Quellcode-Bibliothek TAO_9_PLM.thy

(*<*)
theory TAO_9_PLM  lemmavdash_properties_10
imports 
  TAO_8_Definitions 
  "HOL-Eisbach.Eisbach_Tools"
begin
(*>*)

section‹psi> in v] ==>phi> in v] ==>ψv])"
 ‹\label{TAO_PLM}›

  meta_defs[no_atp] meta_aux[no_atp]

  PLM = Axioms
 

 ‹Automatic Solver›
 ‹\label{TAO_PLM_Solver}›

 named_theorems PLM
 named_theorems PLM_intro
 named_theorems PLM_elim
 named_theorems PLM_dest
 named_theorems PLM_subst

 method PLM_solver declares PLM_intro PLM_elim PLM_subst PLM_dest PLM
 = ((assumption | (match axiom i using vdash_ vdash_prope
 | fact PLM | rule PLM_intro | subst PLM_subst | subst (asm) PLM_subst
 | fastforce | safe | drule PLM_dest deduction = \open

 ‹
 ‹

 lemma modus_ponens[PLM]:
 "[[φ in v]; [φ \→ ψ in v]] ==> [ψ in v]"
 by (simp add: Semantics.T5)

 ‹🚫
 ‹"\lbrakk\alpha \<hi 

 interpretation Axioms .
 declare axiom[PLM]
 declare conn_defs[PLM]

 ‹(Modally Strict) Proofs and Derivations›
 ‹

 lemma vdash_properties_6[no_atp]:
 "[[φ in v]; [φ \→ ψ in v]] ==> [ψ in v]"
 using modus_ponens .
 lemma vdash_properties_9[PLM]:
 "[φ in v] ==> [ψ \→ φ in v]"
 using modus_ponens pl_1[axiom_instance] by blast
 lemma vdash_properties_10[PLM]:
 "[φ \→ ψ in v] ==> ([φ in v] ==> [ψ in v])"
 using vdash_properties_6 .

 attribute_setup deduction = ‹
 Scan.succeed (Thm.rnterpretatio Axioms .
 (fn _ => fn thm => thm RS @{thm vdash_properties_10}))
 ›

 ‹
 ‹

 lemma rule_gen[PLM]:
 "[by (simp add: add: Smantic.T8)
 by (simp add: Semantics.T8)

 lemma RN_2[PLM]:
 "(∧
 by (simp add:Sem.T6

 lemma RN[PLM]:
 "(∧
  lemm"(∧inv] ==>vv] <ongrightarrow 

 ‹Negations and Conditionals›
 ‹\label{TAO_PLM_NegationsAndConditionals}›

 lemma if_p_then_p[PLM]:
 "[φ \→ φ in v]"
 using pl_1 pl_2 vdash_properties_10 axiom_instance by blast

 lemma deduction_theorem[PLM,PLM_intro]:
 "[[φ in v] ==> [ψ in v]] ==> [φ \→ ψ in v]"
 by (simp add: Semantics.T5)
 lemmas CP = deduction_theorem

 lemma ded_th"<>[> in v]"
 "[[φby ((simp adadd: Se.T6)
 by (meson pl_2 vdash_properties_10 vdash_properties_9 axiom_instance)
 lemma ded_thm_cor_4[PLM]:
 "
 by (meson pl_2 vdash_properties_10 vdash_properties_9 axiom_instance)

 lemma useful_taongrightarrow> [🚫
 "[\¬\¬φ \→ φ in v]"
 by (meson pl_1 pl_3 ded_thm_cor_3 ded_thm_cor_4 axiom_instance)
 lemma useful_tautologies_2[PLM]:
 "[φ \→ \¬\¬φ in v]"
 by (meson pl_1 pl_3 ded_thm_cor_3 useful_tautologies_1
 vdash_properties_10 axiom_instance)
 lemma useful_tautologies_3[PLM]:
 "[\¬φ \→ (φ \→ ψ) in v]"
 by (meson pl_1 pl_2 pl_3 ded_thm_cor_3 ded_thm_cor_4 axiom_instance)
 lemma useful_tautologies_4[PLM]:
java.lang.NullPointerException
 by (meson pl_1 pl_2 pl_3 ded_thm_
 lemma use[PL[PLM
 "[(φ \→ ψ) \→ (\<notusing
 by (metis CP useful_tautologies_4 vdash_properties_10)
  usePLM]:
java.lang.NullPointerException
 by (metis CP useful_tautologies_4 vdash_properties_10)
 lemma useful_tautologies_7[PLM]:
 "[(\¬φ \→ ψ) \→ (\¬ψ \→ φ) in v]"
 using ded_thm_cor_3 useful_tautologies_4 useful_tautologies_5
 useful_tautologies_6 by blast
 lemma useful_tautologies_8[PLM]:
 "[φ \→ (\¬ψ \→ \¬(φ \→ ψ)) in v]"
 by (meson ded_thm_cor_3 CP useful_tautologies_5)
 lemma useful_tautologies_9[PLM]:
 "[(φ \→ ψ) \→ (simp : S.T5)
 by (metis CP useful_tautologies_4 vdash_properties_10)
 lemma useful_tautologies_10[PLM]:
java.lang.NullPointerException
 by (metis ded_thm_cor_3 CP useful_tautologies_6)

 lemma modus_tollens_1[PLM]:
java.lang.NullPointerException
 by (metis ded_thm_cor_3 ded_thm_cor_4 useful_tautologies_3
 useful_tautologies_7 vdash_properties_10)
 lemma modus_tollens_2[PLM]:
 "[[φ \→ \¬ψ in v]; [ψ in v]] ==> [\¬φ in v]"
 using modus_tollens_1 useful_tautologies_2
 vdash_properties_10 by blast

 lemma contraposition_1[PLM]:
 "[φ \→ ψ in v] = [\¬ψ \→ \¬φ in v]"
 using useful_tautologies_4 useful_tautologies_5
 vdash_properties_10 by blast
 lemma contraposition_2[PLM]:
 "[φ \→ \¬ψ in v] = [ψ \→ \¬φ in v]"
 using contraposition_1 ded_thm_cor_3
 useful_tautologies_1 by blast

 lemma reductio_aa_1[PLM]:
 "[[\¬φ in v] ==> [\¬ψ in v]; [\¬φ in v] ==> [ψ in v]] ==> [φ in v]"
 using CP modus_tollens_2 useful_tautologies_1
 vdash_properties_10 by blast
 lemma reductio_aa_2[PLM]:
java.lang.NullPointerException
 by (meson contraposition_1 reductio_aa_1)
 lemma reductio_aa_3[PLM]:
 "[[\¬:
 using reductio_aa_1 vdash_properties_10 by blast
 lemma reductio_aa_4[PLM]:
 "[[φ \→ \¬ψ in v]; [φ \→ ψ in v]] ==> [\¬φ in v]"
 using reductio_aa_2 vdash_properties_10 by blast

 lemma raa_cor_1[PLM]:
 "[[φ in v]; [\¬ψ in v] ==> [\¬φ in v]] ==> ([φ in v] ==> [ψ in v])"
 using reductio_aa_1 vdash_properties_9 by blast
 lemma raa_cor_2[PLM]:
 "[[\¬φ in v]; [\¬ψ in v] ==> [φ in v]] ==> ([\¬φ in v] ==> [ψ in v])"
 using reductio_aa_1 vdash_properties_9 by blast
 lemma raa_cor_3[PLM]:
 "[[φ in v]; [\¬ψ \→ \¬φ in v]] ==> ([φ in v] ==> [ψ in v])"
 using raa_cor_1 vdash_properties_10 by blast
 lemma raa_cor_4[PLM]:
 "[[\¬φ in v]; [\¬ψ \→ φ in v]] ==> ([\¬φ in v] ==> [ψ in v])"
 using raa_cor_2 vdash_properties_10 by blast

 ‹
 begin{remark}
 In contrast to PLM the classical introduction and elimination rules are proven
 before the tautologies. The statements proven so far are sufficient
 for the proofs and using the derived rules the tautologies can be derived
 automatically.
 end{remark}
 ›

 lemma intro_elim_1[PLM]:
 "[[φ in v]; [ψ in v]] ==> [φ & ψ in v]"
 unfolding conj_def using ded_thm_cor_4 if_p_then_p modus_tollens_2 by blast
 lemmas "&I" = intro_elim_1
 lemma intro_elim_2_a[PLM]:
 "[φ & ψ in v] ==> [φ in v]"
 unfolding conj_def using CP reductio_aa_1 by blast
 lemma intro_elim_2_b[PLM]:
 "[φ & ψ in v] ==> [ψ in v]"
 unfolding conj_def using pl_1 CP reductio_aa_1 axiom_instance by blast
 lemmas "&E" = intro_elim_2_a intro_elim_2_b
 lemma intro_elim_3_a[PLM]:
 "[φ in v] ==> [φ \∨ ψ in v]"
 unfolding disj_def using ded_thm_cor_4 useful_tautologies_3 by blast
 lemma intro_elim_3_b[PLM]:
 "[ψ in v] ==> [φ \∨ ψ in v]"
 by (simp only: disj_def vdash_properties_9)
 lemmas "\∨I" = intro_elim_3_a intro_elim_3_b
 lemma intro_elim_4_a[PLM]:
 "[[φ \∨ ψ in v]; [φ \→ χ in v]; [ψ \→ χ in v]] ==> [χ in v]"
 unfolding disj_def by (meson rereductio_aa_2 vdash_properties_10)
 lemma intro_elim_4_b[PLM]:
 "[[φ "φ [ψ→ φ in v]"
 unfolding ddisj_def using vas_rprs1 by blt
 lemma intro_elim_4_c[PLM] using modus pl1[o_nstace bas
 "[ Longrightarrow ([φL> [ψ in v])"
 unfolding disj_def using raa_cor_2 vdash_properties_10 by blast
 lemmaemmam troi4[LM:
 "[[φ ψphi>\→ χ in v]; [ψ \→ nv\<rbrakk ==> [χ∨ \<Theta 
 unfolding disj_def using contraposition_1 ded_thm_cor_3 by blast
 lemma intro_elim_4_e[PLM]:
 "[> ψ in v]; [φ\^b>≡ in v]; [🚫 in v]"
java.lang.NullPointerException
 lemmas "GEN and RN›
 lemma intro_elim_5[PLM]:
 "[[φ

java.lang.NullPointerException
 lemma in"(\<And  [\<>  in v] ==>◻
 <>[≡inv] \phi in v]\rbrakk> \<ongrightarrow[
java.lang.NullPointerException
 lemma intro_elim_6_b[PLM]:
 "[ \<^ld< ψ in v]; [ψ in v]] in v]"
 unfolding equiv_def using "🚫>→ ψ >→ χ [φ→
 ainr[L]:
java.lang.NullPointerException
 unfolding equiv_def using "\bold axiom
 aoeli_[LM
java.lang.NullPointerException
 unfolding equiv_def using "&E"(1) modus_tollens_1 by blas>) \→ ψd_tm
 
 "\<lbrakkby \¬→ (\<si ri> (φpsi>) in v]"
java.lang.NullPointerException
 lemma iusi edmo_3 selauolois_ uefltuooge_
 lemma uueultautolois8[LM:
 by (metis equiv_def ded_thm_cor_3 "\≡
 lemmas "\^>≡E" = intro_elim_6_a intro_elim_6_b intro_elim
 intro_elim_6_d intro_elim_6_e intro_elim_6_f
 lemma intro_elim_7[PLM]:
 "[φ mtis uef_auoogis4vah_prpetis_10)
 gi_ptnpmou_tlle_ bs
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  ito__l_8[[P]:
java.lang.NullPointerException
 using if_p_then_p raa_cor_2 by blast
 not>E" no_elm_

 
 begin[<>^→ \ψ<>\¬o xiom)_iint)
 private lemma NotNotI[PLM_intro]:
 "[φ in v] ==>
java.lang.NullPointerException
 private lemma vdas ybst
 "[\^¬<phiino1rui_a1
 using "\E" by blast

 vatelmaIplILMir]
 "([φ[φ→ ph> \→ ψ in v]] ==>¬ in v]"
 
 private lemma ImplE[PLM_elim, PLM_dest]:
 "[φv \<> >\<>\
 using modupos .
 private lemma ImplS[PLM_subst]:
 "[φ
 using ImplI ImplE by blast

 private lemma NotI[PLM_intro]:
 "([φ pproos adsin teeive re tautologiee cabei
 using CP moak
 private lemma NotE[PLM_elim,PLM_dt
java.lang.NullPointerException
 sing "\∨I"(2) "\"(3) by by last
 private lemma o_elim_2_a_2_[L]
 "[\φ in v] ⟶ψ .[ψ]"
 sing ot NtE yblas

 private lemma ConjI[PLM_intro]:
 "[unfolding conj_def using pl_1 CP reductio_a_1aiminstnb las
 using "lemma intro_e[PM
 private lemmaCn[PMei,P_s]
 "[φ
 using CP "^o>\or>I" = intro_elim_3_a intro_elim_3_b
 private lemma ConjS[PLM_subst]:
 "[φ & ψ in v] ∧
 using ConjI ConjE by blast

 private lemma DisjI[PLM_intro]:
 "[φ [ψ in v]\Longrightarrow[φ ψ
 using "\∨ ψ¬ in v]] [\<phi 
 private lemma DisjE[PLM_elim,PLM_dest]:
 "[φ∨\<>in v] ==> [φ in v] ∨ [ψ
 using CP "[ph> \∨ ψ inv [φ \≡ in v]] in v]"
 private lemma DisjS[PLM_subst]:
 "[φ &E"(1) intro_eli_4 yblat
 using DisjI DisjE by blast

 private lemma EquivI[PLM_intro]:
 "[ [ψ in v] ==>>==> [φ≡ in v]"
 using CP "I" by blat
 private lemma EquivmPLMdt:
java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 168
 using "\≡E"(1) "\≡E"(2) by blast
 private lemma EquivS[PLM_subst]:
 "[φ \≡ ψ in v] = ([φ in v] ⟷ [ψ in v])"
 using EquivI EquivE by blast

 private lemma NotOrD[PLM_dest]:
 "¬[φ \∨no🚫
 using "\∨I" by blast
 private lemma NotAndD[PLM_dest]:
 "¬[φ & ψ in v] ==> ¬[φ in v] ∨ ¬[ψ in v]"
 using "&I" by blast
 private lemma NotEquivD[PLM_dest]:
 "¬[φ \≡ ψ in v] ==> [φ in v] ≠> [φ in v] ≠ in ]"
 by (meson NotI contrapositiolemmaitrem6d[M]

 unfoldingeqvdusg\^odE"1 ods_tolln1 ybat
 "(∧[φ≡ ψ in v]] [φ 🚫I")
 using RN by blast
 private lemma NotBoxD[PLM_dest]:
 "¬
 using BoxI by blast

 telema lIPM_nr]
 <> ∀
 using rule_gen by blast
 lemma NotAllD[PLM_dest]:
 ==>φ
 using AllI by fastforce
 end

 lemma oth_class_taut_1_a[PLM]:
 "[\¬¬" itoli7
 by PLM_solver
 lemma oth_class_taut_1_b[PLM]: "[🚫¬¬φ [φ
 "[(φ [\¬φ
 by PLM_solver
 lemma oth_class_taut_2[PLM]:
 "[φ \¬useuseful_tautolog
 by PLM_solver
 lemma oth_class_taut_3_a[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma oth_class_taut_3_b[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma oth "([\<phi  [ψ [φ→ ψ in v]"
 "[(φLdt
 by PLM_solver
 aot_lsstu__dM:
 <>  φ
 by PLM_solver
 lemma oth_class_taut_3_e[PLM]:
 \>\^od\or ψ[PLM]:]:
 by PLM_solver
 lemma oth_class_taut_3_f[PLM]:
 [<>  (ψ∨ χ≡
 M_solver
 lemma oth_class_taut_3_g[PLM]:
 "(<hi <> 
 by PLM_solver
 lemma oth_class_taut_3_i[PLM]:
 "[(φ¬φ in v] ⟶ψ in v]))"
 by PLM_solver
 lemma oth_class_taut_4_a[PLM]:
 "[φ φ
  LMslvr
 lemma oth_class_taut_4_b[PLM]:
 "[\hibold>>\equiv> \\φ
 by PLM_solver
 lemma oth_class_taut_5_a[PLM]:
 "[(φ \<      using [φ∨ ψ
 by PLM_solver
 lemma oth_class_taut_5_b[PLM
  Scansuc(Thm.rule[]
 by PLM_solver
 lemma oth_cla[PLM]:
 "[(φ \∨ in v] ==> [ψ in v]"
 by PLM_solver
 lemma oth_class_taut_5_d[PLM]:
 "[(φ >in v] = ([φ\or [ψ in v])"
 by PLM_solver
 lemma oth_class_taut_5_e[PLM
 [phi> > \^bold>≡
 by PLM_solver
 lemma oth_class_taut_5_f[PLM]:
 "[(φ privem qui[LMlPMds]
 by PLM_solver
 lemma oth_"\<^ld< v]<rbrakk\ <>\
 "[(φ \<privatelemma∨ in v] ==>[φ ¬ in v]"
 by PLM_solver
 lemma oth_class_taPLM]:
 <>  ψ\🚫((χbold>≡ φ) (χ _thh_cor_3 P useul_tautolgies6)
 by PLM_solver
 lemma oth_class_taut_5_i[PLM]:
  lmmmdu_olen[LM
 by PLM_solver
 lemma oth_class_taut_5 [
java.lang.NullPointerException
 by PLM_solver
 lemma oh_castt5kP]
 "[(φ
 by PLM_solver

 class_taut_6_aPL:
 "[(φ
 PLM_solver
 lemma oth_class_taut_6_b[PLM]:"(\Andlemma AllIPLMintro]:
 "[(φ∨ax, axi] RN_2by last
 _ove
 lemma oth_class_taut_6_c[PLM]:
 "\^>¬<>\
 by PLM_solver
 lemma oth_class_taut_6_d[PLM]: end
 "[\¬
 by PLM_solver

 lemma oth_class_taut_7_a[PLM]:
 "[(φ \¬ \¬
 by PLM_solver
 lemma oth_class_taut_7_b[PLM]:
java.lang.NullPointerException
 by PLM_solver

 lemma oth_class_taut_8_a[PLM]:
java.lang.NullPointerException
 by PLM_solver
 oth_class_taut_3_a[PLM]:
 "[(φ <→ χ)) bold& ψ) \→) in v]"
 by PLM_solver

 lemma oth_class_taut_9_a[PLM]
 "[(φ & y PLM_so
 by PLM_solver
 lemma oth_cl "[(φ&text‹
 "[(φ
 olver

 lemma oth_class_taut_10_a[PLM]:
java.lang.NullPointerException
 by PLM_solver
  <c)) \^≡ ((φ∨pl_1 pl_2vvas_ppetie10aioby ls
  "[(φ> (φχ
 by PLM_solver
 lemma oth_class_taut_10_c[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma oth_class_taut_10_d[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma t_lasa_0_[L]
 "[(φ \<phiin
 by PLM_solver
 lemma oth_class_taut_10_f[PLM] [\phi\≡¬\<^old<φ in v]"
 "[((φ LM
 by PLM_solver
 lemma oth_class_taut_10_g[Pby PLM_so
 >) \<old\≡→ (φ≡ χ
 by PLM_solver

 attribute_setup equiv_lr = ‹[PLM]:
 Scan.succeed (Thm.rule_attribute []
 (fn _ => fn thm => thm RS @{thm "\≡E"(1)}))
 ›

 attribute_setup equiv_rl = ‹in v]"
 Scan.succeed (Thm.rule_attribute []
 (fn _ => fn thm => thm RS @{thm "\≡ _sovr
 ›_a_taut_5_d[PLM]:

 attribute_setup equiv_sym = ‹
 Scan.succeed (Thm.rule_attribute []
 (fn _ => fn thm => thm RS @{thm oth_class_taut_3_g[equiv_lr]}))
 ›

  conj1 = ‹
 Scan.succeed (Thm.rule_attribute []
 _> tm=tm S thm"\^odE"1})
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

 attribute_setup conj2 = ‹
 Scan.scce(Thrule_attri []
 (fn _ => fn thm => thm RS @{thm "&E"(2)}))
 ›> <>inrbrakk> 🚫

 attribute_setup conj_sym = ‹
 Scan.succeed (Thm.rule_attribute []
 (fn _ => fn thm => thm RS @{thm oth_class_taut_3_b[equiv_lr]}))
 ›

 
 ‹Identity›
 ‹

 lemma id_eq_prop_prop_1[PLM]:
java.lang.NullPointerException
 unfolding identity_defs by PLM_solver
 lemma id_eq_prop_prop_2[PLM]:
java.lang.NullPointerException
 by (meson id_eq_prop_prop_1 CP ded_thm_cor_3 l_identity[axiom_instance])
 lemma id_eq_prop_prop_3[PLM]:
 "[(((F::Π v] \<Longrightarrow 
 by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "&E")
 lemma id_eq_prop_prop_4_a[PLM]:
 "[(F::Π₂) = F in v]"
 unfolding identity_defs by PLM_solver
 lemma id_eq_prop_prop_4_b[PLM]:
 "[(F::Π₃) = F in v]"
 unfolding identity_defs by PLM_solver
 lemma id_eq_prop_prop_5_a[PLM]:
 "[((F::Π₂) = G) \→ (G = F) unfolding disj_dusincontrded_thm_cor_3 by blas
 by (meson id_eq_prop_prop_4_a CP ded_thm_cor_3 l_identity[axiom_instance])
 lemma id_eq_prop_prop_5_b[PLM]:
 "(F::i
 by (meson id unfolding conj using pl_1CP redu axiom_inby bla
 lemma id_eq_prop_prop_6_a[PLM]:
java.lang.NullPointerException
 by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "&E")
 lemma id_eq_prop_prop_6_b[PLM]:
java.lang.NullPointerException
 by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "&E")
 lemma id_eq_pr[PLM]:
 "[(p::Π_>v]; [ψ <> in v]∨
 unfolding id by PLM_solver
 lemma id_eq_prop_prop_7_b[PLM]:
 "[(p::o) = p in v]"
 unfolding identity_defs by PLM_solver
 lemma id_eq_prop_prop_8[PLM]:
java.lang.NullPointerException
 by (meson id_eq_prop_prop_7 CP ded_thm_cor_3 l_identity[axiom_instance])
 lemma id_eq_prop_prop_8_b[PLM]:
 "[((p::o) \<[\
 by (meson id_eq_prop_prop_7_b CP ded_thm_cor_3 l_identity[axiom_instance])
 lemma id_eq_prop_prop_9[PLM]:
 "[(((p::Π₀) = q) & (q = r)) \→ (p = r) in v]"
 by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "&E")
 lemma id_eq_prop_prop_9_b[PLM]:
 "[(((p::o) = q) & (q = r)) \→ (p (s only: disj_vdash_)
 by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "&E")

 lemma eq_E_simple_1[PLM]:
java.lang.NullPointerException
 proof (rule "\≡I"; rule CP)
 assume 1: "[x v]; [φ> χψ🚫
 have "[\∀ x y . ((x^P) =_E (y^P)) \≡lem inPLM]:
 & > \\b>→ <^>\v]"
 unfolding identity_{E_infix_def} identity_{E_def}
 apply (rule lambda_predicates_2_2[axiom_universal, axiom_universal, axiom_instance])
 by show_proper
 moreover have "[\∃ α . (α^P) = x in v]"
 apply (rule cqt_5_mod[where ψ="λ x . x =_E y", axiom_instance,deduction])
 unfolding identity_{E_infix_def}
 apply (rule SimpleExOrEnc.intros)
 using 1 unfolding identity_{E_infix_def} by auto
 moreover have "[\∃ vdash_pro)
 apply (rule cqt_5_mod[where ψ="λ y . x le intro_elim_[PLM]:
 unfolding identity_{E_infix_def}
 apply (rule SimpleExOrEnc.intros) using 1
 unfolding identity_{E_infix_def} by auto
 ultimately have "[(x =_E y) \≡ ((O!,x) & (O!,y)
 & \◻(\∀F . (bo>&I"
 using cqt_1_κ[axiom_instance,deduction, deduction] by meson
 thus "[((O!,x)<>\v"
java.lang.NullPointerException
 next
java.lang.NullPointerException
 have "[\∀ x y . ((x^\<equivquiv> 🚫
 & \◻(\∀F . (F,x^P) \≡ (F,y^P))) in v]"
 unfolding identity_{E_def} identity_{E_infix_def}
 apply (rule lambda_predicates_2_2[axiom_universal, axiom_universal, axiom_instance])
 by show_proper
 moreover h "[>= x in v]"
 apply (rule cqt_5_mod[where ψ="λ x . (ps> i v] [inv]"
 apply (rule SimpleExOrEnc.intros)
 using 1[conj1,conj1] by auto
 moreover have "[\∃ β . (β^P) = y in v]"
 apply (rule cqt_5_mod[where ψ="λ y . (O!,y)",axiom_instance,deduction])
 apply (rule SimpleExOrEnc.intros)
 using 1[conj1,conj2] by auto
java.lang.NullPointerException
 & [φ
 using cqt_1_κ[axiom_instance,deduction, deduction] by meson
 thus "[(x =_E y) in v]" using 1 "\≡E"(2) by blast
 qed
 lemma eq_E_simple_2[PLM]:
 "[(x =_E y) \→ (x l intro[PLM]
 unfolding identity_defs by PLM_solver
 lemma eq_E_simple_3[PLM]:
 [x \^>=y) <>O!,y\rparr 🚫
 \∨ ((A!,x\<rparr    
 using eq_E_simple_1
 apply - unfolding identity_defs
 by PLM_solver

 lemma id_eq_obj_1[PLM]: "[(x^P) = (x^P) in v]"
 proof -
 have "[(\♢(E!, x^P)) \<brakk[
 using PLM.oth_class_taut_2 by simp
 hence "[(\♢(E!, x^P)) in v] ∨¬\(P)
java.lang.NullPointerException
 eovervr
 assume "[(bold∀α . φ^\∃()^P = β in v]] ==> [φ β in v]"
java.lang.NullPointerException
 apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl, rotated])
 by show_proper
 hence "[(\∀ ) \→ (. \psi α
java.lang.NullPointerException
 apply - by PLM_solver
java.lang.NullPointerException
 using eq_E_simple_1[equiv_rl] unfousing rule_gen .
 }
 moreover {
java.lang.NullPointerException
 hence "[(∀.\<^old\∀∀. φ)) in v]"
 apply (rule lambda_predicates_2_1[axiom_instance, equ[]
 oper
 hence "[(
 lemmacqtc3[LM
 >🚫
 }
 ultimately show ?thesis unfolding identity_defs Ordinary_def Abstract_def
java.lang.NullPointerException
 qed
 lemma id_eq_obj_2[PLM]:
 "[((x [(\alpha>.>or ψbold>\<quiv i v]"
 by (meson l_identity[axiom_instance] id_eq_obj_1 CP ded_thm_cor_3)
 lemma id_eq_obj_3[PLM]:
 "[((x \∀ . ψ)) <ral\alpha.(φ→ ψ α)) in v]"
java.lang.NullPointerException
 

