apply (rule "rule=E"[rotated, OF a[THEN id_sym]])
    apply (rule "rule=E"[rotated, OF b[THEN id_sym]])
    apply (rule "identity:4"[THEN "\<equiv>\<^sub>d\<^sub>fI", OF "&I", rotated])
    using b_prop "&E" apply blast
    apply (safe intro!: "&I")
    by (simp add: "log-prop-prop:2")+
  AOT_hence \<open>A!a & A!b & a \<noteq> b & [\<lambda> [R]a] = [\<lambda> [R]b]\<close>
    using 1 a_prop[THEN "&E"(1)] b_prop[THEN "&E"(1), THEN "&E"(1)]
          "&I" by auto
  AOT_hence \<open>\<exists>y (A!a & A!y & a \<noteq> y & [\<lambda> [R]a] = [\<lambda> [R]y])\<close> by (rule "\<exists>I")
  AOT_thus \<open>\<exists>x\<exists>y (A!x & A!y & x \<noteq> y & [\<lambda> [R]x] = [\<lambda> [R]y])\<close> by (rule "\<exists>I")
qed

AOT_theorem aclassical2: \<open>\<exists>x\<exists>y (A!x & A!y & x \<noteq> y & \<forall>F ([F]x \<equiv> [F]y))\<close>
proof -
  AOT_have \<open>\<exists>x \<exists>y ([A!]x & [A!]y & x \<noteq> y &
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zx] =
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zy])\<close>
    by (rule "aclassical:1"[THEN "\<forall>E"(1)[where \<tau>="\<guillemotleft>[\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]\<guillemotright>"]])
       "cqt:2"
  then AOT_obtain x where \<open>\<exists>y ([A!]x & [A!]y & x \<noteq> y &
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zx] =
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zy])\<close>
    using "\<exists>E"[rotated] by blast
  then AOT_obtain y where 0: \<open>([A!]x & [A!]y & x \<noteq> y &
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zx] =
               [\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zy])\<close>
    using "\<exists>E"[rotated] by blast
  AOT_have \<open>[\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zx]x\<close>
    by (auto intro!: "\<beta>\<leftarrow>C"(1) "cqt:2"
             simp: "&I" "ex:1:a" prod_denotesI "rule-ui:3"
                   "oth-class-taut:3:a" "universal-cor")
  AOT_hence \<open>[\<lambda>z [\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]zy]x\<close>
    by (rule "rule=E"[rotated, OF 0[THEN "&E"(2)]])
  AOT_hence \<open>[\<lambda>xy \<forall>F ([F]x \<equiv> [F]y)]xy\<close>
    by (rule "\<beta>\<rightarrow>C"(1))
  AOT_hence \<open>\<forall>F ([F]x \<equiv> [F]y)\<close>
    using "\<beta>\<rightarrow>C"(1) old.prod.case by fast
  AOT_hence \<open>[A!]x & [A!]y & x \<noteq> y & \<forall>F ([F]x \<equiv> [F]y)\<close>
    using 0 "&E" "&I" by blast
  AOT_hence \<open>\<exists>y ([A!]x & [A!]y & x \<noteq> y & \<forall>F ([F]x \<equiv> [F]y))\<close> by (rule "\<exists>I")
  AOT_thus \<open>\<exists>x\<exists>y ([A!]x & [A!]y & x \<noteq> y & \<forall>F ([F]x \<equiv> [F]y))\<close> by (rule "\<exists>I"(2))
qed

AOT_theorem "kirchner-thm:1":
  \<open>[\<lambda>x \<phi>{x}]\<down> \<equiv> \<box>\<forall>x\<forall>y(\<forall>F([F]x \<equiv> [F]y) \<rightarrow> (\<phi>{x} \<equiv> \<phi>{y}))\<close>
proof(rule "\<equiv>I"; rule "\<rightarrow>I")
  AOT_assume \<open>[\<lambda>x \<phi>{x}]\<down>\<close>
  AOT_hence \<open>\<box>[\<lambda>x \<phi>{x}]\<down>\<close> by (metis "exist-nec" "vdash-properties:10")
  moreover AOT_have \<open>\<box>[\<lambda>x \<phi>{x}]\<down> \<rightarrow> \<box>\<forall>x\<forall>y(\<forall>F([F]x \<equiv> [F]y) \<rightarrow> (\<phi>{x} \<equiv> \<phi>{y}))\<close>
  proof (rule "RM:1"; rule "\<rightarrow>I"; rule GEN; rule GEN; rule "\<rightarrow>I")
    AOT_modally_strict {
      fix x y
      AOT_assume 0: \<open>[\<lambda>x \<phi>{x}]\<down>\<close>
      moreover AOT_assume \<open>\<forall>F([F]x \<equiv> [F]y)\<close>
--> --------------------

--> maximum size reached

--> --------------------

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