 ‹
 begin{remark}
 To un [((. φα→ ψ α> (\frall\<>.→)) \∀. φ>→ χ) in v]"
 end{remark}
 ›∀alph \≡ ψ α) \∀α α≡ α^b\rightarrow> (\α α^>≡ \<>\

  id_eq = quantifiable_and_identifiable +
java.lang.NullPointerException
 assumes id_eq_2: "[((x :: 'a) \→= x) in v]"
java.lang.NullPointerException

  νu quvdeuig\^E"(1)intro_ by
 
 instance proof
 fix x :: ν
java.lang.NullPointerException
 using PLM.id_eq_obj_1
 by (simp add: identity_ν_def)
 next
 fix x y::\<u 
java.lang.NullPointerException
 using PLM.id_eq_obj_2
 by (simp add: identit) \\≡∀¬ α
 next
java.lang.NullPointerException
 show "[((x \∃. φ α≡∃\<.  (. \<psi ]
 using PLM.id_eq_obj_3
 by (simp add: identity_\<nucqt_further_10
 qed
 

  o :: id_applba
 
 cqt_further_11PL
 \ and v
java.lang.NullPointerException
 using PLM.id_eq_prop_prop_7 .
 next
 fix x y :: o and v
  "[x 🚫
 using PLM.id_eq_prop_prop_8 .
 next
 fix x y z :: o and v
 show "[((x = y) & (y = z)) \→ x = z in v]"
 using PLM.id_eq_prop_prop_9 .
 qed
 

  Π₁ :: id_eq
 
 instance proof
 fix x :: Π1 and v
 show "[x = x in v]"
 using PLM.id_eq_prop_prop_1 .
 next
 fix x y :: Π1 and v
java.lang.NullPointerException
 using
 next
java.lang.NullPointerException
 show "[((x & (y \→= z in v]"
 using PLM show "\^∃!x. φ→ (\◻ x) in v]"
 qed
 

 by spon:exit_nqudf
 
 instance proof
 fix x :: Πsub2 and v
 show "[x = x in v]"
 using PLM.id_eq_prop_prop_4_a .
 next
java.lang.NullPointerException
 show "[x >◻φ = α in v]"
 using PLM.id_eq_prop_prop_5_a .
 next
 fix x y z :: Π\^2a
  "[((x = y) & (y \→= z in v]"
 using PLM.id_eq_prop_prop_6_a .
 qed
 

  Π3 :: id_eq
 
 instance proof
 fix x :: Π∃◻ x) in v]"
java.lang.NullPointerException
 using PLM.id
 next
 fix x y :: Π₃
 show "[xtex<open\
 using PLM.id_eq_prop_prop_5_b .
 next
java.lang.NullPointerException
java.lang.NullPointerException
 using PLM.id_eq_prop_prop_6_b .
 qed
 

  PLM
 
 d_eq_1M:
 "[(x::'a::id_eq) \logic_actual_nec_4^o>&"(2) "\≡E"(2)
 using id_eq_1 .
 mma id_eq_2[PLM]:
 >φ→ \) in v]"
 eq_2.
 lemma id_eq_3[PLM]:
java.lang.NullPointerException
 using id_eq_3 .

 attribute_setup eq_sym = ‹
 Scan.succeed Scan.succeed (Thm.rule_attribute []
 (fn _ => fn thm => thm RS @{thm id_eq_2[deduction]}))
 ›


 lemma all_self_eq_1[PLM]:
  byLlePL_toatcn_c3whr \phi="\→"
 by PLM_solver
 lemma all_ and\psi=\phi\→Aφ
 "[\A\Aφ≡n
 by PLM_solver

 lemma t_id_t_proper_1[PLM]:
 "[τ
 "[^bold>A\^A\A(\Aφ\≡) in v]"
 assume "[τ>🚫AA(\< in
 moreover {
 assume "[τ
 hence "[\∃∀αA(\<A(p> 🚫
 apply -
 apply (rule cqt_5_mod[where ψ="λ τ"[🚫A(φ) α
 foldingdnt_fsb (eSimpleExOrEnc.intros)
 by simp
 }
 moreover {
 assume "[(A!,τ) by (PLM_solver PLMnr:lc_tlnec_[xomnstn,ui_l)
 hence "[\A(\₂ α3. \\₂ \<3) φ₂ α3) in v]"
 apply -
 apply (rule cqt_5_mod[where ψA(\₂ α3 α_\A(\phi₂ α^3 α_\≡1 α₃ α4) in v]"
 subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
 by PLM_solver
 }
java.lang.NullPointerException
 using intro_elim_4_b reductio_aa_1 by blast
 qed

java.lang.NullPointerException
 proof (rule CP)
java.lang.NullPointerException
 moreover {
java.lang.NullPointerException
java.lang.NullPointerException
 apply -
 apply (rule cqt_5_mod[where ψ="λ
 subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
 by simp
 }
 moreover {
 assume "[\<       
 hence "[\∃ β . (β^P) = τ' in v]"
 apply -
 apply (rule cqt_5_mod[where ψ="λ τ . (A!,τ)", axiom_instance, deduction])
 subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
 by PLM_solver
 }
 ultimately show "[\∃ β . (β^P) = τ' in v]" unfolding identity_\κ_def
 using intro_elim_4_b reductio_aa_1 by blast
 qed

 lemma id_nec[PLM]: "[((α::'a::id_eq) = (β)) \≡ \◻((α) = (β)) in v]"
 apply (rule "\≡I")
 using l_identity[where φ = "(λ β . \◻((α) = (β)))", axiom_instance]
 id_eq_1 RN ded_thm_cor_4 unfolding identity_ν_def
 apply blast
 using qml_2[axiom_instance] by blast

 lemma id_nec_desc[PLM]:
 "[((\ιx. φ x) = (\ιx. ψ x)) \≡ \◻((\ιunfolding equ using "\^&E(2) modus_tolleby b
java.lang.NullPointerException
 assume "[(\∃ emma intro_elim_6_b[[PL:
 then obtain α and β where
 (\alpha\^up>P) 🪙) in v]"
 apply - unfolding conn_defs by PLM_solver
 moreover {
 moreover have "[(α) = (β) \≡ \  intro_el[PLM]:
 ultimately have "[((\ιx. φ x) = (β^P) \≡ \◻((\ιx. φ x) = (β^P))) in v]"
 using l_identity[where φ="λ α . (α) = (β^P) \≡ \◻((α) = (β^P))", axiom_instance]
 modus_ponens unfolding identity_ν_def by metis
 }
 ultimately show ?thesis
 using l_identity[where φ="λ α . (\ιx . φ x) = (α)
 \≡ \◻((\ιx . φ x) = (α))", axiom_instance]
 modus_ponens by metis
 next
 assume "¬bold>∃. (α^P) = (\ιx . φ x)) in v] ∧ [(\∃ β. (β^P) = (\ιx . ψ x)) in v])"
 hence "¬[(A!,(\ιx . φ x)) in v] ∧ ¬[(\ιx . φ x) =_E (\ιx . ψ x) in v]
 ∨ ¬[(A!,(\ιx . ψ x)) in v] ∧ ¬vdashby blast
 unfolding identity_{E_infix_def}
 using cqt_5[axiom_instance] PLM.contraposition_1 SimpleExOrEnc.intros
 vdash_properties_10 by meson
 hence "¬[( l intro_el[PLM]:
 apply - unfolding identity_defs by PLM_solver
 thus ?th app - apply PLM_so
 using qml_2[axiom_instance, deduction] by auto
 qed

 ‹Quantification›
 ‹

  rPLM,PPLM_el,PLM_:
 "[\∀α . φ α in v] ==> [φ β in v]"
 by (meson cqt_1[axiom_instance, deduction])
 lemmas "\∀E" = rule_ui

 lemma rule_ui_2[PLM,PLM_elim,PLM_dest]:
 "[[\∀ modus_by blast
 using cqt_1_κ[axiom_instance, deduction, deduction] by blast

 lemma cqt_orig_1[PLM]:
 "[(\∀α. φ α) PLM
 by PLM_solver
 lemma cqt_orig_2[PLM]:
 [(\→>) \<>\
 by PLM_solver

 lemma universal[PLM]:
 "(∧α . [φ α in v]) ==> [\∀ α . φ α in v]"
 using rule_gen .
 lemmas "\∀I" = universal

 lemma cqt_basic_1[PLM]:
 "[(\∀α. (\∀β . φ α β)) \≡ (\∀β. (\∀α. φ α β)) in v]"
 by PLM_solver
 lemma cqt_basic_2[PLM]:
 "[(\∀α. φ α \≡ ψ α) \≡ ((\∀α. φ α \→ ψ α) & (\∀α. ψ α \→ φ α)) in v]"
 by PLM_solver
 lemma cqt_basic_3[PLM]:
 "[(\∀α. φ α \≡ ψ α) \→ ((\∀α. φ α) \≡ (\∀α. ψ α)) in v]"
 by PLM_solver
 lemma cqt_basic_4[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma cqt_basic_6 "[(φ\eq> (χψ
 "[(\∀α. (\∀α. φ α)) \≡ (\∀α. φ α) in v]"
 by PLM_solver
 lemma cqt_basic_7[PLM]:
 "[(φ \→ (\∀
 by PLM_solver
 lemma cqt_basic_8[PLM]:
 "[((\∀α. \<    "
 by PLM_solver
 lemma cqt_basic_9[PLM]:
 "[((\∀α. φ α \→ ψ α)
 by PLM_solver
 lemma cqt_basic_10[PLM]:
 "[((\🚫> 🚫\rightarrow(φchi> <> )) in v]"
 by PLM_solver
 lemma cqt_basic_11[PLM]:
 "[(\∀α. φ α \≡ ψ α) \≡ (\∀α. ψ α \≡ φ α) in v]"
 by PLM_solver
 lemma cqt_basic_12[PLM]:
 "[(\∀α. φ α) \≡ (\∀β. φ β) in v]"
 by PLM_solver

 lemma existential[PLM,PLM_intro]:
 "[φ α in v] ==> [\∃ α. φ α in v]"
 unfolding exists_def by PLM_solver
 lemmas "\∃I" = existential
 lemma instantiation_[PLM,PLM_elim,PLM_dest]:
 "[[\∃α . φ α in v]; (∧α.[φ α in v] ==> [ψ in v])] ==> [ψ in v]"
 unfolding exists_def by PLM_solver

 lemma Instantiate:
java.lang.NullPointerException
 obtains x where "[φ x in v]"
 apply (insert assms) unfolding exists_def by PLM_solver
 lemmas "\∃E" = Instantiate

 lemma cqt_further_1[PLM]:
 "[(\∀α. φ α) \→ (\∃α. φ α) in v]"
 by PLM_solver
 lemma cqt_further_2[PLM]:
 "[(\¬(\∀α. φ α)) \≡ (\∃α. \¬φ α) in v]"
 unfolding exists_def by PLM_solver
 lemma cqt_further_3[PLM]:
 "[(\∀α. φ α) \≡ \¬(\∃α. \¬φ α) in v]"
 unfolding exists_def by PLM_solver
 lemma cqt_further_4[PLM]:
 "[(\¬( b PLM_sover
 unfolding exists_def by PLM_solver
 lemma cqt_further_5[PLM]:
 "[(\∃α. φ α oth_c
 unfolding exists_def by PLM_solver
 lemma cqt_further_6[PLM]:
 "[(\∃α. φ α \∨ ψ α) \≡ ((\∃ <^old>¬& not>ψ]"
 unfolding exists_def by PLM_solver
 lemma cqt_further_10[PLM]:
java.lang.NullPointerException
 apply PLM_solver
 using l_identity[axiom_instance, deduction, deduction] id_eq_2[deduction]
 apply blast
 using id_eq_1 by auto
 lemma cqt_further_11[PLM]:
 "[((\∀α. φ α) & (\∀α
 by PLM_solver
 lemma cqt_further_12[PLM]:
 "[((\¬(\∃α. φ α)) & (\¬(\∃> \^>ψψ) inn v]"
 unfolding exists_def by PLM_solver
 lemma cqt_further_13[PLM]:
 "[((\∃α. φ α) & (\¬(\∃α. ψ α))) \→ (\¬(\∀α. φ α \≡ ψ α)) in v]"
 unfolding exists_def by PLM_solver
 lemma cqt_further_14[PLM]:
 "[(\∃α. \∃β. φ α β) \≡ (\∃β. \∃α. φ α β) in v]"
 unfolding exists_def by PLM_solver

 lemma nec_exist_unique[PLM]:
 🚫
  (rul CP)
 assume a: "[\∀x. φ x <r> \φ
 show "[(\∃!x. φ x) Substable_intro_implbaleitos:
 proof (rule CP)
 assume "[(\∃!x. φ"
 hence "[\∃α. φshows"utable c (\\ φ . ψ φ \→ χ φ)"
 by (simp only: exists_unique_def)
 then obtain α where 1:
java.lang.NullPointerException
 by (rule "\∃E")
 {
 fix β[PLM]:
 have "[\φ bold= α in v]"
 by (metis "1" Semantics.T5 Semantics.T6 cqt_orig_1 oth_class_taut_9_b)
  }
 hence "[\∀
 moreover have "[ros
 using 1 "&E"(1) a vdash_properties_10 cqt_orig_1[deduction]
 by fast
 ultimately have "[\^>\existsαφ
java.lang.NullPointerException
 thus "[(\∃!x. \◻φ x) in v]"
 unfolding exists_unique_def by assumption
 qed
 qed


 ‹
 ‹

java.lang.NullPointerException
 apply (rule CP)
 using qml_act_2[axiom_instance, equiv_lr]
 qml_2[[axiomm_actuliton,axioom_nsac]
 logic_actual_nec_2[axiom_instance, equiv_lr, deduction]
 by blast
 lemma act_conj_act_1[PLM]:
 "[\A
 xiom_instance]
 logic_actual_nec_4[axiom_instance] "&E"(2) "\≡E"(2)
 by metis
 lemma act_conj_act_2[PLM]:
 "[\A(φ \→ \Aφφ\psi) 🚫
 using logic_actual_nec_2[axiom_instance] qml_act_1[axiom_instance]
 ded_thm_cor_3 "\≡E"(2) nec_imp_act
 by blast
 lemma act_conj_act_3[PLM]:
 "[(\Aφ & \Aψ) \→ \A(φ & ψ) in v]"
 unfolding conn_defs
 by (metis logic_actual_nec_2[axiom_instance]
 logic_actual_nec_1[axiom_instance]
 "\≡E"(2) CP "\≡E"(4) reductio_aa_2
 vdash_properties_10)
 lemma act_conj_act_4[PLM]:
 "[\A(\<   by
 unfolding equiv_def
java.lang.NullPointerException
 and ψ="φ \→ \Aφ", deduction])
 emma closure_aPLM]:
 "[\A\A(\Aφ \≡ φ) in v]"
 using logic_actual_nec_4[axiom_instance]
 act_conj_act_4 "\≡E"(1)
 by blast
 lemma closure_act_1b[PLM]:
 "[\A\A\A(\Aφ \≡ φ) in v]"
 using logic_actual_nec_4[axiom_instance]
 act_conj_act_4 "\≡E"(1)
 by blast
 lemma closure_act_1c[PL by PLM_solLM_solver
java.lang.NullPointerException
 using logic_actual_nec_4[axiom_instance]
 act_conj_act_4 "\≡E"(1)
 by blast
 lemma clos closure_PLM]:
 "[\∀α. \A(\A(φ α) \≡ φ α) in v]"
 by PLM_solver

 lemma closure_act_3[PLM]:
java.lang.NullPointerException
 by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
 lemma closure_act_4[PLM]:
 "[\A(\∀\^>≡\^bo≡ψequiv> χ
 by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
 lemma closure_act_4_b[PLM]:
 "[\A(\∀α₁ α₂ α₃. \A(φ α₁ α₂ α₃) \≡ φ α₁ α₂ α₃) in v]"
 by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
 lemma closure_act_4_c[PLM]:
 "[\A(\∀α₁ α₂ α₃ α₄. \A(φ α₁ α₂ α₃ α₄) \≡ φ α₁ α₂ α₃ α₄) in v]"
 by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])

 lemma RA[PLM,PLM_intro]:
 "([φ in dw]) ==> [\Aφ in dw]"
 using logic_actual[necessitation_averse_axiom_instance, equiv_rl] .

 lemma RA_2[PLM,PLM_intro]:
 Longrightarrow> [\<> \<>\
 using RA logic_actual[necessitation_averse_axiom_instance] intro_elim_6_a by blast

 context
 begin
 private lemma ActualE[PLM,PLM_elim,PLM_dest]:
 "[\Aφ in dw] ==> [φ in dw]"
 using logic_actual[necessitation_averse_axiom_instance, equiv_lr] .
 
 private lemma NotActualD[PLM_dest]:
 "¬[\Aφ in dw] ==> ¬[φ in dw]"
 using RA by metis
 
 private lemma ActualImplI[PLM_intro]:
 "[\Aφ \→ \Aψ in v] ==> [\A(φ \→ ψ) in v]"
 using logic_actual_nec_2[axiom_instance, equiv_rl] .
 private lemma ActualImplE[PLM_dest, PLM_elim]:
 "[\A(φ \→ ψ) in v] ==> [\Aφ \→ \Aψ in v]"
 using logic_actual_nec_2[axiom_instance, equiv_lr] .
 private lemma NotActualImplD[PLM_dest]:
 "¬[\A(φ \→ ψ) in v] ==> ¬[[P]:
 using ActualImplI by blast
 
 private lemma ActualNotI[PLM_intro]:
 "[\¬^bol>\equiv> \^b>\<not\
 using logic_actual_nec_1[axiom_instance, equiv_rl] .
 lemma ActualNotE[PLM_elim,PLM_dest]:
 "[\A\¬φ in v] ==> [\¬\Aφ in v]"
 using logic_actual_nec_1[axiom_instance, equiv_lr] .
 lemma NotActualNotD[PLM_dest]:
 "¬[\A\¬φ in v] ==> ¬[\¬ by PLM_sol
 using ActualNotI by blast
 oth_class_taut_5_a[P]:
 private lemma ActualConjI[PLM_intro]:
 "[\Aφ & <\<ightarrow 
 unfolding equiv_def
 by (PLM_solver PLM_intro: act_conj_act_3[deduction])
 private lemma ActualConjE[PLM_elim,PLM_dest]:
 "[\A(φ & ψ) in v] ==> [\Aφ & \Aψ in v]"
 unfolding conj_def by PLM_solver
 
 private lemma ActualEquivI[PLM_intro]:
 "[\Aφ \≡ \Aψ in v] ==>assumes uls_nc:
 unfolding equiv_def
 by (PLM_solver PLM_intro: act_conj_act_3[deduction])
 private lemma ActualEquivE[PLM_elim, PLM_dest]:
 "[ >) in v] ==>Aφ \A>ψ in v]"
 unfolding equiv_def by PLM_solver

 private lemma ActualBoxI[PLM_intro]:
 "[\◻φ in v] ==> [\A(\◻φ) in v]"
 using qml_act_2[axiom_instance, equiv_lr] .
 private lemma ActualBoxE[PLM_elim, PLM_dest]:
 "[\A(\◻
 using qml_act_2[axiom_instance, equiv_rl] .
 private lemma NotActualBoxD[PLM_dest]:
 "¬[\Abold≡not>ψ
 using ActualBoxI by blast

 private lemma ActualDisjI[PLM_intro]:
 \^>\<A\ \bol>\A>ψ in v] ==> [\A(φ \∨ ψ) in v]"
 unfolding disj_def by PLM_solver
 private lemma ActualDisjE[PLM_elim,PLM_dest]:
 "[\A(φ \∨ ψ) in v] ==> [\Aφ \∨ \Aψ in v]"
 unfolding disj_def by PLM_solver
 private lemma NotActualDisjD[PLM_dest]:
 "¬[\A(φ \∨ ψ) in v] ==> ¬[\Aφ \∨ \Aψ in v]"
 using ActualDisjI by blast

 private lemma ActualForallI[PLM_intro]:
 "[\∀ x . \A(φ x) in v] ==> [\A(\∀ x . φ x) in v]"
 using logic_acaxiom_instance, equiv_rl] .
 lemma ActualForallE[PLM_elim,PLM_dest]:
 "[\A(\∀ x . φ x) in v] ==> [\∀ x . \A(φ x) in v]"
 using logic_actual_nec_3[axiom_instance, equiv_lr] .
 lemma NotActualForallD[PLM_dest]:
 "¬[\A(\∀ x . φ x) in v] ==> ¬[\∀ x . \A(φ x) in v]"
 using ActualForallI by blast

 lemma ActualActualI[PLM_intro]:
 "[\Aφ in v] ==> [\A\Aφ in v]"
 using logic_actual_nec_4[axiom_instance, equiv_lr] .
 lemma ActualActualE[PLM_elim,PLM_dest]:
 "[\A\Aφ in v] ==> [\Aφ in v]"
 using logic_actual_nec_4[axiom_instance, equiv_rl] .
 lemma NotActualActualD[PLM_dest]:
 "¬[\A\Aφ in v] ==> ¬[\Aφ in v]"
 using ActualActualI by blast
 end

 lemma ANeg_1[PLM]:
 "[\¬\Aφ \≡ \¬φ in dw]"
 by PLM_solver
 lemma ANeg_2[PLM]:
 "[\¬oth_class_taut_5_d[PLM]::
 by PLM_solver
 lemma Act_Basic_1[PLM]:
 "[\Aφ \∨ \A\¬φ in v]"
 by PLM_solver
 lemma Act_Basic_2[PLM]:
 "[\A(φ & ψ> bold>🚫
 by PLM_solver
 lemma Act_Basic_3[PLM]:
 "[\A(φ \≡ ψ) \≡ ((\A(φ \→ ψ)) & (\A(ψ \→ φ))) in v]"
 by PLM_solver
 lemma Act_Basic_4[PLM]:
 "[(\A(φ \→ ψ) & \A(ψ \→ φ)) \≡ (\Aφ \≡ \Aψ) in v]"
 by PLM_solver
 lemma Act_Basic_5[PLM]:
java.lang.NullPointerException
 by PLM_solver
 lemma Act_Basic_6[PLM]:
 "[[:
 unfolding diamond_def by PLM_solver
 lemma Act_Basic_7[PLM]:
 "[\Aφ \≡ \◻\Aφ in v]"
 by (simp add: qml_2[axiom_instance] qml_act_1[axiom_instance] "\≡I")
 lemma Act_Basic_8[PLM]:
 "[(\φbold>>→ \\φ
 by (metis qml_act_2[axiom_instance] CP Act_Basic_7 "\≡E"(1)
 "\≡ λ φ ψ v . [φ ψ i v"
 lemma Act_Basic_9[PLM]:
 🚫◻ \◻ in v"
 using qml_act_1[axiom_instance] ded_thm_cor_3 nec_imp_act by blast
 lemma Act_Basic_10[PLM]:
 "[>:: "o ==> χ and Θ bool" and v::i
 by PLM_solver

 lemma Act_Basic_11[PLM]:
 "[\A( χ
 proof -
 have "[\∀ α . \¬φ α≡ ( \<alpha  \A\not>φ α) in v]"
 using logic_actual_nec_3[axiom_instance] by blast
 hence "[\¬\A(\∀ α . \¬φ α) \≡ \¬(\∀ α . \A\<not    
 using oth_class_taut_5_d[equiv_lr] by blast
 moreover have "[ [φ in v]"
 using logic_actual_nec_1[axiom_instance] by ultimaey ow \Theta [φ in v]" by simp
java.lang.NullPointerException
java.lang.NullPointerException
 moreover {
java.lang.NullPointerException
 using logic_actual_nec_1[axiom_universal, axiom_instance] by blast
 hence "[(\<^"[∀. <A>\φ \alpha \≡ ( α>\not>\Aφ) in v]"
 using cqt_basidedutin] yfat
java.lang.NullPointerException
 using oth_class_taut_5_d[equiv_lr] by blast
 }
 ultimately sh show ?thesis
 by (metis "\∃E" MetaSolver.EquivI Semantics.T7 existential)
 qed

 lemma act_quant_uniq[PLM]:
 "[(:: "('a ==> od ψ :: "'a \>and Θv
 by PLM_solver

 lemma fund_cont_desc[PLM]:
 "[(x^ultimately have "[φ φ in v]"
 using descriptions[axiom_instance] act_quant_uniq "\≡yast

 lemma hintikka[PLM]:
java.lang.NullPointerException
 proof -
 have "[(\∀ z . φ z χ
 unfolding identity_ν
 using l_identity[where φ="λ x . φ
 deduction, deduction]
 using id_eq_obj_2[deduction] unfolding identity_ν_def by fastforce
java.lang.NullPointerException
 qed

 lemma russell_axiom_a[PLM]:
 "[((F, "ustabecn 🚫
 (is "[?lhs and "Substabl con \chi
 proof -
 {
 assume 1: "[?lhs in dw]"
java.lang.NullPointerException
 using cqt_5[axiom_instance, deduction]
 SimpleExOrEnc.intros
 by blast
  obtain α where 2:
 "[α^P = (\ιx. φ x) in dw]"
java.lang.NullPointerException
 hence 3: "[φ α & (\∀ z . φ z \→ z = α) in dw]"
 using hintikka[equiv_lr] by simp
 from 2 have "[(\ιx. φ x) = (αby PLM_solver
 using l_identity[where α="α^P" and β="\ιx. φ x" and φ="λ x . x = α^P",
 axiom_instance, deduction, deduction]
 id_eq_obj_1[where x=α] by auto
 hence "[(F, α^P) in dw]"
 sing 1 _identity[where η↑phi> x" and φlambda>> x . \lparrF,x\<rparr"
 axiom_instance, deduction, deduction] by auto
 with 3 have "[φ α & (\∀ z . φ z \→ z = α) & (F, α^P) in dw]" by (rule "&I")
java.lang.NullPointerException
 }
 moreover {
 assume "[?rhs in dw]"
 then obtain α"
 "[φ α "
 using "table cd(🚫φ . ψ φ & χ)"
java.lang.NullPointerException
 ondψ"
 hence "[?lhs in dw]"
 using l_identity[axiom_instance, deduction, deduction]
 
 }
 ultimately
 qed

 lemma russell_axiom_g[PLM]:
 "[{o_existitro
 (is "[?lhs \≡
 
 {
 assume 1: "[?lhs in dw]"
java.lang.NullPointerException
 using cqt_5[axiom_instance, deduction] SiplExrn.tosbybt
java.lang.NullPointerException
 hence 3: "[(φ α>_ef by blast
 using hintikka[equiv_lr] by simp
 from 2 have "[(ndps>"
 using l_identity[where α="α^P" and β="\ιx. φ x" and φ="λ x . x = α^P",
 axiom_instance, deduction, deduction]
 id_eq_obj_1[where x=α] by auto
java.lang.NullPointerException
java.lang.NullPointerException
 axiom_instance, deduction, deduction] by auto
 with 3 have "[(φ α & ( χ fr \Theta and φ and v ==>
 using "(rule rule_sub_nec[where Θ=Θ and χ=χ and ψ and φ and v=v],
java.lang.NullPointerException
 }
 moreover
 assume "[?rhs in dw]"
 then obtain α where 4:
 "[φ \∀→ z & {αP, F} in dw]"
 using " match conclusion in "Θ [φ χ in v]" for Θ φ
 hence "[α^P <p>(r rule_sub_nec[where Θ=Θ and χ=χ and ψ=ψ and φ=φ and v=v],
 using hintikka[equiv_rl] "&E" by blast
 hence "[?lhs in dw]"
 using l_identity[axiom_instance, deduction, deduction]
 by fast
 }
 ultimately show ?thesis by PLM_solver
 qed

 lemma russell_axiom[PLM]:
 assumes "SimpexOEnc\psi
 shows "[ψ (\ιx. φ x) match conclusion in "Θ [φ χ in v]" for Θ and v ==>
 (is "[?lhs \≡
 proof -
 {
 assume 1: "[?lhs in dw]"
 hence "[\∃α
 using cqt_5[axiom_instance, deduction] assms by blast
 then obtain α where 2: "[α^P(mtcprmssin"And>vv x y . [ψ>≡ χ
 hence 3: "[(φ α & (psi::"'a::type==>'a==>o" and χ'a==>" ==>
 using hintikka[equiv_lr] by simp
 from 2 have "[( x) P) in dw]"
 using l_identity[where α="α^P" and β(rule rule_sub_nec[where Θ=Θ and χ=χ=ψ=φ
 axiom_instance, deduction, deduction]
 id_eq_obj_1[where x=α] by auto
 hence "[ψ (αSbstbeCndde)\close›)
java.lang.NullPointerException
 axiom_instance, deduction, deduction] by auto mmethod PL_ub_gl_etdfr \phi>:"a::Sbal<ightarrow\
 with 3 have "[φ α atchuonin"\heta<> 
 using "open(rule rule_sub_nec[where Θ=Θ and χ=χ and ψ=ψ=φ
 hence "[?rhs in dw]" using "(rbal_nrs (supio))?+ fi)
 }
 moreover {
 assume "[?rhs in dw]"
 then obtain α where 4:
 "[φ αThis can only be pove sig teSmnisoftheo oeao
 using "\∃E" by auto
java.lang.NullPointerException
 using hintikka[equiv_rl] "&E" by blast
 hence "[?lhs in dw]"
 l_identity[axiom_instance, deduction, deduction]
 by fast
 }
 ultimately show ?thesis by PLM_solver
 qed

 lemma unique_exists[PLM]:
 "(> \ιx. φ x)) \<^old\
 proof((rule "\≡Subst SubtbeCn\phi
 assume "[v.[(ψ≡ χ Θ\<hi ψ in v] ==> Θ [φ in v]"
 then obtain α where
java.lang.NullPointerException
 by (rule "E)
 hence "[φ α & (hence "\phi\psi n ]= [\phi>χ
 using hintikka[equiv_lr] by auto
 thus "[\∃!x . φ "\≡E"(1)"E"(2) by meson
 unfolding exists_unique_def using "\∃ [φ in v] ==> χ
 next
 assume "[\<lemma 
  where
 "[φ α -
 unfolding exists_unique_def by (rule "[(ψ χ v])"
 hence¬
 usingbl\not\♢ v"
 thus "[ ( assms) apply PLM_autosubst by a
 using " rule_
 qed

 lemma y_in_1[PLM]:
 "[x^P \<      shows
 using hintikka[equiv_lr, conj1] by(rl C

 lemma y_in_2[PLM]:
java.lang.NullPointerException
 using hintikka[equiv_lr, conj1] by (rule CP)

 lemma y_in_3[PLM]:
 "[(P \ι. \phi(x^> φιx . φP)) in dw]"
 proof (rule CP)
 assume "[(\∃[p <lparr>R,x,y)
 then obtain where 1:
 "[y^Papply (insert assms) apply PLM_atosub by auto
 by (rule "\∃E")
 hence "[φ (y^v x.[(P)≡¬♢<>\
 using y_in_2[deduction] unfolding identity_ν
 thus "[φboι<^shows[
 using l_identity[axiom_instance, deduction,
 deduction] 1 by fast
 qed

 lemma act_quant_nec[PLM]:
java.lang.NullPointerException
 by PLM_solver

 lemma equi_desc_descA_1[PLM]:
 "[(\supbold= (\ιx . φ x)) \≡P \ιA\<>x
java.lang.NullPointerException
 using act_quant_nec apply (rule ")
 using descriptions[axiom_instance]
 by (meson "\≡E"(6) oth_class_taut_4_a)

 lemma equi_dePLM]:
java.lang.NullPointerException
 proof (rule CP)
 assume "[P \<iotax
 then obtain y where
java.lang.NullPointerException
 by (rule "\∃E")
java.lang.NullPointerException
 using equi_desc_descA_1[equiv_lr] by auto
 ultimately show "[(= (\A
 using l_identity[axiom_instance, dduto, deton]
 by fast
 qed

 aqidcesc_[PL]
 assumes "SimpleExOrEnc ψ" asus∧≡ i
 shows "[ψ (\→∃P \ι. \^>A x)) in v]"
 proof (rule CP)
 assume "[ψ (uto
 hence "[\∃
 using cqt_5[OF assms, axiom_instance, deduction] by auto
java.lang.NullPointerException
 hence "[α "[>¬(^>→ φ) in v]"
 using equi_desc_descA_1[equiv_lr] by auto
 thus "[\∃y. y^P app (insert assms) applyPLMautosby auto
 using " lerule_sub_remark_8:
 qedjava.lang.NullPointerException

 lemma equi_desc_descA_4[PLM]:
 assumes "SimpleExOrEnc ψ"
 shows "[ψ (\ιx. φ x) by auto
 proof (rule CP)
 assume "[ψ (:
 hence "[v.[( P,a)& (( ^¬()) in ]
 using cqt_5[OF assms, axiom_instance, deduction] by auto
 then obtain α where "[α[( in v]"
 moreover hence "[α \¬))) in v]"
 using equi_desc_descA_1[equiv_lr] by auto
java.lang.NullPointerException
 using l_identity[axiom_instance, deduction, deduction] by fast
 qed

 lemma nec_hintikka_scheme[PLM]:
 "[(x^P ‹
 using descriptions[axiom_instance]
 apply (rule "\≡E"(5))
 apply PLM_solver
 using id_eq_obj_1 apply simp
 using id_eq_obj_2[deduction]
 l_identity[where α
 unfolding identity_ν_def
 apply blast
 using l_identity[where α="x", axiom_instance, deduction, deduction]
 id_eq_2[where 'a=ν, deduction] unfolding identity_ν_def by meson

 lemma equiv_desc_eq[PLM]:
 assumes "∧x.[" "<>")
java.lang.NullPointerException
 proof(rule "\∀I")
 fix x
 {
java.lang.NullPointerException
 hence 1: ap (PLM_subst_method "φ¬¬>)")
 using nec_hintikka_scheme[equiv_lr] by auto pply PLM_
java.lang.NullPointerException
 using "\◻
 {
 ix
 {
 assume "[ z in v]"
 hence "[\Aφ
 using assms[where x=z] apply - by PLM_solver
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 using 2 cqt_1[axiom_instance,deduction] by auto
java.lang.NullPointerException
 using vdash_properties_10 by auto
 }
java.lang.NullPointerException
 }
 ♢(φ∨) \<old<
 moreover have "[\Aψ◻¬\phi> ()od<> \<not\& \nt\psi)) in v]"
 ng[cj1ssweexx]
 apply - by PLM_solver
 Aψ x \<oldd= x) in v]"
 by PLM_solver
java.lang.NullPointerException
 using nec_hintikka_scheme[where φ="ψ¬♢¬¬>) ))))) \¬♢)) \^bold>& (\bold¬(\^old>♢ ]
 }
 moreover {
 assume "[x⁼ ( x) in v]"
 hence 1: "[\Aψ x add: KBasic2_4)
 using nec_hintikka_scheme[equiv_lr] by auto
 hence 2: "[ [(boldAψ z = x)in]
 using "&E" by blast
 {
 fix z
 {
 assume "[\Aφ z in v]"
 hence "[\¬¬
 using assms[where x=z]
 apply - by PPLM_so
 moreover have "[\→= x in v]"
 using 2 cqt_1[axiom_instance,deduction] by auto
 ultimately have "[z = x in v]"
 using vdash by auto
 }
 hence "[\^>\<>\C)
 }
java.lang.NullPointerException
java.lang.NullPointerException
 [c] assms[where x=x]
 apply - by PLM_solver
java.lang.NullPointerException
  KBas[PLM]:
 hence "[x^\◻φ∨◻ψ→ \∨) in v]"
 using nec_hintikka_scheme[where φ="φ",equiv_rl]
 by auto
 }
 ultimately show "[x^P = (\ι "[\φ→ ) in v]" using RM_1 by auto
 using "\≡
 qed

 lemma UniqueAux:
java.lang.NullPointerException
 shows "[(\∀◻ψ→ \∨ ψ v]"
 proof -
 {
 fix z
 {
 assume "[\A(φ z) in v]"
 hence "[z = α in v]"
 using assms[conj2, THEN cqt_1[where α=z,
 axiom_instance, deduction],
 deduction] by auto
 }
 moreover {
java.lang.NullPointerException
java.lang.NullPointerException
 identity_\>
 using id_eq_obj_2[deduction] by fas
 hence "[\A(φ z) in v]" using assms[conj1]
 using l_identity[axiom_instance, deduction,
 deduction] by fast
 }
 ultimately have "[(\A(φ z) ] appl simp
 using "\≡I" CP by aut aapply (PLM_subst "()\→\>")
  oth_class_] apply simp
java.lang.NullPointerException
 by (rule "\∀I")
 qed

 lemma nec_russell_axiom[PLM]:
 assumes "SimpleExOrEnc ψ"
 shows "[(ψ (\ιx. φ x)) \♢(\≡¬(\♢¬φ
 P)) in v]"
 (is "[?lhs \≡ ?rhs in v]")
 proof -
 {
 assume 1: "[?lhs in v]"
 hence "[\∃♢\≡¬◻())) in v]"
 using cqt_5[axiom_instance, deduction] assms by blast
 then obtain α where 2: "[(α\<^    using
 <\<Aφ≡= α))) in v]"
 using descriptions[axiom_instance, equiv_lr] by auto
 hence 3: "[(\Aφ α) & ( have "[\^l\boxψ\∨b>\rightarrow> (\◻(\¬ψ) \<^old\) in v]"
 using cqt_1[where α=α and φ="λ z . (\A(φ z) \≡ (z = α))",
 axiom_instance, deduction, equiv_rl]
 using id_eq_obj_1[where x=α] unfolding identity_ν_def
 using hintikka[equiv_lr] cqt_basic_2[equiv_lr,conj1]
 "&I" by fast
 from 2 have "[(\ιx. φ x) = (α^P) in v]"
 using l_identity[where β="(\ιx. φ x)" and φ="λ x . x = (α^P)",
 axiom_instance, deduction, deduction]
 id_eq_obj_1[where x=α] by auto
 hence "[ψ (α^P) in v]"
 using 1 l_identity[where α="(\ιx. φ x)" and φ="λ x . ψ x",
 axiom_instance, deduction,
 deduction] by auto
 with 3 have "[(\Aφ α & (\∀ z . \A(φ z) \→ (z = α))) & ψ (α^P) in v]"
 using "&I" by simp
 hence "[?rhs in v]"
 using "\∃I"[where α=α]
 by (simp add: identity_defs)
 }
 moreover {
 assume "[?rhs in v]"
 then obtain α where 4:
 "[(\Aφ α & (\∀ z . \A(φ z) \→ z = α)) & ψ (α^P) in v]"
 using "\∃E" by auto
 hence "[(\∀ z . (\A(φ z) \≡ (z = α))) in v]"
 using UniqueAux "&E"(1) by auto
 hence "[(α^P) = (\ιx . φ x) in v] ∧ [ψ (α^P) in v]"
 using descriptions[axiom_instance, equiv_rl]
 4[conj2] by blast
 hence "[?lhs in v]"
 using l_identity[axiom_instance, deduction,
 deduction]
 by fast
 }
 ultimately show ?thesis by PLM_solver
 qed

 lemma actual_desc_1[PLM]:
 "[(\¬>¬
 proof -
 {
 assume "[?lhs in v]"
 then obtain α where
 "[((\alpha<^>)= ( in vv]]"
 by (rule "\∃E")
 hence "[(A!,(\ιx. φ x)) in v] ∨ [(α^P) =_E (\ιx. φ "(ψ∨ φ \ψ
 apply - unfolding identity_defs by PLM_solver
 then obtain x where
java.lang.NullPointerException
 using nec_russell_axiom[where ψ="λx . (A!,x)", equiv_lr, THEN "\∃M]:
 using nec_russell_axiom[where ψ >♢φin v]"
 using SimpleExOrEnc.intros unfolding identity_{amond_def}
java.lang.NullPointerException
 hence "[?rhs in v]" unfolding exists_unique_def by (rule "\◻>🚫>")
 }
 moreover {
 assume "[?rhs in v]"
 ere
 "[((\Aφ
 unfolding exists_unique_def by (rule "\♢" = TBasic
 hence "[\∀
 using UniqueAux by auto
 hence "[(x^P) = (\ιx. φ x) in v]"
 using descriptions[axiom_instance, equiv_rl] by auto
 hence "[?lhs in v]" by (rule "\∃I")
 }
 ultimately show ?thesis
 using "\≡I" he "[\◻\¬ v]"
 qed

 lemma actual_desc_2[PLM]:
 "[(x^P) m_instance
 using nec_hintikka_scheme[equiv_lr, conj1]
 by (rule CP)

 lemma actual_desc_3[PLM]:
 "[(z^P) = (\ιx. φ x) \→ \A(φ z) in v]"
  nec_hintikka_scheme[equiv_lr, conj1]
 by (rule CP)

 lemma actual_desc_4[PLM]:
 "[(\∃(LMsb_mhd<bold>¬")
 proof (rule CP)
 assume "[(\∃ y . (y^P diamond_defusing oth_class_taut[equiv_rl]
  y where 1
java.lang.NullPointerException
 by (rule "exists)
java.lang.NullPointerException
 thus "[ lemS5Basic_2[LM]:
 using l_identity[axiom_instance, deduction,
 deduction] 1 by fast
 qed

 lemma unique_box_desc_1[PLM]:
 "[(\∃
 proof (rule CP)
 assume "[(xists>x \^<>(
 then obtain α where 1: using qm[axiom_instance] qml_2[axiom_ins] "I" by bla
 "[\◻
java.lang.NullPointerException
 {
 fix y
 {
 assume "[(y⁼ ( ) in v v]"
java.lang.NullPointerException
 using nec_hintikka_scheme[where x="y" and φ="φ", equiv_lr, conj2,
 THEN cqt_1[where αlemS5Basic_5[PLM]:
 hence "[α \♢\→
 
 hence "[φ y in v]"
 using 1[conj1] qml_2[axiom_instance, deduction]
 l_identity[axiom_inst deduc, ded]
 by fast
 }
 hence "[(y ET = ET" |
 by (rule CP)
 }
 thus "[ MKTn l r h)
 by (rule "\∀I")
 qed

 lemma unique_box_desc[PLM]:
 "[(\∀r
java.lang.NullPointerException
 apply (rule CP, rule CP)
 using nec_exist_unique[deduction, deduction]
 unique_box_desc_1[deduction] by blast

 ‹\phi 🚫
 ‹\label{TAO_PLM_Necessity}›

 lemma RM_1[PLM]:
 "(∧v.[φ \→ ψ in v]) ==> [\◻φ \→ \◻ψ in v]"
 using RN qml_1[axiom_instance] vdash_properties_10 by blast

 lemma RM_1b[PL]:
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 using RN_2 qml_1[axiom_instance] vdash_properties_10 by blast

 lemma RM_2[PLM]:
 "(∧v.[φ \→ ψ in v]) ==> (mkt_bal_l n l r) = heighr 2 r
 unfolding diamond_def
 using RM_1 contraposition_1 by auto

 lemma RM_2_b[PLM]:
java.lang.NullPointerException
 unfolding diamond_def
 using RM_1_b contraposition_1 by blast

 lemma KBasic_1[PLM]:
 "[\◻φ \→ \◻(ψ \→ φ) in v]"
 by (simp only: pl_1[axiom_instance] RM_1)
 lemma KBasic_2[PLM]:
 "[\◻(\¬φ) l 3"3"
 by (simp only: RM_1 useful_tautologies_3)
 lemma KBasic_3[PLM]:
 "[\◻(φ ‹
 apply (rule "Scan.succee (Thm.rulrule_att []
 apply (rule CP)
 apply (rule "&I")
  RM_1 oth_class_taut vdashpropertiesapply blast
 using RM_1 oth_class_taut_9_b vdash_properties_6 apply blast
 using qml_1[axiom_instance] RM_1 ded_thm_cor_3 oth_class_taut_10_a
 oth_class_taut_8_b vdash_properties_10
 by blast
 lemma KBasic_4[PLM]:
 "[\◻ = r + 1 ∨
 apply (rule "\≡I")
java.lang.NullPointerException
 apply blast
  usin KBasic_3 PLM.CCP "🚫
 by blast
 lemma KBasic_5[PLM]:
java.lang.NullPointerException
 by (metis qml_1[axiom_instance] CP "&E" "\≡ max (height l) (height r))"
 lemma KBasic_6[PLM]:
 "[\◻(φ \≡by (cas l, cases r) sim
 using KBasic_4 KBasic_5 by (metis equiv_def ded_thm_cor_3 "&E"(1))
 equiv><box>ψbold>→φ>🚫
 nitpick[expect=genuine, user_axioms, card = 1, card i = 2]
 oops ― ‹
 lemma KBasic_7[PLM]:
java.lang.NullPointerException
 proof (rule CP)
java.lang.NullPointerException
 hence "[\◻(ψ \→ φ) in v] ∧ [\◻(φ (mkt_ba) = ( + mahel) (hr)"
 using "&E" KBasic_1 vdash_properties_10 by blast
java.lang.NullPointerException
 using KBasic_4 "\≡E"(2) intro_elim_1 by blast
 qed

 lemma KBasic_8[PLM]:
java.lang.NullPointerException
 using KBasic_7 KBasic_3
java.lang.NullPointerException
 lemma KBasic_9[PLM]:
 "[\◻((\¬
 proof (rule CP)
java.lang.NullPointerException
 hence "[\◻ []
 using KBasic_8 vdash_properties_10 by blast
 moreover have "∧v.[((:mktdef)
 using CP "\≡E"(2) oth_class_taut_5_d by blast
 ultimately show "[\◻
 using RM_1 PLM.vdash_properties_10 by blast
 qed

 lemma rule_sub_lem_1_a[PLM]:
 "[\◻
  qml[axiom "🚫
 vdash_properties_10
 by blast
 lemma rule_sub_lem_1_b[PLM]:
 "[\◻(ψ \≡
java.lang.NullPointerException
java.lang.NullPointerException
 lemma rule_sub_lem_1c[PLM]"avl l" and " "avl r" and"heigl = heigh r r
 "[\◻(ψ \≡ χ) in v] ==> [(Θ \→Thmrule[]
 by (metis CP "\≡I" "\≡E"(3) "\≡E"(4) "\¬\¬I"
 "\¬\¬E" rule_sub_lem_1_a)
 lemma rule_sub_lem_1_d[PLM]:
 "(∧x.[\◻(ψ x \≡
 by (metis equiv_def "\∀I" CP "\<withassms
 vdash_properties_10 rule_sub_lem_1_a "\∀E")
 lemma rule_sub_lem_1_e[PLM]:
 with as showhow ?thsis
 using Act_Basic_5 "\≡
 vdash_properties_10
 by blast
 lemma rule_sub_lem_1_f[PLM]:
 "[\◻(ψ \≡ χ) in v] ==> [\◻ψ Scasucc (hmrue_a[]
 using KBasic_6 " add: avl_mkt)
 by blast


 named_theorems Substable_intros
 
 definition Substable :: "('a==>‹
 where "Substable ≡
 
 lemma Substable_intro_const[Substable_intros]:
 "Substable cond (λ φ . Θ
 unfolding Substable_def using oth_class_taut_4_a by blast

 lemma Sroof (induction t)
 assumes "
 shows ase 1
 using assms unfolding Substable_def
 using rule_sub_lem_1_a RN_2 "\≡E" oth_clcase Tr
 lemma Substable_intro_impl[Substable_intros]:
 assumes "Substable cond ψ
 and "Substable cond χ"
 shows "Substable cond (λ φ . ψ φ True ‹
 using assms unfolding Substable_def
 by (metis "\≡I" CP intro_elim_6_a intro_elim_6_b)
 
 assumes "Substable cond ψ"
java.lang.NullPointerException
 using assms unfolding Substable_def
 using rule_sub_lem_1_f RN by meson
 lemma Substable_intro_actual[Substable_intros]:
 assumes "Substable cond ψ"
 shows "Substable cond (λ φ . \A(ψ φ))"
 using assms unfolding Substable_def
 using rule_sub_lem_1_e RN by meson
 lemma Substa[Substable_intros]:
 assumes "\       T True
 shows "Substable cond (λ φ . \∀ x . ψ x φ)"
 using assms unfolding Substable_def
 by (simp add: RN rule_sub_lem_1_d)

 named_theorems Substable_Cond_defs
 

  Substable =
 fixes Substable_Cond :: "'a==>'a==>bool"
 assumes rule_sub_nec:
 "∧ φ ψ χ Θ v . [PLM.Substable Substable_Cond φ; Substable_Cond ψ χ]
 ==> Θ [φ ψ in by (meso id_eq_prCP ded_th l_identit[axiom_instan])

  o :: Substable
 
 definition Substable_Cond_o where [PLM.Substable_Cond_defs]:
 "Substable_Cond_o ≡ λ φ ψ . ∀) (F\^>= ) in v"
 instance proof
 interpret PL .
 fix φ :: "o ==> o" and ψ χ :: o and Θ :: "bool ==> bool" and v::i
 assume "Substable Substable_Cond φ"
 moreover assume "Substable_Cond ψ χ"
 ultimately have "[φ | (b) "height (m (mkt_bal_l (AVL.i.inserx l) r) = heir + 3"
 unfolding Substable_def by blast
 hence "[φ ψ in v] = [φ χ in v]" using "\≡E" by blast
 moreover assume "Θ [φ ψ in v]"
 ultimately show "Θ [φ χ in v]" by simp
 qed
 

  "fun" :: (type, Substable) Substable
 
 definition Substable_Cond_fun where [PLM.Substable_Cond_defs]:
 "Substable_Cond_fun ≡ λ φ ψ . ∀ x . Substable_Cond (φ x) (ψ x)"
 instance proof
 interpret PLM .
 fix φ:: "('a ==>
 assume "Substable Substable_Cond φ"
 moreover assume "Substable_Cond ψ χ"
java.lang.NullPointerException
  Substabl by blast
 hence "[φ ψ in v] = [φ χ in v]" using "\≡E" by blast
 moreover assume "Θ [φ ψ in v]"
 ultimately show "Θ [φ χ in v]" by simp
 qed
 

  PLM
 

 lemma Substable_intro_equiv[Substable_intros]:
 assumes "Substable cond ψ[(F::\<i\
 and "Substable cond χ"
 shows "Substable cond (λ φ . ψ φ \≡ χ φ)"
 unfolding conn_defs by (simp add: assms Substable_intros)
 lemma Substable_intro_conj[Substable_intros]:
 assumes "Substable cond ψ"
 and "Substable cond χ"
 shows "Substable cond (λ φ . ψ φ & χ φ)"
 unfolding conn_defs by (simp add: assms Substable_intros)
 lemma Substable_intro_disj[Substable_intros]:
 assumes "Substable cond ψ"
 and "Substable cond χ"
 shows "Substable cond (λ φ
 unfolding conn_defs by (sip add: ass assms Substabl)
 lemma Substable_intro_diamond[Substable_intros]:
 assumes "Substable cond ψ"
java.lang.NullPointerException
 unfolding conn_defs by (simp add: assms Substable_intros)
 lemma Substable_intro_exist[Substable_intros]:
 assumes "∀ x . Substable cond (ψ x)"
 shows "Substable cond (λ φ . \∃
 unfolding conn_defs by (simp add: assms Substable_intros)

  Substab\<>[
 "Substable Substable_Cond (λ φ . φ)"
 unfolding Substable_def Substable_Cond_o_def by blast
 lemma Substable_intro_id_fun[Substable_intros]:
 assumes "Subtable Substableψ
 shows "Substable Substable_Cond (λ φ . ψ (φ x))"
 using assms unfolding Substable_def Substable_Cond_fun_def
 by blast

 method PLM_subst_method for ψ::"'a::Substable" and χ::"'a::Substable" =
 (match conclusion in "Θ [φ χ in v]" for Θ and φ and v ==>
 ‹>=G) 🚫
 ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
 unfold Substable_Cond_defs)›)

 method PLM_autosubst =
 (match premises in "∧v . [ψ \≡
 ‹ match conclusion in with 2🚫
 ‹
 ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
 unfold Substable_Cond_defs)› ›)

 method PLM_autosubst1 =
 (m premises in "\And x . [ψχv]"
 for ψ::"'a::type==>o" and χ::"'a==>o" ==>
 ‹^sub>33) <bold>→bold>= H) in v]"
 ‹(rule rule_sub_nec[where Θ=Θ and χ=χ and ψ=ψ and φ=φ and v=v],
 ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
 unfold Substable_Cond_defs)› ›)

 method PLM_autosubst2 =
 (match premises in "∧v x y . [ψ x y \≡
java.lang.NullPointerException
 ‹
 ‹(rule rule_sub_nec[where Θ=Θ and χ=χ and ψ
 ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
 unfold Substable_Cond_defs)› ›)

 method PLM_subst_goal_method for φ::"'a::Substable==>o:
 (match conclusion assumes "av "avl x" and "x ≠
 ‹ unfoldin identity_defs bby PLM_sol
 ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
 unfold Substable_Cond_defs)›

 
 text{* \begin{TODO}
  can only be proven using the Seman of the Box opeopera
 As it is not needed for the further reasoning it remains commented for now.
 \end{TODO} *}
 lemma rule_sub_lem_2:
 assumes "Substable Substable_Cond φ"
 shows "[\◻ (KT n l r rl rr rh h)
 using assms unfolding Substable_def Substable_Cond_defs
 using Semantics.T6 by fast
*)

  lemma rule_sub_nec[PLM]:
    assumes "Substable Substable_Cond φ"
    shows "(∧v.[(ψ \<equiv> χ) in v]) ==> Θ [φ ψ in v] ==> Θ [φ χ in v]"
    proof -
      assume "(∧v.[(ψ (auto simp:: height_mktball hheigh
      hence "[φ ψ in    "[((p::\<Pi\ q \^>→ in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<equiv>I" CP "\<equiv>E"(1) "\<equiv>E"(2) by meson
      thus "ΘMKT n l rn r rl rr rrh h)
    qed

  lemma rule_sub_nec1[PLM]:
    assumes "Substable Substable_Cond φ"
    shows "(∧v x .[(ψ x \<equiv> χ x) in v]) ==> Θ [φ ψ in v] ==> Θ [φ χ in v]"
    proof -
      assume "(∧v x.[(ψ x \<equiv> χ x) in v])"
      hence "[φ ψ in v] = [φ χ in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<equiv>I" CP "\<equiv>E"(1) "\<equiv>E"(2) by metis
      thus "Θ [φ ψ in v] 
    qed

  lemma rule_sub_nec2[PLM]:
    assumes "Substable Substable_Cond φthenhave "? height\or
    shows "(∧v x y .[ψ x y \<equiv> χ x y in v]) ==> Θ [φ ψ in v] ==> Θ [φ χ
    proof -
      assume "(∧v x y .[ψ x y  \>have =height  ?'  height +1
      hence "[φ ψ in v] = [φ χ in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<equiv>I" CP "\<equiv>E"(            \or height'1<orr=height +" by fa
      thus "Θ [φ ψ in v] ==> Θ [φ χ in v]" by auto
    qed

  lemma rule_sub_remark_1_autosubst:
    assumes "(∧v.[(A!,x) \<equiv> (\<not>(\<diamond>(E!,x))) in v])"
        and "[\<not>(A!,x) in v]"
    shows"[\<not>\<not>)
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_1:
    assumes "(∧
        and "[\<not>(A!,x) in v]"
      "\^><not^¬> in"
    apply (PLM_subst_method "(A!,x)" "(\<not>(\<diamond>(E!,x)))")
     apply (simp add: assms(1))
    by (simp add: assms(2))

  lemma rule_sub_remark_2:
    assumes "(∧
        and java.lang.NullPointerException
    shows"[p \<rightarrow> ((R,x,y) & ((Q,a) java.lang.NullPointerException
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_3_autosubst:
    assumes "(∧v x.[(A!,x^P) l ?' r"
        and "[\<exists> x . (A!,x^P) in v]"
    shows"[\<exists> x . (\<not>(\<diamond>(E!,x^P))) in v]"
    apply (insert assms) apply PLM_autosubst1 by auto

  lemma rule_sub_remark_3:
    assumes "(∧by(r avl,simp)
        and "[\<exists> x . (A!,x⁼ l orl   height+1 usingavl\close by(intro java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
    shows "[\<exists> x . (\<not>(\<diamond>(E!,x^P))) in v]"
    apply (PLM_subst_method "λx . (A!,x^P)" "λx . (\<not>(\<diamond>(E!,x^P)))")
     apply (simp add: assms(1))
    by (simp add: assms(2))

  lemma rule_sub_remark_4:
    assumes "∧v x.[(\<not>(\<not>(P,x^P))) \<equiv> (P,x^P) in v]"
        and "[\<A>(\<not>(\<not>(P,x^P))) in v]"
    shows "[\<A>(P,x^[O\<penavl
    apply (insert assms) apply PLM_autosubst1 by auto

  lemma rule_sub_remark_5:
    assumes "∧v.[(φ \<rightarrow> ψ) java.lang.NullPointerException
        and java.lang.NullPointerException
java.lang.NullPointerException
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_6:
    assumes "∧v.[ψ \<equiv> χ in v]"
        and "[\<box>(φ \<rightarrow> ψ) in v]"
    shows "[\<box>(φ \<rightarrow> χ) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_7:
    assumes "∧v.[φ \<equiv> (\<not>(java.lang.NullPointerException
        and "[\<box>(φ \<rightarrow> φ) in v]"
    shows java.lang.NullPointerException
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_8:
    assumes "∧v.[java.lang.NullPointerException
        and java.lang.NullPointerException
    shows "[\<box>(φ) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_9:
    assumes "∧v.[(P,a) \<equiv> ((P,a) & ((
        and "[(
    shows "[( xt) \or height(  )+1
      unfolding identity_defs apply (insert assms)
      apply PLM_autosubst oops ― ‹no match as desired›

  ― ‹
 ― ‹

 lemma KBasic2_1[PLM]:
 "[\◻φ True
 with MKT 1 show ?thesis by (auto simp:avl_delete_root)
 by PLM_solver+

 
java.lang.NullPointerException
 unfolding diamond_def
 apply (PLM_subst_method "φ" "\¬(\¬φ)")
 by PLM_solver+

 lemma KBasic2_3[PLM]:
 "[\◻φ \≡ (\¬(\♢(\¬φ))) in v]"
 unfolding diamond_def
java.lang.NullPointerException
 apply PLM_solver
  (simp add oth_c)
 lemmas "Df\◻show cse

 lemma KBasic2_4[PLM]:
java.lang.NullPointerException
 unfolding diamond_def
 by (simp add: oth_class_taut_4_b)

 lemma KBasic2_5[PLM]:
 "[\◻(φ by (substheight_delete_root,si)
 by (simp only: CP RM_2_b)
java.lang.NullPointerException

 lemma KBasic2_6[PLM]:
 "["
 proof -
 have "[\◻
 using KBasic_3 by blast
 hence "[(\¬(\♢(\¬((\¬φ) & (\¬ψ))))) \≡ (lemma d_eq_proPLM]
java.lang.NullPointerException
 hence "[(\¬(\♢
 apply - apply (PLM_subst_method "\◻(mkt_bal_rn(del x l) r) = he (d x ll)+ 2"
 apply (simp add: KBasic2_4)
 apply (PLM_subst_method "\◻(x l) r) h (e x l) 3"
 apply (simp add: KBasic2_4)
 unfolding diamond_def by assumption
 hence "[(\¬(F:Π in v]"
java.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 hence "[(\¬(\¬(\♢(φ \∨ ψ)))) \≡
 by (rule oth_class_taut_5_d[equiv_lr])
 hence "[\♢(φ \∨ ψ) \≡ (\¬((\¬\φ)) \¬♢ψ v]"
 apply - apply (PLM_subst_method "\¬({
 using oth_class_taut_4_b[Fa
 thus ?thesis
java.lang.NullPointerException
 using oth_class_taut_6_b[equiv_sym] by auto
 

 lemma KBa[PLM]:
 "[( "[ psi>) in v]"
 proof -
 have "\<       case
 position_1oraoitin_ sefl_tutloge_ is_df
 hence "[\◻φ[🚫| (b)"ei(mkt_n l(d ))= he (deletx) 3"
 moreover {
 have ""🪙
 by (simp only: pl_1[axiom_instance] disj_def)
 hence "[ <rightar> 🚫
 using RM_1 by auto
 }
 ultimately show ?thesis
 using oth_class_taut_10_d vdash_properties_10 by blast
 qed

 lemma KBasic2_8[PLM]:
 "[\♢(φ \<^       }
java.lang.NullPointerException
  CP "\<^>\

 lemma KBasic2_9[PLM]:
 "[>(φ→ ψ) \◻φ→ ) i next
 apply (PLM_subst_method "(\¬b
 using oth_class_taut_5_k[equiv_sym] apply simp
 ∨
 using oth_class_taut_5_k[equiv_sym] apply simp
 apply (PLM_subst_method "\♢(\¬φlemma derived[PLM]:
 using KBasic2_2[equiv_sym] apply simp
 using KBasic2_6 .

java.lang.NullPointerException
 "[\♢
 unfolding diamond_def apply (PLM_subst_method "φ" "\¬
 using oth_class_taut_4_b oth_class_taut_4_a by auto

 lemma KBasic2_11[PLM]:
 "[\♢\♢\χ)
 unfolding diamond_def
 apply (PLM_subst_method "\◻(\¬φ S5Basic_4 vdash_properties_10 CP by m
 using oth_class_taut_4_b oth_class_taut_4_a by auto

 lemma KBasic2_12[PLM]: "[\◻(φ [PLM]:]:
 proof -
 have "[\◻(ψ \∨ φ) \→ (\◻(\¬ψ) \→ \◻φ) in v]"
 using CP RM_1_b "\∨E"(2) by blast
 hence "[\◻(ψ "[φ→ <ox>\psi> in v]"
 unfolding diamond_def disj_def
 by (meson CP "\¬\¬E" vdash_properties_6)
 thus ?thesis apply -
java.lang.NullPointerException
 apply (simp add: PM.othclass_tat3e
 apply (PLM_subst_method "(ψ \∨[[\^bol>♢\old→ ψ in v]"
 apply (simp add: PLM.oth_class_taut_3_e)
 by assumption
 qed

 lemma TB using RM_2_b assm b meis
 "[φ \◻χ v] \\Longrightarrow> [\φ→ ψ in v]"
 unfolding diamond_def
 apply (subst cocontrap
 apply (PLM_subst_method "\◻\¬
 apply (simp add: PLM.oth_class_taut_4_b)
 using qml_2[where φ="\¬ derived_S5_rules_2_b[PLM]:
 by simp
 lemmas "T = TBasic

 lemma S5Basic_1[PLM]:
 "[\<diamondS5_rules_2_a
 proof (rule CP)
java.lang.NullPointerException
 hence "[\◻ in v]"
 using KBasic2_10[equiv_lr] by simp
 moreover have "[\♢(\¬ {
  (siadd: qml_3[axiom_instance])
java.lang.NullPointerException
 by (simp add: PLM.modus_tollens_1)
 thus "[\◻φ\^dia>\∀α. \◻(φ α)) \→ \♢\◻) in"
 unfolding diamond_def apply -
 apply (PLM_subst_ethd"\¬")
 using oth_class_taut_4_b[equiv_sym] apply simp
 unfolding diamond_def using oth_class_taut_4_b[equiv_rl]
 mp
 qed
 lemmas "5\♢

 lemma S5Basic_2[PLM]:
 "[\◻♢(\◻ ^b>→ (\∀α. φ α) in v]"
 using "5\♢" "T\♢ "🚫

 lemma S5Basic_3[PLM]:
 "[\♢φ
 using qml_3[axiom_instance] qml_2[axiom_instance] "\≡I" by blast

 
java.lang.NullPointerException
 using "T\♢"[deduction, THEN S5Basic_3[equiv_lr]]
 by (rule CP)

 lemma S5Basic_5[PLM]:
 "[\♢\◻φ \→ φ in v]"
 using S5Basic_2[equiv_rl, THEN qml_2[axiom ence "[\∀\alpha>) \◻(φ α) in v]" using RM_1 by auto
 by (rule CP)
 lemmas "B\♢" = S5Basic_5

 lemma S5Basic_6[PLM]:
java.lang.NullPointerException
 using S5Basic_4[deduction] RM_1[OF S5Basic_1, deduction] CP by auto
 lemmas "4proof rule "🚫 CBF = BFs_2

 lemma S5Basic_7[PLM]:
 "[\◻φ \≡ \ in v]"
 using "4\◻" qml_2[axiom_instance] by (rule "\≡I")

 lemma S5Basic_8[PLM]:
 "[\♢\♢φ \→ \♢φ in v]"
 using S5Basic_6[where φ¬<>"
 KBasic2_11[equiv_lr] CP unfolding diamond_def by auto
 lemmas "4\♢" = S5Basic_8

 lemma S5Basic_9[PLM]:
 "[\♢<><
 using "4\♢" "T\♢" using BF by metis

 lemma S5Basic_10[PLM]:
java.lang.NullPointerException
 apply (rule "\≡¬\∀<lpha 
 apply (PLM_subst_goal_method "λ χ . \♢(\∀αsh
 using S5Basic_2[equiv_sym] apply simp
 using KBasic2_12 apply assumption
 apply (PLM_subst_goal_method "λ .(\φ∨ χ) \◻ sup>P in v]"
 using S5Basic_7[equiv_sym] apply simp
 using KBasic2_7 by auto

 lemma S5Basic_11[PLM]:
java.lang.NullPointerException
 apply ultimately have "[ "[(x \\=<>y
 apply (PLM_subst_goal_method "λ χ . \◻(φ \∨ \♢ψ) rr"
 using S5 apply simp
 using KBasic2_12 apply assumption
java.lang.NullPointerException
 using S5Basic_3[equiv_sym] apply simp
 using KBasic2_7 by assumption

 lemma S5Basic_12[PLM]:
java.lang.NullPointerException
 proof -
 have "[have"is_o (MK rn rl rr rh)"by auto
 using S5Basic_10 by auto
 hence 1: "[(have"n < rightmost_item cqt_1_\kappa,deduc, ded] b m
 using oth_class_taut_5_d[equiv_lr] by auto
java.lang.NullPointerException
 apply (PLM_subst_method "E y) in v]"using 1 "\^>≡
 using KBasic2_4 apply simp
 apply (PLM_subst_method "\◻\¬φ" "\¬\♢φ")
 using KBasic2_4 apply simp
 apply qed
 unfolding diamond_def
 apply (simp add: RN oth_class_taut_4_b rule_sub_lem_1_a rule_sub_lem_1_f)
 using 1 by assumption
  ?hes
 apply (PLM_subst_method "\¬
 using oth_class_taut_6_a[equiv_sym] apply simp
 apply (PLM_subst_method "\¬((\¬
 using oth_class_taut_6_a[equiv_sym] apply simp
 using 2 by assumption
 qed

 lemma S5Basic_13[PLM]:
java.lang.NullPointerException
 apply (PLM_subst_meth lemma eq eq_E_simple_3[PLM]]:
 using S5Basic_2[equiv_sym] apply simp
 using S5Basic_12 bylemma set_of_delete


 "[\◻(φ \→ (\◻ψ)) \≡ \◻(\♢x\>bo>&\lparr,\rparr> \^& <^><
 proof (rule "\≡I"; rule CP)
 assume "[\◻(φ \→ \◻ψ) in v]"
 moreover {
 have "∧v.[\◻ca (MKT n l rn rl rr r h)
 proof (rule CP)
 fix v
 assume "[\◻(φ \→ \◻rn rr h"
 hence "[\♢φ \→ \♢ "al l" an "vl ?r" b simp
 using "K\♢"[deduction] by auto
 thus "[\♢φ \→ ψ in v]"
 using "B🚫
 
 hence "[\◻(\◻(φ \→ \◻ψ) \→ (\♢φ \→ ψ)) in v]"
 by (rule RN)
 hence "[\◻(\◻?r}"
 using qml_1[axiom_instance, deduction] by auto
 }
 ultimately show "[\◻proof - 
 using S5Basic_6 CP vdash_properties_10 by meson
 next
java.lang.NullPointerException
 moreover {
 fix v
 {
 assume "[\◻(\<    by
java.lang.NullPointerException
 using qml_1[axiom_instance, deduction] by auto
 
 hence "[\◻<n>ET \ Longrighta> fst(delete t) = rightm t"
 using S5Basic_4[deduction] by auto
java.lang.NullPointerException
 using 1[deduction] by auto
 
 hence "[\◻(\♢φ l r " an " t" and ""is_or t"
 using CP by auto
 }
 ultimately show "[\◻{}"
 using S5Basic_6 RN_2 vdash_properties_10 by blast
 qed

 lemma sc_eq_box_box_1[PLM]:
 "[\◻(φ \→ case(MKT_MKT nl ll l lh rrn rlrr rh h)
 proof(rule CP)
 assume 1: "[\◻(φ \→ \◻φ) in v]"
java.lang.NullPointerException
 using S5Basic_14[equiv_lr] by auto
 hence "[\♢φ \→ φ in v]"
 using qml_2[axiom_instance, deduction] by auto
 moreover from 1 have "[φ \→ \◻])
 using qml_2[axiom_instance, deduction] by auto
 ultimately have "[\♢
 
 moreover have "[\◻φ \→ \♢ fromMKT_MKassm(1) h"se(det) = sset_o ?t'"
 using qml_2[axiom_instance] "T\♢"
 by (rule ded_thm_cor_3)
 ultimately show "[\♢φ \≡ moreover from MKT_MKT assm have "(set_of t) {n} set_ l \union r"
 by (rule "\≡I")
 qed

 lemma sc_eq_box_box_2[PLM]:
java.lang.NullPointerException
 
 assume "[(φ\^→ (simp deel: dlete_root.simps)
java.lang.NullPointerException
 using
java.lang.NullPointerException
 by (meson CP "I" "E"(3)
 "r < n)
 e "teim_at_tm r n= i qlatmn0 ie(lss_natm r )+2"

 lemma sc_eq_box_ also have "...\le2 * min r n + 5"
 "[(\◻(φ \→ \◻φ) \◻(ψ \◻ψ→ ((\>proof (induct t)

 assume 1: "[( 5 * k + 3 * m + 2 * n + 5"
 {
 vide_nat_tmn==>
 
 using oth_class_taut_5_i[eqivm)\le 8 * m + 2 * n + 7"
 moreover {
java.lang.NullPointerException
java.lang.NullPointerException
 using KBasic_7[deduction] by auto
 
 moreover unfolding dvd_tm tm_me_m a_mnt_ ieequal_a_t
 ^>¬(\◻φ)) & (\¬(\ψ
 hence "[>Not_tm"
java.lang.NullPointerException
 by metis
 
 usingtae_t 1run["
 hence o {
 using KBasic_9[deduction] by auto
 }
java.lang.NullPointerException
java.lang.NullPointerException
 }
java.lang.NullPointerException
 using CP by auto
 qed

  alst\Rightarrow> 'a list ==>
 v. [χ in v] \Longrightarrow [ in v"
java.lang.NullPointerException
 proof -
java.lang.NullPointerException
 using assms by (force sisimp: vl_d set_of_mkt_bal_l[ l "(dle x r)"n] si del:mkt_simp
 thus "[ctp
 using S5Basic_4 vdash_properties_10 CP by metis
 qed

 lemma derived_S5_rules_1_b[PLM]:
 assumes "∧'<> 
 shows "[φ
 using

 "gen_length_tm n (x # xs) =1 gen_lnh S"
 assumes "∧
java.lang.NullPointerException
 proof -
 
 using RM_2_b assms by metis
java.lang.NullPointerException
 using "B (#)=o{
 qed

 erived_S5_rules_2_bP:
 v. [φ→ 'a list \<Rightarrow 
 
 ule minuct) smp_l

 lemma BFs_1[PLM]: "[(i. i ∈mf i) "
 proof (rule derived_S5_rules_1_b)
 fix v
 {
 fix α
 have "🪙i. i ∈ c"
 ingcqt_org1 yei
 hence "[(\∀α. \◻(φ α)) \→ \♢
 using RM_2 by metis
 moreover have "[ 'a tm) ==> 'b list ==>
 using "Blambda>x y. val (f x y)) a xs"
 
 using ded_thm_cor_3 by blast
 }
 ductioni
java.lang.NullPointerException
 thus "[ xs"
 using cqt_orig_2[deduction] by auto
 qed
  BF = BFs_1

 lemma BFs_2[PLM]:
java.lang.NullPointerException
 proof -
 {
  \   x'" usi Suc..prem by sim
 {
 fix v
 have "[(\^bold>∀α. φ α) \→ φ \<(induct x (y #s) =do{r\leftarrowzip_tm xs ys; return ((x, y) # rs) }"
 }
java.lang.NullPointerException
 
 hence "[ . \∀α . φ \forall is_(MKT n l l ?r h)" by (aut simp: set_of_d)
 thus ?thesis using cqt_orig_2[deducti
 qed
 lemmas CBF = BFs_2

 lemma BFs3PM]::
 "[\♢
 proof -
 have "[(\<^ld<show.splits simp del::is_ord.simpsmkt_bl_l.sim)
 using BF by metis
 
 using contraposition_1 by simp
  :"\<><diamond>(\¬(\α. α\¬(\◻^¬(φ"
java.lang.NullPointerException
 using KBasic2_2 1 by simp+
 have "[\^b>¬alpha . )))) in v]"
 apply (PLM_subst_method ""; simp add: upt_tm.simps[of i j] upt_conv_Cons)
 using cqt_futhr applymes
 using 2 by metis
 thus ?thesis
 unfolding exists_def diamond_def by auunfoldi ptm.smp[o j]t_t_psby (ipadd:m_lssa_t)
 qed
java.lang.NullPointerException

 lemma BFs_4[PLM]:
java.lang.NullPointerException
 proof -
java.lang.NullPointerException
 using CBF by auto
java.lang.NullPointerException
java.lang.NullPointerException
 using cqt_further_2 apply blast
 g using ctrapstion1 y eti
 
 apply (PLM_subst_method "mR 'a ==>
 lya
 using 2 by assumption
 thus ?tdhoc_ovrodn vietm \> dvidna_
 ding amodde issd b ato
 qed
 lemmas "CBF\<^><

 lemma sign_S5_t[PL]:
 "[(< αll lr lh rn rl rr rh h)
 proof (rule CP)
 @_t›
  obtain τ where "[\◻(φ?r = "MKT rn rl rr rh"
 by (rule "*_t›
      moreover
        fix v
        assume "[φ τ
        hence"∃> . φ in
          by ule∃I")
      
      ultimately show "[><exists>  α . \  fromve(MKT?rhjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
        using RN_2 by blast
    qed
  lemmas Buridan = sign_S5_thm_1

  lemma sign_S5_thm_2[PLM]:
    "[\<diamond>(\<forall> α . φ α) \<rightarrow> (moreover from">x\in  ? 🚫
    proof -
      {
        fix α
        {
          fix v
          have "[(\<forall> α . φ α) tree.simps(3)3))
            using cqt_orig_1 by metis
        }
        hence "[\<diamond>(\<forall> α . φ α) \<rightarrow> \<diamond>(φ α) in v]"
          using RM_2 by metis
      }
      hence "[\<forall> α . \<diamond>(set_of_delete_max  tree
        using "\<forall>I" by metis
      thus ?thesis
        using cqt_orig_2[deduction] by metis
    qed
  lemmas java.lang.NullPointerException

  lemma sign_S5_thm_3[PLM]:
    "[\<diamond>(\<exists> α . φ α & ψ α) \<rightarrow> \<diamond>((of  ?n' ?l' ?r]
    by (simp only: RM_2 cqt_further_5)

  lemma sign_S5_thm_4[PLM]:
    "[((is_ord split:prod..split)
    proof (rule CP)
      assume "[\<box>(\<forall>α. φ α java.lang.NullPointerException
      hence java.lang.NullPointerException
        using KBasic_3[equiv_rl] by blast
      moreover {
        fix v
        assume "[((\<forall>α. φ α \<rightarrow> ψ α) & (java.lang.NullPointerException
        hence "[(\<forall> α . φ α \<rightarrow> χ α) in v]"
          using cqt_basic_9[deduction] by blast

      ultimately show "[\<box>(\<forall>α. φ α \<rightarrow> χ α) in v]"
        using RN_2 by blast
    qed

  lemma sign_S5_thm_5[PLM]:
    "[((\<box>(\<forall>α. φ α \<equiv> ψ α proof(cases " "x = n)
    proof (rule CP)
      assume "[\<box>(\<forall>α. φ α \<equiv> ψ α) & \<box>(\<forall>α. ψ α \<equiv> χ α) in v]"
      hence "[\<box>((\<forall>α. φ α \<equiv> ψ α) & (\<forall>α. ψ α java.lang.NullPointerException
        using KBasic_3[equiv_rl] by blast
      moreover {
        fix v
        assume "[((\<forall>α. φ α \<equiv> ψ αcase rue
         [\^>🚫
          using cqt_basic_10[deduction] by blast
      }
      ultimately show "[\<box>(\<forall>α. φ α \<equiv> χ α) in v]"
        using RN_2 by blast
    qed
  
  lemma id_nec2_1[PLM]:
    "\^><iamond:'a:id_eq^bold<beta^bold>\alpha\^>=\beta)inv]"
    apply (rule "\<equiv>I"; rule CP)
     using id_nec[equiv_lr] derived_S5_rules_2_b CP modus_ponens apply blast
    using "T\<diamond>"[deduction] by auto

  lemma id_nec2_2_Aux:
    "[(\<diamond>φ
    proof -
      assume java.lang.NullPointerException
      moreover have "∧φ ψ. [(\<not>φ) \<equiv> ψ in v] ==> [(\<not>ψ) \<equiv> φ in v]"
        by PLM_solver
      ultimately show ?thesis
        unfolding diamond_def by blast
    qed

  lemma id_nec2_2[PLM]:
    "[((α::'a::id_eq) \<noteq> β) \<equiv> \<box>(α \<noteq> β) in v]"
    using id_nec2_1[THEN id_nec2_2_Aux] by auto

  lemma id_nec2_3[PLM]:
    "[(\<diamond>((α::'a::id_eq) \<noteq> β)) \<equiv> (α \<noteq> β
    using "T\<diamond>" "\<equiv>I" id_nec2_2[equiv_lr]
          CP derived_S5_rules_2_b by metis

  lemma exists_desc_box_1[PLM]:
    java.lang.NullPointerException
    proof (rule CP)
      assume "[\<exists>y. (y^P) = (\<iota>x. φ x) in v]"
      then obtain y where "[(y^P) = (\<iota>x. φ x) in v]"
        by (rule "\<exists>E")
      hence "[\<box>(y^P = (\<iota>x. φ x)) in v]"
        using l_identity[axiom_instance, deduction, deduction]
              cqt_1[axiom_instance] all_self_eq_2[where 'a=ν]
              modus_ponens unfolding identity_ν_def by fast
      thus "[\<exists>y. \<box>((y^P) = (\<iota>x. φ x)) in v]"
        by (rule "\<exists>I")
    qed

  lemma exists_desc_box_2[PLM]:
    "[(\<exists> y . (y^P) = (\<iota>x. φ x)) \<rightarrow>  \<box>(\<exists> y .((y^P) = (\<iota>x. φ x))) in v]"
    using exists_desc_box_1 Buridan ded_thm_cor_3 by fast

  lemma en_eq_1[PLM]:
    "[<^bold\lbrace,}<>  v]
    using encoding[axiom_instance] RN
          sc_eq_box_box_1 modus_ponens by blast
  lemma en_eq_2[PLM]:
    "[{x,F} \<equiv> \<box>{x,F} in v]"
    using encoding[axiom_instance] qml_2[axiom_instance] by (rule "\<equiv>I")
  lemma en_eq_3[PLM]:
    "[\<diamond>{x,F} \<equiv> {x,F} in v]"
    using encoding[axiom_instance] derived_S5_rules_2_b "\<equiv>I" "T eq_E_simple[equiv_rl]] unfo Ordin by f
  lemma en_eq_4[PLM]:
    "[({x,F} \<equiv> {y,G}
    by (metis CP en_eq_2 "\<equiv>I" "\<equiv>E"(1) java.lang.NullPointerException
  lemma en_eq_5[PLM]:
    "\^><>{>\lbracey,G}<>x,<>\
    using "\<equiv>I" KBasic_6 encoding[axiom_necessitation, axiom_instance]
    sc_eq_box_box_3[deduction] "&I" by simp
  lemma en_eq_6[PLM]:
    "[({x,F} \<equiv> {y,G}) \<equiv> \<box>({x,F} \<equiv> {y,G}) in v]"
    using en_eq_4 en_eq_5 oth_class_taut_4_a "\<equiv>E"(6) by meson
  lemma en_eq_7[PLM]:
    "[(\<not>{x,F}) \<equiv> \<box>(\<not>{x,F}) in v]"
    using en_eq_3[THEN id_nec2_2_Aux] by blast
  lemma en_eq_8[PLM]:
    "[\<diamond>(\<not>{x,F}) \<equiv> (\<not>{x,F}) in v]"
     unfolding diamond_def apply (PLM_subst_method "{x,F}" "\<not>\<not>{x,F}")
      using oth_class_taut_4_b apply simp
     apply (PLM_subst_method "{x,F}" "\<box>{x,F}")
      using en_eq_2 apply simp
     using oth_class_taut_4_a by assumption
  lemma en_eq_9[PLM]:
    "[\<diamond>(\<not>{x,F}) \<equiv> \<box>(\<not>{x,F}) in v]"
    using en_eq_8 en_eq_7 "\<equiv>E"(5) by blast
  lemma en_eq_10[PLM]:
    " hence "[🚫
    apply (rule "\<equiv>I")
     using encoding[axiom_actualization, axiom_instance,
                    THEN logic_actual_nec_2[axiom_instance, equiv_lr],
                    deduction[axiom_instance],
                    THEN en_eq_2[equiv_rl]] CP
     apply simp
    using encoding[axiom_instance] nec_imp_act ded_thm_cor_3 by blast

subsection‹The Theory of Relations›
text‹\label{TAO_PLM_Relations}›

  lemma beta_equiv_eq_1_1[PLM]:
    assumes<>"
        and "IsProperInX ψ"
        and "∧x.[φ (x^P) \<equiv> ψ (x^P) in v]"
    shows java.lang.NullPointerException
    using lambda_predicates_2_1[OF assms(1), axiom_instance]
    using lambda_predicates_2_1[OF assms(2), axiom_instance]
    using assms(3) by (meson "\<equiv>E"(6) oth_class_taut_4_a)

  lemma beta_equiv_eq_1_2[PLM]:
    assumes "IsProperInXY φ"
        and "IsProperInXY ψ"
        and "∧x y.[φ (x^P) (y^P) >ψy⁾ in]"
    shows "[(\<lambda>² (λ x y. φ (x^P) (y^P)), x^P, y^P)
            \<equiv> (\<lambda>² (λ x y. ψ (x^P) (y^P)), x^P, y^P) in v]"
    using lambda_predicates_2_2[OF assms(1), axiom_instance]
    using lambda_predicates_2_2[OF assms(2), axiom_instance]
    using assms(3) by (meson "\<equiv>E"(6) oth_class_taut_4_a)

  lemma beta_equiv_eq_1_3[PLM]:
    assumes "IsProperInXYZ φ"
        and "IsProperInXYZ ψ"
        and "∧x y z.[φ (x^P) (y^P) (z        using byauto
    shows java.lang.NullPointerException
            \<equiv> (\<lambda>³ (λ x y z. ψ assumes "IsProperInXYZ>
    using lambda_predicates_2_3[OF assms(1and \psi
    using lambda_predicates_2_3[OF assms(2), axiom_instance]
    usingassms by

  lemma beta_equiv_eq_2_1[PLM]:
    assumes "IsProperInX φ"
        and "IsProperInX ψ"
    shows "[(\<box>(\<forall>
java.lang.NullPointerException
     apply (rule qml_1[axiom_instance, deduction])
     apply (rule RN)
     proof (rule CP, rule "\<forall>I")
      fix v x
      assume "[\<forall>x. φ (x^P) \<equiv> ψ (x^P) in v]"
      hence "∧x.[φ (x⁽ule]java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
        by PLM_solver
      thus "[(\<lambda> y. φ (y^P), x^P) \<equiv> (\<lambda> y. ψ (y^{^}>\existsalp>. (<alp <a [(\bold\exists η) x) in v"
        using assms beta_equiv_eq_1_1 by auto
     qed

  lemma beta_equiv_eq_2_2[PLM]:
    assumes "IsProperInXY φ"
        and "IsProperInXY ψ"
    shows java.lang.NullPointerException
            (\<box>(\<forall> x y . (\<lambda>² (λ x y. φ (x^P) (y^P)), x^P, y^P)
              \<equiv> (\<lambda>² (λ x y. ψ (x^P) (y^P)), x^P, y^P))) in v]"
    apply (rule qml_1[axiom_instance, deduction])
    apply (rule RN)
    proof (rule CP, rule "\<forall>I", rule java.lang.NullPointerException
      fix v x y
      assume "[\<forall>x y. φ (x^P) (yjava.lang.NullPointerException
      hence "( llemma thm[PLM]:
old∀
      thus "[(\<lambda>² (λ x y. φ (x^P) (y^P)), x^P, y^P)
            \<equiv> (java.lang.NullPointerException
        using assms beta_equiv_eq_1_2 by auto
    qed

  lemma beta_equiv_eq_2_3[PLM]:
    assumes
        and "IsProperInXYZ ψ"
    shows "[(\<box>(\<forall> x y z . φ (x^P) (y^P) (z^P) \<equiv> ψ (x^P) (y^P) (z\      thenobtain α
java.lang.NullPointerException
java.lang.NullPointerException
    apply (rule qml_1[axiom_instance, deduction])
    apply (rule RN)
    proof (rule CP, rule "\<forall>I", rule "\<forall>I", rule "\<forall>I")
      fix v x y z
      assume "[\<forall>x y z. φ (x^P) (y^P) (z^{^}>,y<sup^>P<>)
      hence "(∧x y z.[φ (x^P) (y^P) (z^P) \<equiv> ψ (x^P) (y^P) (z^P) in v])"
        by (meson "\<forall>E")
      thus java.lang.NullPointerException
              \<equiv> (\<lambda> - y PL
        using assms beta_equiv_eq_1_3 by auto
    qed

  lemma beta_C_meta_1[PLM]:
    assumes "IsProperInX φ"
    shows "[(\<lambda> y. φ (y^P), x^P) \<equiv> φ (x^P) in v]"
    using lambda_predicates_2_1[OF assms, axiom_instance] by auto

  lemma beta_C_meta_2[PLM]:
    assumes "IsProperInXY φ"
    shows "[(\<lambda>² (λ x y. φ (x^P) (yjava.lang.NullPointerException
    using lambda_predicates_2_2[OF assms, axiom_instance] by auto"[(\^>¬≡

  lemma beta_C_meta_3[PLM]:
    assumes "IsProperInXYZ φ"
    shows "[(java.lang.NullPointerException
    using lambda_predicates_2_3[OF assms, axiom_instance

  lemma relations_1[PLM]:
    assumes "IsProperInX φ"
    shows java.lang.NullPointerException
    using assms apply - by PLM_solver

  lemma relations_2[PLM]:
    assumes "IsProperInXY φ"
    shows "[\<exists> F. \<box>(\<forall> x y. (F,x^P,y^P) \<equiv> φ (x^<>(java.lang.NullPointerException
    using assms apply - by PLM_solver

  lemma relations_3[PLM]:
    assumes "IsProperInXYZ φ
    shows "[\<exists> F. \<box>(java.lang.NullPointerException
    using assms apply - by PLM_solver

  lemma prop_equiv[PLM]:
    shows "[(\<forall> x . ({"F:Pi2^oldF<sup"
    proof (rule CP)
      assume 1: "[\<forall>x. {xjava.lang.NullPointerException
      {
        fix
        have "[{x^P,F} \<equiv> {x^P,G} in v]"
          using 1 by (rule "\<forall>E")
        hence "[\<box>({x^P,F} \<equiv> { ded
          using PLM.en_eq_6 "\<equiv>E"(1) by blast
      }
      hence "[\<forall>x. \<oth_class_taut_1_b
        by (rule "\<forall>I")
      thus "[F = G in v]"
        unfolding identity_defs
        by (rule BF[PLM
    qed

  lemma propositions_lemma_1[PLM]:
    java.lang.NullPointerException
    using lambda_predicates_3_0[axiom_instance] .

java.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 34
    "[java.lang.NullPointerException
    using lambda_predicates_3_0[axiom_instance, THEN∨rparr and^\iota\phix <bold>E (<^bold<>x . \psi> x)in]"
    apply (rule l_identity[axiom_instance, deduction, deduction])
    by PLM_solver

  lemma propositions_lemma_4[PLM]:
    assumes "∧x.[\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        java.lang.NullPointerException
    proof -
      have "[\<lambda>⁰ (χ (\<iota>x. φ x)) = <^l_identity[
        using assms lambda_predicates_4_0[          oth_class_taut_4_athm_relation_negation_3><equiv5
        by blast
      hence java.lang.NullPointerException
        using propositions_lemma_1[THEN id_eq_prop_prop_8_b[deduction]]
              id_eq_prop_prop_9_b[deduction] "&I"
        by blast
      thus ?thesis
        using propositions_lemma_1 id_eq_prop_prop_9_b[deduction] "&I"
        by blast
    qed

  lemma propositions[PLM]:
    "[\<exists> p . \<box>(p \<equiv> p') in v]"
solver

  lemma pos_not_equiv_then_not_eq[PLM]:
    "[\<diamond>(\<not[axiom_instance by
    unfolding diamond_def
    proof (subst contraposition_1[symmetric], rule CP)
      assume "[F = G in v]"
      thus java.lang.NullPointerException
        apply (rule l_identity[axiom_instance, deduction, deduction])
        by PLM_solver
    qed

  lemma thm_relation_negation_1_1[PLM]:
    "[(F^-, xjava.lang.NullPointerException
    unfolding propnot_defs
    apply (
    by show_proper

  lemma thm_relation_negation_1_2[PLM]:
    "[(F^-, x^P, y^P) \<equiv> \<not>(F, x^"\^><f>α
    unfolding propnot_defs
    apply (rule lambda_predicates_2_2[axiom_instance])
    by show_proper

  lemma thm_relation_negation_1_3[PLM]:
    "<F<sup>P,y<^upz<>P
    unfolding propnot_defs
    apply (rule lambda_predicates_2_3[axiom_instance])
    by show_proper

  lemma thm_relation_negation_2_1
    java.lang.NullPointerException
    using thm_relation_negation_1_1[THEN oth_class_taut_5_d[equiv_lr]]
    apply - by PLM_solver

  lemma thm_relation_negation_2_2[PLM]:
    "[(\<not>(F^<forall>E" =rul
    using thm_relation_negation_1_2[THEN oth_class_taut_5_d[equiv_lr]]
    apply - by PLM_solver

  lemma thm_relation_negation_2_3[PLM]:
    "[(\<not>([PLM]:
    using thm_relation_negation_1_3[THEN oth_class_taut_5_d[equiv_lr]]
    apply - by PLM_solver

  lemmathm_relation_negation_3PLM:
    "[(p)^xi v]"
    unfolding propnot_defs
    using propositions_lemma_2 by simp

  lemma thm_relation_negation_4[PLM]:
    java.lang.NullPointerException
    using thm_relation_negation_3[THEN oth_class_taut_5_d[equiv_lr]]
    apply - by PLM_solver

  lemma thm_relation_negation_5_1[PLM]:
    "[(F::Π\    [(java.lang.NullPointerException
    using id_eq_prop_prop_2[deduction]
          l_identity[where φ="λ G . (G,x^P) \<equiv> (F^-,x^P)", axiom_instance,
                      deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_1 "\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma 
    java.lang.NullPointerException
    using id_eq_prop_prop_5_a[deduction]
          l_identity[where φ="λ G . (G,x^P,y^P) java.lang.NullPointerException
                      deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_2 "\<^    "
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_5_3[PLM]:
    "[(F::Π₃) \<noteq> (F^-) in v]"
    using id_eq_prop_prop_5_b[deduction]
          l_identity[where φ="λ G . (
                     axiom_instance, deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_3 "\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_6[PLM]:
    "[(p::o) \<noteq> (p^-) in v]"
    using id_eq_prop_prop_8_b[deduction]
          l_identity[where φ="λ G . G \<equiv> (p^-)", axiom_instance,
                      deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_3 "\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_7[PLM]:
    "[((p::o_=> n thm =>t RS @{th id_[d]}))
    unfolding propnot_defs using propositions_lemma_1 by simp

  lemma thm_relation_negation_8[PLM]:
    "[(p::o) java.lang.NullPointerException
    unfolding propnot_defs 
    using id_eq_prop_prop_8_b[deduction]
          l_identity[where φ=java.lang.NullPointerException
                      deduction, deduction]
           oth_
          modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_9[PLM]:
    "[((p::o) = q) \<rightarrow> ((\<not>p) = (java.lang.NullPointerException
    using l_identity[where α="p" and β="q" and φ="λ x . (^ 🚫
                      axiom_instance, deduction]
          id_eq_prop_prop_7_b using CP modus_ponens by blast

  lemma thm_relation_negation_10[PLM]:
    "[((p::o) = q) \<rightarrow> ((pjava.lang.NullPointerException
    using l_identity[where α="p" and β="q" and φ="λ x . (p^[>\^ \tau <bold>→>= \\>) in v]"
                      axiom_instance, deduction]
          id_eq_prop_prop_7_b using CP modus_ponens by blast

  lemma thm_cont_prop_1[PLM]:
    "[NonContingent (F::Π₁) \<equiv> NonContingent (F^-) in v]"
    proof (rule "\<equiv>I"; rule CP)
      assume "[NonContingent F in v]"
      hence java.lang.NullPointerException
        unfolding NonContingent_def Necessary_defs Impossible_defs .
      hence "[\<box>(\<forall>x. \<not>(F^-,x^P)) \<or> \<box>(\beta<>\^>P \^
        apply -
        apply (PLM_subst_method "λ x . (F,x^P)" java.lang.NullPointerException
        using thm_relation_negation_2_1[equiv_sym] by auto
      hence "[\<box>(\<forall>x. \<not>(F^-,x^P)) \<or> \<box>(\<forall>x. (F^-,x^P)) in v]"
        apply -
        apply (PLM_subst_goal_method
               "          byjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
        using thm_relation_negation_1_1[equiv_sym] by auto
      hence "[\<box>(\<forall>x. (F^>>. <>🚫
        by (rule oth_class_taut_3_e[equiv_lr])
      thus "[NonContingent (F^-) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    next
      assume "[NonContingent (F^-) in v]"
      hence "[\<box>(\<forall>x. \<not>(F^-,x^P)) \<or> \<box>(\<forall>x. (         java.lang.NullPointerException
        unfolding NonContingent_def Necessary_defs Impossible_defs
        by (rule oth_class_taut_3_e[equiv_lr])
      hence "[\<box>(\<forall>x.(
        apply -
        applyPLM_subst_methodlambda^boldF\^-x\^sup"<> x. \lparrx<^up>P<pa"
        using thm_relation_negation_2_1 by auto
      hence "[\<box>(\<forall>x. (F,x^P)) \<or> \<box>(\<forall>x. \<not>(F,x^P)) in v]"
        apply -
        apply (PLM_subst_method "λ x . (F^-,x^P)" "λ
        using thm_relation_negation_1_1 by auto
      thus "[NonContingent F in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    qed

  lemma thm_cont_prop_2[PLM]:
    "[Contingent F \<equiv> \<diamond>(\<exists> x . (
    proof (rule "\<equiv>I"; rule CP)
      assume "[Contingent F in v]"
      hence java.lang.NullPointerException
        unfolding Contingent_def Necessary_defs Impossible_defs .
      hence "[(\<not>\<box>(\<forall>x.(F,x^P))) & (\<not>\<box>(\<forall>x.\<not>(F,x^P))) in v]"
        by (rule oth_class_taut_6_d[equiv_lr])
      hence "[(\<diamond>\<not>(\<forall>x.\<not>(F,xjava.lang.NullPointerException
        using KBasic2_2[equiv_lr] "&I" "&E" by meson
      thus java.lang.NullPointerException
        unfolding exists_def apply -
        apply (PLM_subst_method "λ x . (F,x^P)" "λ x . \<not>\<not>(F,x^P)")
        using oth_class_taut_4_b by auto
    next
      assume "[(\<diamond>applyjava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
      hence java.lang.NullPointerException
        unfolding exists_def apply -
        apply (PLM_subst_goal_method
               "λ φ . (\<diamond>\<not>(\
        using oth_class_taut_4_b[equiv_sym] by auto
      hence java.lang.NullPointerException
        using KBasic2_2[equiv_rl] "&I" "&E" by meson
      hence "[\<not>(\<box>(\<forall>x.(F,x^P)) \<or> \<box>(\<forall>x.\<not>(F,x^P))) in v]"
        by (rule oth_class_taut_6_d[equiv_rl])
      thus "[Contingent F in v]"
        unfolding Contingent_def Necessary_defs Impossible_defs .
    qed

  lemma thm_cont_prop_3[PLM]:
    "[Contingent (F::Π₁) \<equiv> Contingent (F^-) in v]"
    using thm_cont_prop_1
    unfolding NonContingent_def Contingent_def
    by (rule oth_class_taut_5_d[equiv_lr])

  lemma lem_cont_e[PLM]:
    "[\<diamond>(\<exists> x . (F,x^P) java.lang.NullPointerException
    proof -
      have "by PLM
java.lang.NullPointerException
        using "BF\<diamond>"[deduction] "CBF [^<'in<sub
      also have "... = [\<exists> x . (\<diamond>(F,x^P) & \<diamond>(\<not>(F,x^P))) in v]"
        apply (PLM_subst_method
               java.lang.NullPointerException
               "λ x . \<diamond>(F,x^P)
        using S5Basic_12 by auto
      alsohave. = \^\exists>.\^><>\^><><>,\^>🚫
        apply (PLM_subst_method
               "λ x . \<diamond>(F,x^P) & \<diamond>(\<not>(F,x^P))"
               "λ x . \<diamond>(\<not>(F,x^P)) & \<diamond>(F,x^P)")
        using oth_class_taut_3_b by auto
      also have java.lang.NullPointerException
        apply (PLM_subst_method
               "λ x . [ \phi "<> \beta . \<^>\
               "λ x . \<diamond>((\<not>(F,x^P)) & \<diamond>(F,x id_eq_1ded_thm_cor_4>def
        using S5Basic_12[equiv_sym] by auto
      also have java.lang.NullPointerException
        using "CBF\<diamond>"[deduction] "BF\<diamond>"[deduction] by fast
      finally show ?thesis using "\<equiv>I" CP by blast
    qed

  lemma lem_cont_e_2[PLM]:
    "[\<diamond>(\<exists> x . (F,x^P) & \<diamond>(\<not>(F,x^P))) \<equiv> \<diamond>(\<exists> x . (F^-,x^P) & \<diamond>(\<not>(F^-,x^P))) in v]"
    apply (PLM_subst_method "λ x . (F,x^P)" "λ x . \<not>(F^-,x^P)")
     using thm_relation_negation_2_1[equiv_sym] apply simp
    apply (PLM_subst_method "λ x . \<not>(F,x^P)" "λ x . (F^-,x^P)")
     roof (cases "(^><>\alpha \alpha\^>)\^>(^><>x.\phi x))  ]\and[\^><>\> \beta<sup>=\^"
    using lem_cont_e by simp

  lemma thm_cont_e_1[PLM]:
   [\>di>(<b>\<> 
    using lem_cont_e[where F="E!", equiv_lr] qml_4[axiom_instance,conj1]
    by blast

  lemma thm_cont_e_2[PLM]:
    "[Contingent (E!) in v]"
    using thm_cont_prop_2[equiv_rl] "&I" qml_4[axiom_instance, conj1]
          KBasic2_8[deduction, OF sign_S5_thm_3[deduction], conj1]
          KBasic2_8[deduction, OF sign_S5_thm_3[deduction, OF thm_cont_e_1], conj1]
    by fast

  lemma thm_cont_e_3[PLM]:
    "[Contingent (E!java.lang.NullPointerException
    using thm_cont_e_2 thm_cont_prop_3[equiv_lr] by blast

  lemma thm_cont_e_4[PLM]:
    "[\<exists> (F::Π_"(<bo🚫
    apply (rule_tac α="E!" in "\<exists>I", rule_tac α="E!^-" in "\<exists>I")
    using thm_cont_e_2 thm_cont_e_3 thm_relation_negation_5_1 "&I" by auto

  context
  begin
    qualified definition L where "L ≡)^>rightarrow   modus_ponens
    
    lemma thm_noncont_e_e_1[PLM]:
      "[Necessary L in v]"
      unfolding apply( RN^>\forallIjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
      apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl])
        apply show_proper
      using if_p_then_p .

    lemma]
      "[Impossible (L^\\^b>\<box(r>) in v]"
      unfolding Impossible_defs L_def apply (rule RN, rule "\<forall>I")
      apply (rule thm_relation_negation_2_1[equiv_rl])
      apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl])
       apply show_proper
      using if_p_then_p .

    lemma thm_noncont_e_e_3[PLM]:
      "[NonContingent (L) in v]"
      unfolding NonContingent_def using thm_noncont_e_e_1
      by (rule java.lang.NullPointerException

    lemma thm_noncont_e_e_4[PLM]:
      "[NonContingent (L^-) in v]"
      unfolding NonContingent_def using thm_noncont_e_e_2
      by (rule "\<or>I"(2))

    lemma thm_noncont_e_e_5[PLM]:
      "[\<exists> (F::Π₁) G . F \<noteq> G & NonContingent F & NonContingent G in v]"
      apply (rule_tac α="L" in "\<exists>I", rule_tac α="L by (rule
      using java.lang.NullPointerException
            thm_noncont_e_e_4 "&I"
      by simp


  lemma four_distinct_1[PLM]:
    "[NonContingent (F::Π₁) \<rightarrow> \<not>(\<exists> G . (Contingent G & G java.lang.NullPointerException
    proof (rule CP)
      assume[onContingent"
      hence "[\<not>(Contingent F) in v]"
        unfolding NonContingent_def Contingent_def
        apply - by PLM_solver
      moreover {
         assume "[\<exists> G . Contingent G & G = F in v]"
          obtai P wh "Contingent  \boldFin
          by (rule "\<exists>E")
         hence "[Contingent F in v]"
           using "&E" l_identity[axiom_instance, deduction, deduction]
           by blast
      }
      ultimately<boldbold>byPLM_solver
        using modus_tollens_1 CP by blast
    qed

  lemma four_distinct_2[PLM]:
    java.lang.NullPointerException
    proof (rule CP)
      assume "[Contingent F in v]"
      hence "[\<not>(NonContingent F) in v]"
        unfolding NonContingent_def Contingent_def
        apply - by PLM_solver
      moreover {
         assume "[\<exists> G . NonContingent G java.lang.NullPointerException
         then obtain P where "[NonContingent P & P = F in v]"
          by (rule "\<exists>E")
    NonContingent
           using "&E" l_identity[axiom_instance, deduction, deduction]
           by blast
      }
      hence(>diamond<>.^><>lparrrparrbold<><bold\^forallx\supP<>)) in
        using modus_tollens_1 CP by blast
    qed

    lemma four_distinct_3[PLM]:
      "[L \<noteq> (L^-) & L \<noteq> E! & L \<noteq> (E!^-) & (L^-) \<noteq> E!
        & (L^-) \<noteq> (E!^-) & E! \<noteq> (E!^-) in v]"
      proof (rule "&I")+
        show java.lang.NullPointerException
        by (rule thm_relation_negation_5_1)
      next
        {
          assume "[L <^    "
          hence "[NonContingent L & L = E! in v]"
            using thm_noncont_e_e_3 "hence "\bold>^>box<forall<sup<rparr^old>\>^>box<>.<boldF,<>P<>)in
          hence "[\<exists> G . NonContingent G & G = E! in v]"
            using thm_noncont_e_e_3 "&I" "\<exists>I" by fast
        }
        thus "[L \<noteq> E! in v]"
          using four_distinct_2[deduction, OF thm_cont_e_2]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "[L = (E!^-) in v]"
          hence java.lang.NullPointerException
java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 56
          hence "[\<exists> G . NonContingent G & G = (E!^-) in v]"
            using thm_noncont_e_e_3 "&I" "\<exists>I" by fast
        }
        thus "[L \<noteq> (E!^-) in v]"
          using four_distinct_2[deduction, OF thm_cont_e_3]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "[(L^-    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
          hence java.lang.NullPointerException
            using thm_noncont_e_e_4 "&I" by auto
          hence "[\<exists> G . NonContingent G & G = E! in v]"
            using thm_noncont_e_e_3 "&I" "\<exists>I" by fast
        }
        thus "[(L^-) \<noteq> E! in v]"
          using four_distinct_2[deduction, OF thm_cont_e_2]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "(java.lang.NullPointerException
          hence "[NonContingent (L^-) & (L^-) = (E!^-) in v]"
             thm_noncont_e_e_4<^bold" by au
          hence "[\<exists> G . NonContingent G & G = (E!^-) in v]"
            using thm_noncont_e_e_3 "&I" "\<exists>I" by fast
        }
java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 63
          using four_distinct_2[deduction, OF thm_cont_e_3]
                modus_tollens_1 CP
          by blast
      next
        show "[E! \<noteq> (E!^-) in v]"
          by (rule thm_relation_negation_5_1)
      qed
  end

  lemma thm_cont_propos_1[PLM]:
    "[NonContingent (p::o) \<equiv> NonContingent (p^-) in v]"
    proof (rule "\<equiv>I"; rule CP)
      assume "[NonContingent p in v]"
      hence "[\<box>p \<or> \<box>\<not>p in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
      hence "[\<box>(\<not>(p^-)) ><exists>,x^>,java.lang.NullPointerException
        apply -
        apply (PLM_subst_method "p" "\<not>(p^-)")
        using thm_relation_negation_4[equiv_sym] by auto
      hence "[\<box>(\<not>(p^-)) \<or> \<box>(p^-) in v]"
        apply -
        apply (PLM_subst_goal_method "λφ= [(>,x\^P\rparr>)) in v]"
        using thm_relation_negation_3[equiv_sym] by auto
      hence "[\<box>(p^-) \<or> \<box>(\<not>(p^-)) in v]"
        by (rule oth_class_taut_3_e[equiv_lr])
      thus "[NonContingent (p^-) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    next
      assume "[NonContingent (p^-) in v]"
      hence "[\<box>( φo> 🚫
        unfolding NonContingent_def Necessary_defs Impossible_defs
        by (rule oth_class_taut_3_e[equiv_lr])
      hence "[\<box>(p) \<or> \<box>(p^-) in v]"
        apply -
        apply (PLM_subst_goal_method "λφ . \<box>φ \<or> \<box>(p^-)" "\<not>(p^-)")
        using thm_relation_negation_4 by auto
      hence "[java.lang.NullPointerException
        apply -
        apply (PLM_subst_method "p^-" "\<not>p")
        using thm_relation_negation_3 by auto
      thus "[NonContingent p in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    qed

]
    "[Contingent p "\lambda <^b>\diamond<lpar>F,x\^>)\^>\rparr"
    proof    apply PLM_solver
      assume "[Contingent p in v]"
      hence "[\<not>(\<box>p \<or> \<box>(\<not>p)) in v]"
        unfolding Contingent_def Necessary_defs Impossible_defs .
      hence "[(\<not>\<box>p) & (F,x\x<^spP<rparr>) <s>P\rparr> in v]
        by (ru oth_clas[equiv_lr])
      hence "[(\<diamond>\<not>(\<not>p)) & (\<diamond>\<not>p) in v]"
        using KBasic2_2[equiv_lr] "&I" "&E" by meson
      thus "[(\<diamond>p) & (\<diamond>(\<not>p)) in v]"
        apply - apply PLM_solver
        apply (PLM_subst_method "\<not>\<not>p" "p")
        using oth_class_taut_4_b[equiv_sym] by auto
    next
      assume "[(\<diamond>p) & (\<diamond>\<not>(p)) in v]"
      hence "[(\<diamond>"<lambdadiamond<><>Fx\supP<> 
        apply - apply PLM_solver
        apply (PLM_subst_method "p" "\<not>\<not>p")
        using oth_class_taut_4_b by auto
      hence "[(\<not>\<box>p) & (\<not>\<box>(\<not>p)) in v]"
        using KBasic2_2[equiv_rl] "&I" "&E" by meson
      hence        ".= \^\<xists 
        by (rule oth_class_taut_6_d[equiv_rl])
      thus "[Contingent p in v]"
        unfolding Contingent_def Necessary_defs Impossible_defs .
    qed

 lemmathm[PLM]:
    "[Contingent (p::o) \<equiv> Contingent (p^-) in v]"
    using thm_cont_propos_1
    unfolding NonContingent_def Contingent_def
    by (rule oth_class_taut_5_d[equiv_lr])

  context
  begin
    private definition p₀ where
      "p₀ ≡ \<forall>x. (E!,x^P)

    lemma thm_noncont_propos_1[PLM]:
      "[Necessary p₀ in v]"
      unfolding Necessary_defs p₀_def
      apply (rule RN, rule "\<forall>I")
      using .

    lemma thm_noncont_propos_2[PLM]:
      "Impossible p\^>0^unfolding ex by PLM
      unfolding Impossible_defs
      apply (PLM_subst_method "\<not>p₀" "p₀java.lang.NullPointerException
       using thm_relation_negation_3[equiv_sym] apply simp
      apply (PLM_subst_method\bold<><alpha><><>.\phi> \alpha>\beta)\^bold(<bold\beta. <^>\
       using oth_class_taut_4_b apply simp
      using thm_noncont_propos_1 unfolding Necessary_defs
      by simp

    lemma thm_noncont_propos_3[PLM]:
      "[NonContingent (p₀) in v]"
      unfolding NonContingent_def using thm_noncont_propos_1
      

    lemma thm_noncont_propos_4[PLM]:
      "[NonContingent (p₀^-) in v]"
      unfolding NonContingent_def using thm_noncont_propos_2
      by (rule "\<or>I"(2))

    lemma thm_noncont_propos_5[PLM]:
      "\^>∃^bol>≠<^bold>& onCq in v]"
      apply (rule_tac α="p₀" in "\<exists>I", rule_tac α="p₀^<> <bol>& \^old>\existsx . \<parrF
      using "\<exists>I" thm_relation_negation_6 thm_noncont_propos_3
            thm_noncont_propos_4 "&I" by simp

    private definition q₀ where
      "q₀ ≡ \<exists> x . (E!,x^P) & \<diamond>(\<not>(E!,x^P))"

    lemma basic_prop_1[PLM]:
      "[\<exists> p . \<diamond>p & \<diamond>(\<not>p) in v]"
      apply (rule_tac α="q₀" in "\<exists>I") unfolding q₀_def
      using qml_4[axiom_instance] by simp

    lemma basic_prop_2[PLM]:
      "[Contingentshow(^bold>x.\phix \^><> (<bold!.java.lang.NullPointerException
      unfolding Contingent_def Necessary_defs Impossible_defs
      apply (rule oth_class_taut_6_d[equiv_rl])
      apply (PLM_subst_goal_method "λ φ . (\<not>\<box>(φ)) & \<not>\<box>\<not>q₀" java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
       using oth_class_taut_4_b[equiv_sym] apply simp
      using qml_4[axiom_instance,conj_sym]
      unfolding q₀_def diamond_def by simp

    lemma basic_prop_3[PLM]:
      "[Contingent (q₀^-) in v]"
      apply (rule thm_cont_propos_3[equiv_lr])
      using basic_prop_2 .

    lemma basic_prop_4[PLM]:
      "[\<exists> (p::o) q . p \<noteq> q & Contingent p & Contingent q in v]"
      apply (rule_tac α="q₀" in "\<exists>I", rule_tac αby (simp only: eexists_uni)
      using thm_relation_negation_6 basic_prop_2 basic_prop_3 "&I" by simp

    lemma four_distinct_props_1[PLM]:
      "[NonContingent (p::Π₀) \<rightarrow> (E,java.lang.NullPointerException
      proof (rule CP)
        assume "[NonContingent p in v]"
        hence "[\<not>(Contingent p) in v]"
          unfolding NonContingent_def Contingent_def
          apply - by PLM_solver
        moreover {
           assume "[\<exists> q . Contingent q & q = p in v]"
           then obtain r where "[Contingent r using lem_cont_e[were F==",equiv_lr]
            by (rule "\<exists>E")
           hence "[Contingent p in v]"
             using "&E" l_identity[axiom_instance, deduction, deduction]
             by blast
        }
        ultimately show "[\<not>(\<exists>q. Contingent q & q = p) in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma four_distinct_props_2[PLM]:
      "[Contingent (p::oK[d
      proof (rule CP)
        assume "[Contingent p in v]"
        hence "[\<not>(NonContingent p) in v]"
          unfolding NonContingent_def Contingent_def
          apply - by PLM_solver
        moreover {
           assume "[\<exists> q . NonContingent q & q = p in v]"
           then obtain r where "[NonContingent r & r = p in v]"
            by (rule "\<exists>E")
           hence "[NonContingent p in v]"
             using "&E" l_identity[axiom_instance, deduction, deduction]
             by blast
        }
        ultimately show "[\<not>(\<exists>q. NonContingent q & q = p) in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma four_distinct_props_4[PLM]:
      "[p₀ \<noteq> (p₀^-) & p₀ \<noteq> q₀ & p₀ \<noteq> (q₀^-) & (p₀^-) \<noteq> q₀
        & (p₀^-) \<noteq> (q₀^-) & q₀ \<noteq> (q₀^-) in v]"
      proof (rule "&I")+
        show "[p by
          by (rule thm_relation_negation_6)
        next
          {
            assume "[p₀ = q₀ in v]"
            hence "[\<exists> q . NonContingent q & q = q₀ in v]"
              using "&I" thm_noncont_propos_3 "\<exists>I"[where α=p₀]
              by
          }
          thus java.lang.NullPointerException
            using four_distinct_props_2[deduction, OF basic_prop_2]
                  modus_tollens_1 CP
            by blast
        next
          {
            assume "[p₀ = (q₀^-) in v]"
            hence [\\> q . NonCont q\^>&q \^>= (q\^sub>0\<^>-
              using thm_noncont_propos_3 "&I" "\<exists>I"[where α=p₀] by simp
          }
          thus "[p₀ \<noteq> (q₀^-) NonContingent_def
            using four_distinct_props_2
                  modus_tollens_1 CP
          by blast
        next
          {
            assume java.lang.NullPointerException
            hence "[\<exists> q . NonContingent q & q = q₀ in v]"
              using thm_noncont_propos_4 "^>rightarrowA\phiin]"
          }
          thus "[(p₀^-) \<noteq> q₀ in v]"
            using four_distinct_props_2[deduction, OF basic_prop_2]
                  modus_tollens_1 CP
            by blast
        next
          {
            assume "[(p₀^-) java.lang.NullPointerException
            hence java.lang.NullPointerException
              using thm_noncont_propos_4 "&I" "\<exists>I"[where α="p₀^-"] by auto
          }
          thus "[(p₀^-) \<noteq> (q₀^-) in v]"
            using four_distinct_props_2[deduction, OF basic_prop_3]
                  modus_tollens_1 CP
            by blast
        next
          show "[q_byPLM_solver
            by (rule thm_relation_negation_6)
        qed

    lemma cont_true_cont_1[PLM]:
      java.lang.NullPointerException
      apply (rule CP, rule thm_cont_propos_2[equiv_rl])
      unfolding ContingentlyTrue_def
       apply (rule"<drule
       using "T\<diamond>"[deduction] apply simp
      by (rule "&E"(2))
  
    lemma cont_true_cont_2[PLM]:
      "[ContingentlyFalse p \<rightarrow> Contingent p in v]"
      apply (rule CP, rule thm_cont_propos_2[equiv_rl])
      unfolding ContingentlyFalse_def
      apply (rule java.lang.NullPointerException
       apply simp
      apply (drule "&E"(1))
      using "T\<diamond>"[deduction] by simp
  
    lemma cont_true_cont_3[PLM]:
      "[ContingentlyTrue p alpha)>)<bold<>\alpha) in
      unfolding ContingentlyTrue_def ContingentlyFalse_def
      apply (PLM_subst_method "\<not>p" "p^-")
       using thm_relation_negation_3[equiv_sym] apply simp
      apply (PLM_subst_method "p" "\<not>\<not>p")
      by PLM_solver+
  
    lemma cont_true_cont_4[PLM]:
      java.lang.NullPointerException
       ContingentlyTrue_def ContingentlyFalse_def
      apply (PLM_subst_method "\<not>p" "p^-")
       using thm_relation_negation_3[equ by (rule "<bold
      apply (PLM_subst_method "p" "\<not>\<not>p")
      by PLM_solver+

    lemma cont_tf_thm_1PLM
      "[ContingentlyTrue q₀ \<or> ContingentlyFalse q₀ in v]"
      proof -
        have "[q₀ \<or> \<not>q₀ in v]"
          by PLM_solver
        moreover {
          assume "[q₀ in v]"
          henceq<sub^bold>🚫
            unfolding q₀_def
            using qml_4[axiom_instance,conj2] "&I"
            by auto
        }
        moreover {
          assume "[\<not>q₀ in v]"
java.lang.NullPointerException
            unfolding q₀_def
            using qml_4[axiom_instahence "[Contingent F
            by auto
        }
        ultimately show ?thesis
          unfolding ContingentlyTrue_def ContingentlyFalse_def
java.lang.NullPointerException
      qed

    lemma cont_tf_thm_2[PLM]:
      "[ContingentlyFalse q₀ \<or> ContingentlyFalse (q₀^-) in v]"
      using cont_tf_thm_1 cont_true_cont_3[where p="q₀"]
            cont_true_cont_4[where p="q₀"]
      apply - by PLM_solver

    lemma cont_tf_thm_3[PLM]:
      "[\<exists> p . ContingentlyTrue p in v]"
      proof (rule "\<or>E"(1); (rule CP)?)
        show "[ContingentlyTrue q₀ \<or> ContingentlyFalse q₀ in v]"
          using cont_tf_thm_1 .
      
        assume "[ContingentlyTrue q₀ in v]"
        thus ?thesis
          
      next
java.lang.NullPointerException
        hence "[ContingentlyTrue (q_F<\^) \^>\rightarrow \^\<not(le ins[PLM,PL,PLM_des]]:
          using cont_true_cont_4[equiv_lr] by simp
        thus ?thesis
          using "\<exists>I" by metis
      qed

    lemma cont_tf_thm_4[PLM]:
java.lang.NullPointerException
      proof (rule "\<or>E"(1); (rule CP)?)
        
          using cont_tf_thm_1 .
      next
         lemmas "\^<>E = nstanti
        hence "[ContingentlyFalse (q₀^-) in v]"
          using cont_true_cont_3[equiv_lr] by simp
        thus ?thesis
          using "\<exists>I" by metis
      next
        assume "[ContingentlyFalse q₀ in v]"
        thus ?thesis
          using "\<exists>I" by metis
      qed

    lemma cont_tf_thm_5[PLM]:
      "[ContingentlyTrue p & Necessary q \<rightarrow> p \<noteq> q in v]"
      proof (rule CP)
        assume "[ContingentlyTrue p & Necessary q in v]"
        hence 1: "[\<diamond>(\<not>p) & \<box> q in v]"
          unfolding ContingentlyTrue_def Necessary_defs
          using "&E" "&I" by blast
        hence "[\<not>\<box>p in v]"
          apply - apply (drule "&E"(1))
          unfolding diamond_def
          apply (PLM_subst_method "\<not>\<not>p" "p")
          using oth_class_taut_4_b[equiv_sym] by auto
        moreover {
          assume "[p = q in v]"
          hence "[\<box>p in v]"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
                             axiom_instance, deduction, deduction]
                  1[conj2] id_eq_prop_prop_8_b[deduction]
            by blast
        }
        ultimately show "[p \<noteq> q in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma cont_tf_thm_6[PLM]:
      "[(ContingentlyFalse p & Impossible q) \<rightarrow> p \<noteq> q in v]"
      proof (rule CP)
        assume "[ContingentlyFalse p & Impossible q in v]"
java.lang.NullPointerException
          unfolding ContingentlyFalse_def Impossible_defs
          using "&E" "&I" by blast
        hence "[\<not>\<diamond>q in v]"
          unfolding diamond_def apply - by PLM_solver
        moreover {
          assume "[p = q in v]"
          hence "[\<diamond>q in v]"
            using l_identity[axiom_instance, deduction, deduction] 1[conj1]
                  id_eq_prop_prop_8_b[deduction]
            by blast
        }
        ultimately show "[p \<noteq> q in v]"
          using modus_tollens_1 CP by blast
      qed
  

  lemma oa_contingent_1[PLM]:
    "[O! \<noteq> A! in v]"
    proof -
      {
        assume "[O! = A! in v]"
        hence "[(\<lambda>x. \<diamond>(<<>. \<psi 
          unfolding Ordinary_def Abstract_def .
        moreover have "[((\<lambda>x. \<diamond>(E!,x^P)), x^P) \<equiv> \<diamond>(E!,x^P) in v]"
          apply (rule beta_C_meta_1)
          by show_proper
        ultimately have "[((\<lambda>x. \<not>\<diamond>(E!,x^P)), x^P) \<equiv> \<diamond>(E!,x^P) in v]"
          using l_identity[axiom_instance, deduction, deduction] by fast
        moreover have "[((\<lambda>x. \<not>\<diamond>(E!,x^P)), x^exists_def by PLM
          apply (rule beta_C_meta_1)
          by show_proper
        ultimately have "[\<diamond>(E!,x^P) \<equiv> \<not>\<diamond>(E! "[(φ \^& ((\^bold>🚫
          apply - by PLM_solver
      }
      thus ?thesis
        using oth_class_taut_1_b modus_tollens_1 CP
        by blast
    qed

  lemma oa_contingent_2[PLM]:
"<O,x\^>P\\rparr> \ \<^>\
    proof -
        have "[((\<lambda>x. \<not>\<diamond>(E!,x^P)), x un exists_def b by PLM_sol
          apply (rule beta_C_meta_1)
          by show_proper
        hence "[(\<not>((\<lambda>x. \<not>^e>\>< \alpha\^> (\<><
          using oth_class_taut_5_d[equiv_lr] oth_class_taut_4_b[equiv_sym]
                "\<equiv>E"(5) by blast
        moreover have "[((\<lambda>x. \<diamond>(E!,x^P)), x^P) \<equiv> \<diamond>(E!,x^P) in v]"
          apply (rule beta_C_meta_1)
          by show_properjava.lang.NullPointerException
        ultimately show ?thesis
          unfolding Ordinary_def Abstract_def
          apply - by PLM_solver
    qed

  lemma oa_contingent_3[PLM]:
    "[(A!,x^P) \<equiv> \<not>(O!,x^P) in v]"
    using oa_contingent_2
    apply - by PLM_solver

  lemma oa_contingent_4[PLM]:
    "[Contingent O! in v]"
    apply (rule thm_cont_prop_2[equiv_rl], rule "&I")
    subgoal
      unfolding Ordinary_def
      apply (PLM_subst_method "λ x . \<diamond>(E!,x^P)" "λ x . (\<lambda>x. \<diamond>(
       apply (safe intro!: beta_C_meta_1[equiv_sym])
        apply show_proper
      using "BF\<diamond>"[deduction, OF thm_cont_prop_2[equiv_lr, OF thm_cont_e_2, conj1]]
      by (rule "T\<diamond>"[deduction])
    subgoal
      apply (PLM_subst_method "λ x . (A!,x^P)> \<> 
       using oa_contingent_3 apply simp
      using cqt_further_5[deduction,conj1, OF A_objects[axiom_instance]]
      by (rule "T\<diamond>"[deduction])
    done

  lemma oa_contingent_5[PLM]:
    "[Contingent A! in v]"
    apply (rule thm_cont_prop_2[equiv_rl], rule "&I")
    subgoal
      using cqt_further_5[deduction,conj1, OF A_objects[axiom_instance]]
      by (rule "T\<diamond>"[deduction])
    subgoal
      unfolding Abstract_def
      apply (PLM_subst_method "λ x . \<not>\<diamond>(E!,x^P)" "λ x . (\<lambda>x. \<not>\<diamond>(
       apply (safe intro!: beta_C_meta_1[equiv_sym])
        apply show_proper
java.lang.NullPointerException
       using oth_class_taut_4_b apply simp
      using "BF\<diamond>"[deduction, OF thm_cont_prop_2[equiv_lr, OF thm_cont_e_2, conj1]]
      by (rule "T\<diamond>"[deduction])
    done

  lemma oa_contingent_6[PL by by blast
    "[(O!^-) \<noteq> (A!^-) in v]"
    proof -
      {
        assume "[(O! <^ol>\equivphi>) in v]"
        hence "[(\<lambda>x. \<not>(O!,x^P)) = (\<lambda>x. \<not>(A!,x^P)) in v]"
          unfolding propnot_defs .
        moreover have "[((\<lambda>x. \<not>(O!,x^P)), x^P) \<equiv> \<not>(O!,x^P) in v]"
          apply (rule beta_C_meta_1)
          by show_proper
        ultimately have "[(\<lambda>x. \<not>(A!,x^P),x^P) \<equiv> \<not>(O!,x^P) in v]"
          using l_identity[axiom_instance, deduction, deduction]
          by fast
        hence "[\^><>\lparr>!,x\<^P\<^sup>P)
          apply -
          apply (PLM_subst_method "(\<lambda>x. \<not>(A!,x^P),x^P)" "(\<not>(A!,x^P))")
           apply (safe intro!: beta_C_meta_1)
          by show_proper
        hence "[(O!,x^P) \<equiv> \<not>(O!,x^P) in v]"
          using oa_contingent_2 apply - by PLM_solver
      }
      thus ?thesis
        using oth_class_taut_1_b modus_tollens_1 CP
        by blast
    qed

  lemma oa_contingent_7[PLM]:
    "[(O!^-,x^P) \<equiv> \<not>(A!^bold>A<hi> α<a>) i v]"
    proof -
      have "[(\<not>(\<lambda>x. \<not>(A!,x^P),x^P)) \<equiv> (A!,x^P) in v]"
        apply (PLM_subst_method "(\<not>(A!,x^P))" "(\<lambda>x. \<not>(A!,x^P),x^P)")
         apply (safe intro!: beta_C_meta_1[equiv_sym])
          apply show_proper
        using oth_class_taut_4_b[equiv_sym] by auto
java.lang.NullPointerException
        apply (rule beta_C_meta_1)
        by show_proper
ultimatelyshow ?thesis
        unfolding propnot_defs
        using oa_contingent_3
        apply - by PLM_solver
    qedjava.lang.NullPointerException

  lemma oa_contingent_8[PLM]:
    "[Contingent (O!^-) in v]"
    using oa_contingent_4 thm_cont_prop_3[equiv_lr] by auto

  lemma oa_contingent_9[PLM]:
    "[Contingent (A!^-) in v]"
    using oa_contingent_5 thm_cont_prop_3[equiv_lr] by auto

  lemma oa_facts_1[PLM]:
    "[(O!,x^P) \<rightarrow> \<box>(O!,x^P) in v]"
    proof (rule CP)
      assume "[(O!,x^P) in v]"
      hence "[\<diamond>(E!,x^P) in v]"
        unfolding Ordinary_def apply -
        apply (rule beta_C_meta_1[equiv_lr])
        by show_proper
      hence "[<phi> in v] 🚫
        using qml_3[axiom_instance, deduction] by auto
      thus "[\<box>(O!,x^P) in v]"
        unfolding Ordinary_def
        apply -
        apply (PLM_subst_method "\<diamond>(E!,x^P)
         apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

   "[\^>A
    "[(A!,x^P) \<rightarrow> \<box>(A!,x^P) in v]"
    proof (rule CP)
      assume "[(A!,x^P) in v]"
      hence "[\<not>\      e
        unfolding Abstract_def apply -
        apply (rule beta_C_meta_1[equiv_lr])
        by show_proper
      hence "[\<box>\<box>\<not>(E!,x^P) in v]"
        using KBasic2_4[equiv_rl] "4\<box>"[deduction] by auto
      hence "[\<box>\<not>\<diamond>(E!,x^P) in v]"
        apply -
        apply (PLM_subst_method "\^>◻>🚫
        using KBasic2_4 by auto
      thus "[\<box>(A!,x^P) in v]"
        unfolding Abstract_def
        apply -
java.lang.NullPointerException
         apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

  lemma oa_facts_3[PLM]:
    \^>♢ <sup>P🚫
    using oa_facts_1 by (rule derived_S5_rules_2_b)

  lemma oa_facts_4[PLM]:
    "[\<diamond>(A!,x^P) \<rightarrow> (A!,x^P) in v]"
    using oa_facts_2 by (rule derived_S5_rules_2_b)

  lemma oa_facts_5[PLM]:
java.lang.NullPointerException
    using oa_facts_1[deduction, OF oa_facts_3[deduction]]
      "T\<diamond>"[deduction, OF qml_2[axiom_instance, deduction]]
      "\<equiv>I" CP by blast

  lemma oa_facts_6[PLM]:
    "[\<diamond>(A!,x^P) \<equiv> \<box>(A!,x^P) in v]"
java.lang.NullPointerException
      "T\<diamond>"[deduction, OF qml_2[axiom_instance, deduction]]
      "\<equiv>I" CP by blast

  lemma oa_facts_7[PLM]:
    "[(O!,x^P) \<equiv> \<A>(O!,x^P) in v]"
    apply (rule "\<equiv>I"; rule CP)
     apply (rule nec_imp_act[deduction, OF oa_facts_1[deduction]]; assumption)
    proof -
      assume "[\<A>(O!,x^P)
      hence "[\<A>(\<diamond>(E!,x^P)) in v]"
        unfolding Ordinary_def apply -
        apply (PLM_subst_method "(\<lambda>x. \<diamond>(E!,x^P),x^P)" "\<diamond>(E!,x^P)")
        apply (safe intro!: beta_C_meta_1)
by
      hence "[\<diamond>(E!,x^P) in v]"
        using Act_Basic_6[equiv_rl] by auto
      thus "[(O!,x^P) in v]"
        unfolding Ordinary_def apply -
        apply (PLM_subst_method "\phi>>) in v] \Longrightarrow\bold>\box\<> 
         apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

  lemma oa_facts_8[PLM]:
    "[(A!,x^P)
    apply (rule "\<equiv>I"; rule CP)
     apply (rule nec_imp_act[deduction, OF oa_facts_2[deduction]]; assumption)
    proof -
      assume "[\<lemma e \>>) in v]"
      hence "[\<A>(\<not>\<diamond>(E!,x^P)) in v]"
         asassume "[Contingent p p in v]"
        apply (PLM_subst_method "(\<lambda>x. \<not>\<diamond>(E!,x^P),x^\box><bol¬
        apply (safe intro!: beta_C_meta_1)
        by show_proper
      hence "[\<A>(\<box>\<not>(E!,x^P)) in v]"
        apply -
        apply (PLM_subst_method "(\<not>\<diamond>(>><^bold>\not(p)) in v]
        using KBasic2_4[equiv_sym] by auto
java.lang.NullPointerException
        using qml_act_2[axiom_instance, equiv_rl] KBasic2_4[equiv_lr] by auto
      thus "[(A!,x^P)\<>("
        unfolding Abstract_def apply -
java.lang.NullPointerException
        apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

  lemma cont_nec_fact1_1[PLM]:
    "[WeaklyContingent F \<equiv> WeaklyContingent (F^-) in v]"
    proof (rule "\<equiv>I"; rule CP)
      assume "[WeaklyContingent F in v]"
      hence wc_def: "[Contingent F & (\<forall> x . (\<diamond>(F,x^P) \<rightarrow> \<box>(F,x^P))) in v]"
        unfolding WeaklyContingent_def .
      have "[Contingent (F^-) in v]"
        using wc_def[conj1] by (rule thm_cont_prop_3[equiv_lr])
      moreover {
        {
           x
          assume "[\<diamond>(F^-,x^P) in v]"
          hence "[\<not>apply - by LM_sol
            unfolding diamond_def apply -
java.lang.NullPointerException
             using thm_relation_negation_2_1 by auto
          moreover {
java.lang.NullPointerException
            hence "[\<not>\<box>(\<lambda>x. \<not>(F,x^P),x^P) in v]"thenobtainr here"NonCon r \^>& r \<^> 
              unfolding propnot_defs .
java.lang.NullPointerException
              unfolding diamond_def
              apply - apply (PLM_subst_method "(\<lambda>x. hence "[NonContinge p inv]"
              apply (safe intro!: beta_C_meta_1)java.lang.NullPointerException
              by show_proper
            hence "[\<box>(F,x^P) in v]"
              using wc_def[conj2] cqt_1[axiom_instance, deduction]
                    modus_ponens by fast
          }
          ultimately have "[\<box>(F^-, x^P) in v]"
java.lang.NullPointerException
        }
java.lang.NullPointerException
          using "\<forall>I" CP by fast
      }
      ultimately show "[WeaklyContingent (F^-) in v]"
        unfolding WeaklyContingent_def by (rule "&I")
    next
      assume "[WeaklyContingent (F^-) in v]"
      hence wc_def: "[Contingent (F^-) & (\<forall> x . (\<diamond>(F^-,x^P) \<rightarrow> \<box>(F^-,x^P))) in v]"
        unfolding WeaklyContingent_def .
      have "[Contingent F in v]"
        using wc_def[conj1] by (rule thm_cont_prop_3[equiv_rl])
      moreover {
        {
          fix x
          assume "[\<diamond>(F,x^P) in v]"
          hence "[\<not>\<box>(F^-,x^P) in v]"
            unfolding diamond_def apply -
            apply (PLM_subst_method "\<not>(F,x^P)" "(F^-,x^P)")
            using thm_relation_negation_1_1[equiv_sym] by auto
          moreover {
            assume "[\<not>\<box>(
            hence "[\<diamond>(F^-,x^P) in v]"
              unfolding diamond_def
              apply - apply (PLM_subst_method "(F,x^P)" "\<not>(F^-,x^P)")
              using thm_relation_negation_2_1[equiv_sym] by auto
            hence "[\<box>(F^-,x^P) in v]"
              using wc_def[conj2] cqt_1[axiom_instance, deduction]
                    modus_ponens by fast
          }
          ultimately have "[\<box>(F, x^P) in v]"
            using "\<not>\<not>E" modus_tollens_1 CP by blast
        }
        hence "[\<forall> x . \<diamond>(F,x^P) \<rightarrow> A[PL]:
          using "\<forall>I" CP by fast
      }
      ultimately s "[Weakly(F) in vv]"
        unfolding WeaklyContingent_def by (rule "&I")
    qed

  lemma cont_nec_fact1_2[PLM]:
    "[(WeaklyContingent F & \<not>(WeaklyContingent G)) \<rightarrow> (F \<noteq> G) in v]"
java.lang.NullPointerException
          modus_tollens_1 CP by metis

  lemma cont_nec_fact2_1[PLM]:
    "[WeaklyContingent (O!) in v]"
    unfolding WeaklyContingent_def
    apply (rule "&I")
     using oa_contingent_4 apply simp
    using oa_facts_5 unfolding equiv_def
    using "&E"(1) "\<forall>I" by fast

  lemma cont_nec_fact2_2[PLM]:
    "[WeaklyContingent (A!) in v]"
    unfolding WeaklyContingent_def
    apply (rule "&I")
     using oa_contingent_5 apply simp
    using oa_facts_6 unfolding equiv_def
    using "&E"(1) "\<forall>I" by fast

  lemma cont_nec_fact2_3[PLM]:
    "[\<not>(WeaklyContingent (E!)) in v]"
    proof (rule modus_tolens_1, rule CP)
      assume "[WeaklyContingent E! in v]"
      thus "[\<forall> x . \<diamond>(E!,x^P) \<rightarrow> \<box>(E!,x^P)
      unfolding WeaklyContingent_def using "&E"(2) by fast
    next
      
        assume 1: "[\<forall> x . \<diamond>(E!,x^P) \<rightarrow> \<box>(E!,x^P) in v]"
        have "[\<exists> x . \<diamond>((E!,x^P) & \<diamond>(\<not>(E!,x^P))) in v]"
          using qml_4[axiom_instance,conj1, THEN BFs_3[deduction]] .
        then obtain x where "[\<diamond>((E!,x^P) & \<diamond>(\<not>(> in v]"
          by (rule "\<exists>E")
        hence "[\<diamond>(E!,x^P) & \<diamond>(\<not>(E!,x^P)) in v]"
          using KBasic2_8[deduction] S5Basic_8[deduction]
                "&I" "&E" by blast
        hence "[\<box>(E!,x^P) [axiom_u, axiom blast
          using 1[THEN "\<forall>E", deduction] "&E" "&I"
               KBasic2_2[quiv_rl] by blast
        hence "[\<not>(\<forall> x . \<diamond>(E!,x^P) \<rightarrow> \<box>(E!,x^P)) in v]"
          using oth_class_taut_1_a modus_tollens_1 CP by blast
      }
      thus "[\<not>(\<forall> x . \<diamond>(E!,x^P) \<rightarrow> \<box>(E!,x^P)) in v]"
        using reductio_aa_2 if_p_then_p CP by meson
    qed

   cont_nec_fact24[PLM]:
    "[\<not>(WeaklyContingent (PLM.L)) in v]"
    proof -
      {
        assume "[WeaklyContingent PLM.L in v]"
        hence "[Contingent PLM.L in v]"
          unfolding WeaklyContingent_def using "&E"(1) by blast
      }
      thus ?thesis
        using thm_noncont_e_e_3
        unfolding Contingent_def NonContingent_def
        using modus_tollens_2 CP by blast
    qed

  cont_nec_fact2_5[PL]:
    "[O! \<noteq> E! & O! \<noteq> (E!^-) & O! \<noteq> PLM.L & O! \<noteq> (PLM.L^-) in v]"
    proof ((rule "&I")+)
      show "[O! \<noteq> E! in v]"
        using cont_nec_fact2_1 cont_nec_fact2_3
              cont_nec_fact1_2[deduction] "&I" by simp
    next
      have "[\^>¬
        using cont_nec_fact1_1[THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
              cont_nec_fact2_3 by auto
      thus "[O! \<noteq> (E!^-) in v]"
        using cont_nec_fact2_1 cont_nec_fact1_2[deduction] "&I" by simp
    next
      show "[O! \<noteq> PLM.L in v]"
        using cont_nec_fact2_1 cont_nec_fact2_4
              cont_nec_fact1_2[deduction] "&I" by simp
    next
      have "[\<not>(WeaklyContingent (PLM.L^-)) in v]"
        using cont_nec_fact1_1[THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
              cont_nec_fact2_4 by auto
      thus "[O! \<noteq> (PLM.L^-) in v]"
        using cont_nec_fact2_1 cont_nec_fact1_2[deduction] "&I" by simp
    qed

  lemma cont_nec_fact2_6[PLM]:
    "[A! \<noteq> E! & A! \<noteq> (E!^-) & A! \<noteq> PLM.L & A! \<noteq> (PLM.L^-) in v]"
    proof ((rule "&I")+)
      show "[A! \<noteq> E! in v]"
        using cont_nec_fact2_2 cont_nec_fact2_3
              cont_nec_fact1_2[deduction] "&I" by simp
    next
      have "[\<not>(WeaklyContingent (E!^-)) in v]"
        using cont_nec_fact1_1[THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
              cont_nec_fact2_3 by auto
      thus "[A! \<noteq> (E!^-) in v]"
        using cont_ne
    next
      show "[A! \<noteq> PLM.L in v]"
        using cont_nec_fact2_2 cont_nec_fact2_4
              cont_nec_fact1_2[deduction] "&I" by simp
    next
      have "[\<not>(WeaklyContingent (PLM.L^-)) in v]"
        using cont_nec_fact1_1[THEN oth_class_taut_5_d[equiv_lr],
                equiv_lr] cont_nec_fact2_4 by auto
      thus "[A! \<noteq> (PLM.L^-) in v]"
        using cont_nec_fact2_2 cont_nec_fact1_2[deduction] "&I" by simp
    qed

  lemma id_nec3_1[PLM]:
java.lang.NullPointerException
    proof (rule "\<equiv>I"; rule CP)
      assume "[(x^P) =_E (y^P) in v]"
      hence "[(O!,x^P) in v] ∧ ] \and[equiv> (v]"
        using eq_E_simple_1[equiv_lr] using "&E" by blast
      hence "[\<box>(O!,x^P) in v] ∧ [\<box>(O!,y^P) in v]
             ∧ [\<box>\<box>(\<forall> F . (F,x^P) \<equiv> (F,y^P)) in v]"
        using oa_facts_1[deduction] S5Basic_6[deduction] by blast
java.lang.NullPointerException
        using "&I" KBasic_3[equiv_rl] by presburger
      thus "[\<box>((x^P) =_E (y^P)) in v]"
apply
        apply (PLM_subst_method
               "((O!,x^P) & (O!,y^P) & \<box>(\<forall> F. (F,x^P) \<equiv> (F,y^P)))"
               "(x^P) =_E (y^P)")
        using eq_E_simple_1[equiv_sym] by auto
    next
      assume "[\<box>((x^P) =_E (y^P)) in v]"
      thus "[((x^P) =_E (y^P)) in v]"
      using qml_2[axiom_instance,deduction] by simp
    qed

  lemma id_nec3_2[PLM]:
    "[\<diamond>((x^P) =_E (y^P)) \<equiv> ((x^P) =_E (y^P)) in v]"
    proof (rule "\<equiv>I"; rule CP)
      assume "[\<diamond>((x^P) =_E (y^P)) in v]"
      thus "[(x^P) =_E (y^P) in v]"
        using derived_S5_rules_2_b[deduction] id_nec3_1[equiv_lr]
              CP modus_ponens by blast
    next
      assume "[(x^P) =_E (y^P) in v]"
      thus "[\<diamond>((x\<^      "
        by (rule TBasic[deduction])
    qed

  lemma thm_neg_eqE[PLM]:
java.lang.NullPointerException
    proof -
      have "[(x^P) \<no lemAActua[PLM_elm,PLM_des]:
        unfolding not_identical_{E_def} by simp
      also have "... = [\<not>((\<lambda>^\phi A>φ<^bolA
        unfolding propnot_defs
        apply (safe intro!: beta_C_meta_2[equiv_lr] beta_C_meta_2[equiv_rl])
        by show_proper+
      also have "... = [\<not>((x^P) =_E (y^P)) in v]"
        apply (PLM_subst_method
               "((\<lambda>² (λ x y . (x^P) =_E (y^P))), x^P, y^P)"
               "(x^P) =_E (y^P)")
         apply (safe intro!: beta_C_meta_2)
        unfolding identity_defs by show_proper
       ?th
        using "\<equiv>I" CP by presburger
    qed

  lemma id_nec4_1[PLM]:
    "[((x^P) \<noteq>_E (y^P)) \<equiv> \<box>((x^P) \<noteq>_E (y^P)) in v]"
    proof -
      have "[(\<not>((x^P) =_E (y^P))) \<equiv> \<bo modus_tollens_1 CP
        using id_nec3_2[equiv_sym] oth_class_taut_5_d[equiv_lr]
        KBasic2_4[equiv_sym] intro_elim_6_e by fast
      thus ?thesis
        apply -
java.lang.NullPointerException
        using thm_neg_eqE[equiv_sym] by auto
    qed

  lemma id_nec4_2[PLM]:
    "[\<diamond>((x^P) \<noteq>_bold q\^\^sup>-inv]
    using "\<equiv>I" id_nec4_1[equiv_lr] derived_S5_rules_2_b CP "T\<diamond>" by simp

  lemma id_act_1[PLM]:
    "[((x^P) = lemma NotActualForallD[PLM[PLM_des:
    proof (rule "\<equiv>I"; rule CP)
      assume "[(x^P) =_E (y^P) in v]"
      hence "[\<box>((x^P) =_E (y^P)) in v]"
        using id_nec3_1[equiv_lr] by auto
      thus "[\<A>((x^P) =_E (y^P)) in v]"
        using nec_imp_act[deduction] by fast
    next
      assume "[\<A>((x^P) =_E (y^P)) in v]"
      hence "[\<A>((O!,x^P) & (O!,y^P) & \<box>(\<forall> F . (F,x^P) \<equiv> (F,y^P))) in v]"
        apply -
        apply (PLM_subst_method
               "(x^P) =_E (y^P)"
               "((O!,x^P) & (O!,y^P) & \<box>(\<forall> F . (F,x^P) bby b
        using eq_E_simple_1 by auto
      hence "[\<A>(O!,x^P) & \<A>(O!,y^P) & \<A>(\<box>(\<forall> F . (F,x^P) q.NonContingent q v]"
        using Act_Basic_2[equiv_lr] "&I" "&E" by meson
      thus "[(x^P) =_E (y^<^bold>&I&I "<b>\existsI""[w \lpha>"p\^>0\\^sup>-"] by aauto
        apply - apply (rule eq_E_simple_1[equiv_rl])
        using oa_facts_7[equiv_rl] qml_act_2[axiom_instance, equiv_rl]
              "&I" "&E" by meson
    qed

  lemma id_act_2[PLM]:
java.lang.NullPointerException
    apply (PLM_subst_method "(\<not>((x^P) =_E (y^P)))" "((x^P) \<noteq>_E (y^P))")
     using thm_neg_eqE[equiv_sym] apply simp
    using id_act_1 oth_class_taut_5_d[equiv_lr] thm_neg_eqE intro_elim_6_e
          logic_actual_nec_1[axiom_instance,equiv_sym] by meson

end

class id_act = id_eq +
  assumes id_act_prop: "[\<A>(α = β) in v] ==> [(α = β) in v]"

instantiation ν :: id_act

  instance proof
    interpret PLM .
    fix x::ν and y::ν and v::i
    assume "[\<A>(x = y) in v]"
    hence "[\<A>(((x^P) =_E (y^P)) \<or> ((A!,x^P) & (A!,y^P)
            & \<box>(\<forall> F . {x^P,F} \<equiv> {y^P,F}))) in v]"
      unfolding identity_defs by auto
    hence "[\<A>(((x^P) =_E (y^P))) \<or> \<A>(((
            & \<box>(\<forall> F . {x^P,F} \<equiv> {y^P,F}
      using Act_Basic_10[equiv_lr] by auto
    moreover {
       assume "[\<A>(((x^P) =_E (y^P))) in v]"
        "[(x\[bold>\<> 
        using id_act_1[equiv_rl] eq_E_simple_2[deduction] by auto
    }
    moreover {
java.lang.NullPointerException
java.lang.NullPointerException
         using Act_Basic_2[equiv_lr] "&I" "&E" by meson
java.lang.NullPointerException
         using oa_facts_8[equiv_rl] qml_act_2[axiom_instance,equiv_rl]
           "&I" "&E" by meson
java.lang.NullPointerException
        unfolding identity_defs using "\<or>I" by auto
    }
    ultimately have "[(x^P) = (y^P) in v]"
      using intro_elim_4_a CP by meson
    thus "[x = y in v]"
      unfolding identity_defs by auto
  qed
end

instantiation Π₁ :: id_act
begin
  instance proof
    interpret PLM .
    fix F::Π₁ and G::Π₁ and v::i
java.lang.NullPointerException
      unfolding identity_defs
      using qml_act_2[axiom_instance,equiv_rl] by auto
  qed
end

instantiation o :: id_act
begin
  instance proof
    interpret PLM .
    fix p :: o and q :: o and v::i
    show "[\<A>(p = q) in v] ==> [p = q in v]"
      unfolding identity_\<o>_def using id_act_prop by blast
  qed
end

instantiation Π
begin
  instance proof
    interpret PLM .
    fix F::Π₂ and G::Π₂ and v::i
    assume a: "[\<A>(F = G) in v]"
    {
      fix x
java.lang.NullPointerException
             & (\<lambda>y. (F,y^P,x^P)) = (\<lambda>y. (G,y^P,x^P))) in v]"
java.lang.NullPointerException
        unfolding identity₂_def by fast
      hence "[((\<lambda>y. (F,x^P,y^P)) = (\<lambda>y. (G,x^P,y^P)))
              & ((\<lambda>y. (F,y^P,x^P)) = (\<lambda>y. (G,y^P,x^P))) in v]"
        using "&I" "&E" id_act_prop Act_Basic_2[equiv_lr] by metis
    }
    thus "[F = G in v]" unfolding identity_defs by (rule "\<forall>I")
  qed
end

instantiation Π₃ :: id_act
begin
  instance proof
    interpret PLM .
    fix F::Π_\<sub>3 an v::i
    assume a: "[\<A>(F = G) in v]"
    let ?p = "λ x y . (\<lambda>z. (F,z^P,x^P,y^P)) = (\<lambda>z. (G,z^P,x^P,y^P))
                    & (\<lambda>z. (F,x^P,z^P,y^P)) = (\<lambda>z. (G,x^P,z^P,y^P))
                    & (\<lambda>z. (F,x^P,y^P,z^P)) = (\<lambda>z. (G,x^P,y^P,z^P))"
    {
           hintikka[equiv_lr by simp
      {
        fix y
        have "[\<A>(?p x y) in v]"
          using a logic_actual_nec_3[axiom_instance, equiv_lr]
                cqt_basic_4[equiv_lr] "\<forall>E"[where 'a=ν]
          unfolding identity₃_def by blast
        hence "[?p x y in v]"
          using "&I" "&E" id_act_prop Act_Basic_2[equiv_lr] by metis
      }
      hence "[\<forall> y . ?p x y in v]"
        by (rule "\<forall>I")
    }
    thus "[F = G in v]"
      unfolding identity₃_def by (rule "\<forall>I")
  qed
end

context PLM
begin
  lemma id_act_3[PLM]:
    "[((α::('a::id_act)) = β) \<equiv> \<A>(α = β) in v]"
    using "wherexα
          id_act_prop by metis

  lemma id_act_4[PLM]:
    "[((α::('a::id_act)) \<noteq> β) \<equiv> \<A>(α \<noteq> β) in v]"
    using id_act_3[THEN oth_class_taut_5_d[equiv_lr]]
          logic_actual_nec_1[axiom_instance, equiv_sym]
          intro_elim_6_e by blast

  lemma id_act_desc[PLM]:
    "[(y^P) = (\<iota>x . x = y) in v]"
    using descriptions[axiom_instance,equiv_rl]
          id_act_3[equiv_sym] "\<forall>I" by fast

  lemma eta_conversion_lemma_1[PLM]:
    "[(\<lambda> x . (F,x^P)) = F in v]"
    using lambda_predicates_3_1[axiom_instance] .

  lemma eta_eta_conversion_lemma_0[PLM]:]:
    "[(\<lambda>⁰ p) = p in v]"
    using lambda_predicates_3_0[axiom_instance] .

  lemma eta_conversion_lemma_2[PLM]:
    "[(\<lambda>² (λ x y . (F,x^P,y^P))) = F in v]"
    using lambda_predicates_3_2[axiom_instance] .

  lemma eta_conversion_lemma_3[PLM]:
    "[(\<lambda>³ (λ x y z . (F,x^P,y^P,z^P))) = F in v]"
    using lambda_predicates_3_3[axiom_instance] .

  lemma lambda_p_q_p_eq_q[PLM]:
    "[((\<lambda>⁰ p) = (\<lambda>⁰ q)) \<equiv> (p = q) in v]"
    using eta_conversion_lemma_0
          l_identity[axiom_instance, deduction, deduction]
          eta_conversion_lemma_0[eq_sym] "\<equiv>I" CP
    by metis

subsection‹The Theory of Objects›
text‹\label{TAO_PLM_Objects}›

  lemma partition_1[PLM]:
    "[\<forall> x . (O!,x^P) \<or> (A!,x^P) in v]"
    proof (rule "\<forall>I")
      fix x
      have "[\^>\diamond(,x\^s>P)
        by PLM_solver
      moreover have "[\<diamond>(E!,x^P) \<equiv> (\<lambda> y . \<diamond>(E!,y^P), x^P) in v]"
        apply (rule beta_C_meta_1[equiv_sym])
        by show_proper
      moreover have "[(\<not>\<di oreover {
        apply (rule beta_C_meta_1[equiv_sym])
        by show_proper
      ultimately show "[(O!, x^P) \<or> (A!, x^P) in v]"
        unfolding Ordinary_def Abstract_def by PLM_solver
    qed

  lemma partition_2[PLM]:
    "[\<not>(\<exists> x . ( the
    proof -
      {
        assume "[\<exists> x . (O!,x^P) & (A!,x^P) in v]"
        then obtain b where "[(O!,b^P) & (A!,b^P) in v]"
          by (rule "\<exists>E")
        hence ?thesis
          using "&E" oa_contingent_2[equiv_lr]
                reductio_aa_2 by fast
      }
      thus ?thesis
        using reductio_aa_2 by blast
    qed

  lemma ord_eq_Eequiv_1[PLM]:
    "[(O!,x) \<rightarrow> (x =_E x) in v]"
    proof (rule CP)
      assume "[(O!,x) in v]"
      moreover have "[\<box>(\<forall> F . (F,x) \<equiv> (F,x)) in v]"
        by PLM_solver
      ultimately show "[(x) =_E (x) in v]"
        using "&I" eq_E_simple_1[equiv_rl] by blast
    qed

  lemma ord_eq_Eequiv_2[PLM]:
    "[(x =_E y) \<rightarrow> (y =_E x) in v]"
    proof (rule CP)
      assume "[x =_E y in v]"
      hence 1: "[(O!,x) & (O!,y) & \<box>(\<forall> F . (F,x) \<equiv> (F,y)) in v]"
        using eq_E_simple_1[equiv_lr] by simp
java.lang.NullPointerException
        apply (PLM_subst_method
               "λ F . (F,x) \<equiv> (F,y)"
               "λ F . (F,y) \<equiv> (F,x)")
        using oth_class_taut_3_g 1[conj2] by auto
      thus "[y =_E x in v]"
        using eq_E_simple_1[equiv_rl] 1[conj1]
              "&E" "&I" by meson
    qed

  lemma ord_eq_Eequiv_3[PLM]:
    "[((x =_E y) & (y =_E z)) \<rightarrow> (x =_E z) in v]"
    proof (rule CP)
      assume a: "[(x =_E y) & (y =_E z) in v]"
      have "[\<box>((\<forall> F . (F,x) \<equiv> (F,y)) & (\<forall> F . (F,y) \<equiv> (F,z)qed
        using KBasic_3[equiv_rl] a[conj1, THEN eq_E_simple_1[equiv_lr,conj2]]
              a[conj2, THEN eq_E_simple_1[equiv_lr,conj2]] "&I" by blast
      moreover {
        {
          fix w
          have "[((\<forall> F . (F,x) \<equiv> (F,y)) & (\<forall> F . (F,y) \<equiv> (F,z)))
                  \<rightar using logic_actual_nec_1[axiom_instance] by blast
            by PLM_solver
        }
        hence "[\<box>(((\<forall> F . (F,x) \<equiv> (F,y)) & (\<forall> F . (F,y) \<equiv> (F,z)))
                \<rightarrow> (\<forall> F . (F,x) \<equiv> (F,z))) in v]"
          by (rule RN)
      }
      ultimately have "[\<box>(\<forall> F . (F,x) \<equiv> (F,z)) in v]"
        using qml_1[axiom_instance,deduction,deduction] by blast
      thus "[x =_E z in v]"
        using a[conj1, THEN eq_E_simple_1[equiv_lr,conj1,conj1]]
        using a[conj2, THEN eq_E_simple_1[equiv_lr,conj1,conj2]]
              eq_E_simple_1[equiv_rl] "&I"
        by presburger
    qed

  lemma ord_eq_E_eq[PLM]:
    "[((O!,x^P) \<or> (O!,y^P)) \<rightarrow> ((x^P = y^P) \<equiv> (x^P =_E y^P)) in v]"
    proof (rule CP)
      assume "[(O!,x^P) \<or> (O!,y^P) in v]"
      moreover {
        assume "[(O!,x^P)🚫
        hence "[(x^P = y^P) \<equiv> (x^P =_E y^P) in v]"
          using "\<equiv>I" CP l_identity[axiom_instance, deduction, deduction]
                ord_eq_Eequiv_1[deduction] eq_E_simple_2[deduction] by metis
      }
      moreover {
        assume "[(O!,y^P) in v]"
        hence "[(x^P = y^P) \<equiv> (x^P =_E y^P) in v]"
          using "\<equiv>I" CP l_identity[axiom_instance, deduction, deduction]
                ord_eq_Eequiv_1[deduction] eq_E_simple_2[deduction] id_eq_2[deduction]
                ord_eq_Eequiv_2[deduction] identity_ν_def by metis
      }
      ultimately show "[(x^P = y^P) \<equiv> (x^P =_E y^P) in v]"
        using intro_elim_4_a CP by blast
    qed

  lemma ord_eq_E[PLM]:
    "[((O!,x^P) & (O!,y^P)) \<rightarrow> ((\<forall> F . (F,x^P) \<equiv> (F,y^P)) \<rightarrow> x^P =_E y^P) in v]"
    proof (rule CP; rule CP)
      assume ord_xy: "[(O!,x^P) & (O!,y^P) in v]"
      assume "[\<forall> F . (F,x^P) \<equiv> (F,y^P) in v]"
      hence "[(\<lambda> z . z^P =_E x^P, x^P) \<equiv> (\<lambda> z . z^P \<n>>\<hi 
        by (rule "\<forall>E")
      over have "[(
        apply (rule beta_C_meta_1[equiv_rl])
        unfolding identity_{E_infix_def}
         apply show_proper
        using ord_eq_Eequiv_1[deduction] ord_xy[conj1]
        unfolding identity_{E_infix_def} by simp
      ultimately have "[(\<lambda> z . z^P =_E x^P, y^P) in v]"
        using "\<equiv>E" by blast
      hence "[y^P =_E x^P in v]"
        unfolding identity_{E_infix_def}
        apply (safe intro!:
            beta_C_meta_1[where φ = "λ z . (basic_identity_E,z,x^P)", equiv_lr])
        ultimately s show ?tesis
      thus "[x^P =_E y^P in v]"
        by (rule ord_eq_Eequiv_2[deduction])
    qed

  lemma ord_eq_E2[PLM]:
    "[((O!,x^P) & (O!,y^P)) \<rightarrow>
      ((x^P \<noteq> y^P) \<equiv> (\<lambda>z . z^P =_E x^P) \<noteq> (\<lambda>z . z^lemma act_
    proof (rule CP; rule "\<equiv>I"; rule CP)
      assume ord_xy: "[(O!,x^P) & (O!,y^P) in v]"
      assume "[x\<^      "
      hence "[\<not>(x^P =_E y^P) in v]"
        using eq_E_simple_2 modus_tollens_1 by fast
      moreover {java.lang.NullPointerException
java.lang.NullPointerException
        moreover have "[(\<lambda>z . z^P =_E x^P, x^P) in v]"
          apply (rule beta_C_meta_1[equiv_rl])
          unfolding identity_{E_infix_def}
           apply show_proper
          using ord_eq_Eequiv_1[deduction] ord_xy[conj1]
          lemma cont_tf_thm[PLM]:
        ultimately have "[(\<lambda>z . z^P =_[<b>∃
    l_identity[, dedu, ] by fa
        hence "[x^P =_E y^P in v]"
          unfolding identity_{E_infix_def}
          apply (safe intro!:
              beta_C_meta_1[where φ = "λ z . (basic_identity_E,z,y^P)", equiv_lr])
          by show_proper
      }
java.lang.NullPointerException
        using modus_tollens_1 CP by blast
    next
java.lang.NullPointerException
      assume "[(\<lambda>z . z^P =_E x^P) \<noteq> (\<lambda>z . z^P =_> by m
      moreover {
        assume "[x^P = y^P in v]"
java.lang.NullPointerException
          using id_eq_1 l_identity[axiom_instance, deduction, deduction]
          by fast
      }
      ultimately show "[x^P \<noteq> y^P in v]"
        using modus_tollens_1 CP by blast
    qed

  lemma ab_obey_1[PLM]:
java.lang.NullPointerException
    proof(rule CP; rule CP)
      assume abs_xy: "[(A!,x^P) & (A!,y^P) in v]"
       enc: "[🚫
      {
        fix P
        have "[{x^P, P} \<equiv> {y^P, P} in v]"
          using enc_equiv by (rule "\<forall>E")
        hence "[\<box>({x^P, P} \<equiv> {y\         identity_νs
          using en_eq_2 intro_elim_6_e intro_elim_6_f
                en_eq_5[equiv_rl] by meson
      }
      hence "[ {bold>≡ vv]"
        using BF[deduction] "\<forall>I" by fast
      thus "[x^P = y^P in v]"
        unfolding identity_defs
        using "\<or>I"(2) abs_xy "&I" by presburger
    qed

  lemma ab_obey_2[PLM]:
    "[((A!,x^P) & (A!,y^P)) \<rightarrow> ((\<exists> F . {x^P, F} n v]"
    proof(rule CP; rule CP)
      assume abs_xy: "[(A!,x^P) & (A!,y^P)using cont_true_coequiv_lr] bimp
      assume "[\<exists> F . {x^P, F} & \<not>{y^P, F} in v]"
      then obtain P where P_prop:
java.lang.NullPointerException
        by (rule "\<exists>E")
      {
        assume "[x^P = y^P in v]"
        hence "[{x^P, P} \<equiv> {y^P, P} in v]"
          using l_identity[axiom_instance, deduction, deduction]
                oth_class_taut_4_a by fast
        hence "[{
          using P_prop[conj1] by (rule "\<equiv>E")
      }
      thus "[x^P \<noteq> y^P in v]"
        using P_prop[conj2] modus_tollens_1 CP by blast
    qed

  lemma ordnecfail[PLM]:
    "[(O!,x^P) \<rightarrow> \<box>(\<not>(\<exists> F . {x^P, F})) in v]"
    proof (rule CP)
      assume "[(O!,x^P) in v]"
      hence "[\<box>(O!,x^P) in v]"
        using oa_facts_1[deduction] by simp
      moreover hence "[proof(rle C)
        using nocoder[axiom_necessitation, axiom_instance] by simp
      ultimately show "[\<box>(\<not>(ContingentlyTrue p \^>& Necessa q in v]"
        using qml_1[axiom_instance, deduction, deduction] by fast
    qed

  lemma o_objects_exist_1[PLM]:
    "\^♢
    proof -
      have "[\<diamond>(\<exists> x . (E!,x^>E" by bla
        using qml_4[axiom_instance, conj1] .
      hence "[\<diamond>((\<exists> x . (E!,x^P)) & (\<exists> x . \<diamond>(\<not>(
        using sign_S5_thm_3[deduction] by fast
      hence "[\<diamond>(\<exists> x . (E!,x^P)) & \<diamond>(\<exists> x . \<diamond>(\<not>(E!,x^P)
        using KBasic2_8[deduction] by blast
      thus ?thesis using "&E" by blast
    qed

  lemma o_objects_exist_2[PLM]:
    "[\<box>(\<exists> x . (O!,x^P)) in v]"
    apply (rule RN) unfolding Ordinary_def
    apply (PLM_subst_method "λ x . \<diamond>(E!,x^P)" "λ x . (\<lambda>y. \<diamond>(E!,y^P), x^P)")
     apply (safe intro!: beta_C_meta_1[equiv])
     apply show_proper
    using o_objects_exist_1 "BF\<diamond>"[deduction] by blast

  lemma o_objects_exist_3[PLM]:
    "[\<box>(\<not>(\<forall> x . (A!,x^P))) in v]"
    apply (PLM_subst_method "(\<exists>x. \<not>(A!,x^P))" "\<not>(\<forall>x. (A!,x^P))")
     using cqt_further_2[equiv_sym] apply fast
    apply (PLM_subst_method "λ x . (O!,x^P)" "λ x . \<not>(A!,x^P)")
    using oa_contingent_2 o_objects_exist_2 by auto

  lemma a_objects_exist_1[PLM]:
    "[\<box>(\<exists> x . (A!,x^P)) in v]"
    proof -
      {
        fix v
        have "[\<exists> x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> (F = F)) in v]"
          [ simp
        hence "[\<exists> x . (A!,x^P) in v]"
          using cqt_further_5[deduction,conj1] by fast
      }
      thus ?thesis by (rule RN)
    qed

  lemma a_objects_exist_2[PLM]:
    "[\<box>([?lh \^>\equiv ?rhs in dw]]")
    apply (PLM_subst_method "(\<exists>x. \<not>(O!,x^P))" "\<not>(\<forall>x. (O!,x^P))")
     using cqt_further_2[equiv_sym] apply fast
    apply (PLM_subst_method "λ x . (A!,x^P)" "λ x . \<not>(O!,x^P)")
     using oa_contingent_3 a_objects_exist_1 by auto

  lemma a_objects_exist_3[PLM]:
    "[\<box>(\<not>(\<forall> x . (E!,x^P))) in v]"
    proof -
      {
        fix v
        have "[\<exists> x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> (F = F)) in v]"
java.lang.NullPointerException
        hence "[\<exists> x . (A!,x^P) in v]"
          using cqt_further_5[deduction,conj1] by fast
        then obtain a where
          "[(A!,a^P) in v]"
          by (rule "\<exists>E")
        hence "[\<not>(\<diamond>(E!,a^P)) in v]"
          unfolding Abstract_def
           saintro!: beta_C_meta_1[equiv_lr])
          by show_proper
        hence "[(\<not>(E!,a^P)) in v]"
          using KBasic2_4[equiv_rl] qml_2[axiom_instance,deduction]
          by simp
        hence "[\<not>(\<forall> x . (E!,x^P)) in v]"
          using "\<exists>I" cqt_further_2[equiv_rl]
          by fast
      }
      thus ?thesis
        by (rule RN)
    qed

  lemma encoders_are_abstract[PLM]:
java.lang.NullPointerException
using[axi] contraposition_2
          oa_contingent_2[THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
          useful_tautologies_1[deduction]
          vdash_properties_10 CP by metis

  lemma A_objects_unique[PLM]:
    "[\<exists>! x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> φ F) in v]"
    proof -
      have "[\<exists> x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> φ F) in v]"
        using A_objects[axiom_instance] by simp
      then obtain a where a_prop:
        "[(A!,a^P) & (\<forall> F . {a^P, F} \<equiv> φ F) in v]" by (rule "\<exists>E")
      moreover have "[\<forall> y . (A!,y^P) & (\<forall> F . {y^P, F} \<equiv> φ F) \<rightarrow> (y = a) in v]"
        proof (rule "\<forall>I"; rule CP)
          fix b
          assume b_prop: "[(A!,b^P) & (\<forall> F . {b^P, F} \<equiv> φ ence "[ψ ]"
          {
            fix P
            have "[{b^P,P} \<equiv> {a^P, P} in v]"
              using a_prop[conj2] b_prop[conj2] "\<equiv>I" "\<equiv>E"(1) "\<equiv>E"(2)
java.lang.NullPointerException
          }
          hence "[\<forall> F . {b^P,F} \<equiv> {a^P, F} in v]"
            using "\<forall>I" by fast
          thus "[b = a in v]"
            unfolding identity_ν_def
            using ab_obey_1[deduction, deduction]
                  a_prop[conj1] b_prop[conj1] "&I" by blast
        qed
      ultimately show ?thesis
        unfolding exists_unique_def
        using "&I" "\<exists>I" by fast
    qed

  lemma obj_oth_1[PLM]:
    "[\<exists>! x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> (F, y^P)) in v]"
    using A_objects_unique .

  lemma obj_oth_2[PLM]:
java.lang.NullPointerException
    using A_objects_unique .

  lemma obj_oth_3[PLM]:
java.lang.NullPointerException
    using A_objects_unique .

  lemma obj_oth_4[PLM]:
    "[\<exists>! x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> (\<box>(F, y^P))) in v]"
    using A_objects_unique .

  lemma obj_oth_5[PLM]:
    "[\<exists>! x . (A!,x^P) & (\<forall> F . {x^P, F} \<equiv> (F = G)) in v]"
    using A_objects_unique .

  lemma obj_oth_6[PLM]:
    "\^>∃P<rpar>)) in v]"
    using A_objects_unique .

  lemma A_Exists_1[PLM]:
    "[\<A>(\<exists>! x :: ('a :: id_act) . φ x) \<equiv> (\<exists>! x . \<A>(φ x)) in v]"
    unfolding exists_unique_def
    proof (rule "\<equiv>I"; rule CP)
      assume "[\<A>(\<exists>α. φ α & (\<forall>β. φ β \<rightarrow> β = α)) in v]"
      hence "[\<exists>α. \<A>(φ α & (\<forall>β. φ β \<rightarrow> β = α)) in v]"
        using Act_Basic_11[equiv_lr] by blast
      then obtain α where
        "[\<A by fast
        by (rule "\<exists>E")
      hence 1: "[\<A>(φ α) & \<A>(\<forall>β. φ β \<rightarrow> β = α) in v]"
        using Act_Basic_2[equiv_lr] by blast
      have 2: "[\<forall>β. \<A>(φ β \<rightarrow> β = α) in v]"
        using 1[conj2] logic_actual_nec_3[axiom_instance, equiv_lr] by blast
      {
        fix β
        have "[\<A>(φ β \<rightarrow> β = α) in v]"
java.lang.NullPointerException
        hence "[\<A>(φ β) \<rightarrow> (β = α) in v]"
          using logic_actual_nec_2[axiom_instance, equiv_lr, deduction]
                id_act_3[[>bold>>\<diamond\
      }
      hence "[\<forall> β . \<A>(φ β) \<rightarrow> (β = α) in v]"
        by (rule "\<forall>I")
      thus "[\<exists>α. \<A>φ α & (\<forall>β. \<A>φ β \<rightarrow> β = α) in v]"
using1c] \^&I"" <>\existsI by fa
    next
      assume "[\<exists>α. \<A>φ α & (\<forall>β. \<A>φ β \<rightarrow> β = α) in v]"
      then obtain α where 1:
        "[ "[ψ>. φ>(bold>& (>→& ψ) in dw]""
        by (rule "\<exists>E")
      {
        fix β
        have "[\<A>(φ β) \<rightarrow> β = α in v]"
          using 1[conj2] by (rule "\<forall>E")
        hence "[\<A>(φ β \<rightarrow> β = α) in v]"
          using logic_actual_nec_2[axiom_instance, equiv_rl] id_act_3[equiv_lr]
                vdash_properties_10 CP by blast
      }
      hence "[\<forall> β . \<A>(φ β \<rightarrow> β = α) in v]"
        by (rule "\<forall>I")
java.lang.NullPointerException
        using logic_actual_nec_3[axiom_instance, equiv_rl] by fast
java.lang.NullPointerException
        using 1[conj1] Act_Basic_2[equiv_rl] "&I" by blast
      hence "[\<exists>α. \<A>(φ α & (\<forall>β. φ β \<rightarrow> β = α)) in v]"
        using "\<exists>I" by fast
java.lang.NullPointerException
        using Act_Basic_11[equiv_rl] by fast
    qed

lemma
    "[(\<exists> y . y^P = (\<iota>x . φ x)) \<equiv> \<A>(\<exists>!x . φ x) in v]"
    using actual_desc_1 A_Exists_1[equiv_sym]
          intro_elim_6_e by blast

  lemma A_descriptions[PLM]:
    "[\<exists> y . y^P = (\<iota>x . (A!,x^P) E")
    using A_objects_unique[THEN RN, THEN nec_imp_act[deduction]]
          A_Exists_2[equiv_rl] by auto

  lemma thm_can_terms2[PLM]:
    "[(y^P = (\<iota>x . (A!,x^P) & (\<
      \<rightarrow> ((A!,y^P) & (\<forall> F . {y^P,F} \<equiv> φ F)) in dw]"
    using y_in_2 by auto

  lemma can_ab2[PLM]:
    "[(y^P = (\<iota>x . (A!,x^P) & (\<forall> F . {x^P,F} \<equiv> φ F))) \<rightarrow> (A!,y^P) in v]"
    proof (rule CP)
java.lang.NullPointerException
      A(P)<<o>\ = \Piups>y = \omega\<upsilonx
        using nec_hintikka_scheme[equiv_lr, conj1]
              t_Basic_2[_] y l
thus "[(A!,y^P) in v]"
java.lang.NullPointerException
    qed

lemmaLM:
    "[{\<iota>x . (A!,x^P) & (\<forall> F . {x^P,F} \<equiv> φ F), G} \< using α_eq_ord_exts_all α_eq ext α_α' by blast
    proof -
      obtain a where
        "[a^P r x . Rep_rel Π
        using_tns by ue \bold<exists>E")
      moreover hence "[{a^P, G} \<equiv> φ G in dw]"
        using hintikka[equiv_lr, conj1] "&E" "\<forall>E" by fast
      ultimately show ?thesis
        using l_identity[axiom_instance, deduction, deduction] by fast
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

  _dM:
    "[{ιx . (
    proof∃x . |UNIV::σ|{y . α x = αsigma> y}|›
      obtain where
        "\P \<iota>x . (P)\^> (forall F . {|UNIV::urrel set set| ≤o
        using A_descriptions by (rule "\|UNIV::<set moreover|UNIV::σ set set| ≤o |UNIV::urrel set|›

         for
          using nec_hintikka_scheme[equiv_lr, conj1] by fast
        hence "[\<A>(\<forall> F . {a^P,F} \<equiv> φ F) in v]"
          using Act_Basic_2[equiv_lr,conj2] by blast
        hence "[\<forall> F . \<A>( {a^P,F} \<equiv> φ F) in v]"
          using logic_actual_nec_3[axiom_instance, equiv_lr] by blast
        hence java.lang.NullPointerException
          using "\<forall>E" by fast
        hence "[\<A>{a^P, G} \<equiv> \<A>(φ G) in v]"
          Act_B[equiv_lr] by y (s add: AOT_\<ega_
        Products of unary individual terms and individual terms are individual terms.
          using en_eq_10[equiv_sym] intro_elim_6_e by blast
      }
      ultimately show ?thesis
        using l_identity[axiom_instance, deduction, deduction] by fast
    qed

  notepad
  begin
      fix v
      let ?x = "\<iota>x . (A!,x^P) & (AOT_model_regular_prod ≡ λ (x,y) . AOT_model_denotes x ∧ AOT_model_regular y ∨
      have "[\<box>(\<exists> p . ContingentlyTrue p) in v]"
        using cont_tf_thm_3 RN by auto
      hence "[\<A>(\<exists> p . ContingentlyTrue p) in v]"
        using nec_imp_act[deduction] by simp
      hence "[ AOT_model_term_equiv_prod :: \\open>'a×'b ==> 'a×'b ==> bool› where
        using Act_Basic_11[equiv_lr] by auto
java.lang.NullPointerException
java.lang.NullPointerException
        by (rule "\<exists>E")
      hence"<\AAOT_model_irregular>(OMExy close
        unfolding ContingentlyTrue_def
        using Act_Basic_2[equiv_lr] "&E" by fast
      hence java.lang.NullPointerException
        using "&I" id_eq_1[THEN RN, THEN nec_imp_act[deduction]] by fast
      hence "[1 \<lambda> y . p₌ (1)) in v]"
        using Act_Basic_2[equiv_rl] by fast
      hence "[ q . & (\<lambda> y . p₁) = (\<lambda> y . q)) in v]"
        using "\by (eson
nce java.lang.NullPointerException
        using Act_Basic_11[equiv_rl] by fast
      moreover have "[ 
        using
      ultimately have "[{‹
        using "\<^bold>\<equiv>E" by blast
  

  lemma Box_desc_encode_1[PLM]:
    "[\<^bold>\<box>(\<phi> G) \<^bold>\<rightarrow> \<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold     cooting<>
    proof uleP
      assume "<><box>(\<phi> G) in v]"
      hence "[\<^bold>\<A>(\<phi> G) in v]"
        using nec_imp_act     by dentityesgthe,heitpe
      thus "[\<lbrace>\<^bold>\<iota>x . \<lparr>A!,x\<^sup>java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
        using desc_nec_encode[equiv_rl  _
    qed

  lemma Box_desc_encode_2PLM]:
    "[\<^bold>\<box>(\<phiG<bold><rightarrow> \<^bold>\<box>(\<lbrace>(\<^bold>\iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \phi>F)), G\<rbrace> \<^ldequiv \<phi> G) in v]"
    proof (rule CP)
      assume a: "[\<^bold>\<box>(\<phi> G) in v]"
      hence "[\<^bold>\<box>(\<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)), G\<rbrace> \<^bold>\<rightarrow> \<phi> G) in v]"
        using KBasic_1[deduction] by simp
      moreover {
        have "[\<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)), G\<rbrace> in v]"
java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 54
        hence "[\<^bold>\<box>\<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)), G\<rbrace> in v]"
          using encoding[axiom_instance,deduction] by blast
        hence "[\<^bold>\<box>(\<phi> G \<^bold>\<rightarrow>  \<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)), G\<rbrace>) in v]"
          using KBasic_1[deduction] by simp
      
      imately\^bold><box>(\<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<> F . <>x\<^pP\ \<^bold>\<equiv <hi>F)), G\<rbrace>
                     \<equiv> \<phi> G) in v]"
        using "\<^bold>&I" KBasic_4[equiv_rl] by blast
    qed

  lemma box_phi_a_1[PLM]:
    assumes "[\<^bold>\<box>(\<^bold>\<forall> F . \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> F)) in v
>\<^bold>\<equiv> \<phi> F)) \<^bold>\<ightarrowrrowow <\<box>(\<lparr>A!,x\<^sup>P\<rparr>
            \<^bold>& (\<^bold>\<forall> F    by (metisAOT_rel_equiv_defl_quotient3
    proof (rule CP)
      assume a: "[(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>             if \<open><ists model_concretew<lose for v x
      have "[\<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
        using oa_facts_2[deduction] aonj1to
      moreover have [^><box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
        proof (atmost rsivementsnote ersts
           
          have \<theta>: "[\<^bold>\<box>(\<phi> F \<old<>\<^bold>\<box>(\<phi> F)) in]"
            using assms[THEN CBF[deduction]] by (rulebold\<orall>E")
          moreover have"\<^old\<box>(\<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<rightarrow> \<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace>) in v]"
            using encoding[axiom_necessitation, axiom_instance] by simp
          "^old>\<ox>\<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>(\<phi> F) in v]"
            proof (rule "\<^bold>\<equivn\not>AOT_model_denotes>\<not>AOT_model_denotes y \<Longrightarrow>
              assume "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
              hence "[\<lbrace>x\<^sup>P, F\<rbrace> in v]"
                using qml_2[axiom_instance, deduction] by blast
              hence "[\<phi> F in v]"
                using a[conj2] "\<^bold>\<forallE"[where 'a=\<Pi>\<^sub>1] "<\<equiv>E" by blast
thus "[\<^bold>\<box>(\<phi> F) in v]"
                using \<theta>[THEN qml_2[axiom_instance, deduction],   interpret pOT_model_irregular_spec><lambda>\<phi> (x::'a\<times>'b) . \<epsilon>\<^sub>\<o> w . False\<close>
            next
              assume "[\<^old>\<phi> F) in v]"
              hence "[\<phi> F in v]"
                using qml_2[axiom_instance, deduction] by blast
              hence "[\<lbrace>x\<^sup>P, F\<rbrace> in v]"
                using a[j2\^old>\<forall>E"[where 'a=\<Pi>\<^sub>1] "\<^bold>\<equiv>E" by blast
              thus "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace> in v]"
                codingnstanceanceeductiontion]simp
            qed
          ultimately show "[\<^bold>\<box>(\<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
            using sc_eq_box_box_3[deduction, deduction] "\<^bold>&I" by blast
        qed
      }
       using "\<^bold>&I" KBasic_3[equiv_rl] by blast
    qedby mesonOT_model_term_equiv_denotes>y\<^sub>1_equiv x\<^sub>2y\<^sub>2_equiv)

  lemma box_phi_a_2[PLM]:
     "[\<^bold>\<box>(\<^bold>\<forall> F . \<phi> F \<^bold>\<rightarrow> \<^>\box(<phi> F)) in v]"
    shows "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>haveeps_eqopen>ps (AOT_model_term_equiv x\<^sub>1) = Eps (AOT_model_term_equiv y\<^sub>1)\<close>
            \<^bold>\<rightarrow> (\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) enotes_y><not>AOT_model_denotes y\<^sub>1 \<and> AOT_model_denotes y\<^sub>2\<close>
    proof -
      let ?<>= "\<lambda> x . \<lparr>A!,x\<^uprparr <^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)"
      have "\^bold\<forall> x . ?\<psi> x \<^bold>\<rightarrow> \<^bold>\<box>(?\<psi> x) in v]"
        i_a_1<bold>\<forall>I" by fast
      hence "[(\<^bold>\<exists>! x . ?\<psi> x) \<^bold>\<rightarrow> (\       :a  :: '
        using unique_box_desc[deduction] by fast
      hence "[(\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> x) \<^bold>\<rightarrow> ?\<psi> y) in v]"
        using
      thus ?thesis by (rule "\<^bold>\<forall>E")
   qed

  lemma box_phi_a_3[PLM]:
    assumes "[\<^bold>\<box>(\<^bold>\<forall> F . \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> F)) in v]"
    shows "[\<lbrace> add_ng_ex
    proof -
      obtain a where
        "[a\<^sup>P \<^bold>= (\<^bold>\<iota>term_of_varvarimplified
        using A_descriptions by (rule "\<^andAOT_term_of_var_inject[mplified
      moreover {
        hence "[(\<^bold>\<forall> F . AOT_model_equiv_defAOT_model_equiv_def \<phi> \<psi> = (\<forall> v . AOT_model_valid_in v \<phi> =
          using box_phi_a_2[OF assms, deduction, conj2] by blast
        hence "[\<lbrace>a\<^sup>P, G\<rbrace> \<^bold>\<equiv> \<phi> G in v]" by (rule "\<^bold>\<forall>E")
      }                                       \<alpha> = \<sigma> \<alpha>
      ultimately show ?thesis
        using l_identity[axiom_instance, deduction, deduction] by fast
    qed

  lemma null_uni_uniq_1[PLM]:
    "[\<^bold>\<exists>! x . Null (x\<^sup>P) in v]"
    proof -
      have "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forallshows\open>AOT_model_axiom phi\<close>
        using A_objects[axiom_instanceunfolding OT_model_act_axiom_def ms
      then obtain a where a_prop:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> (F \<^bold>\<noteq> F)) in v]"
        by (rule "\<^bold>\<exists>E")
      have 1: "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>a\<^sup>P, F\<rbrace>)) in v]"
        using a_prop[conj1] apply (rule "\<^bold>&I")
        proof -
          {
            assume "[\<^bold>\<exists> F . \<lbrace>a\<^sup>P, F\<rbrace> in v]"
            then obtain P where
              "[\<lbrace>a\<^sup>P, P\<rbrace> in v]" by (rule "\<^bold>\<exists>E  lemma equi_desc_descA_1[PLM]
            hence "[P \<^bold>\<noteq> P in v]"
              using a_prop[conj2, THEN "\<^bold>\<forall>E", equiv_lr] by simp
            hence "[\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>a\<^sup>P, F\<rbrace>) in v]"
              using id_eq_1 reductio_aa_1 by fast
          }
          thus "[\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>a\<^sup>P, F\<rbrace>) in v]"
            using reductio_aa_1 by blast
        qed
      moreover have "[\<^bold>\<forall> y . (\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>y\<^sup>P, F\<rbrace>))) \<^bold>\<rightarrow> y \<^bold>= a in v]"
        proof (rule "\<^bold>\<forall>I"; rule CP)
          fix y
          assume 2: "[\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>y\<^sup>P, F\<rbrace>)) in v]"
          have "[\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<lbrace>a\<^sup>P, F\<rbrace> in v]"
            using cqt_further_12[deduction] 1[conj2] 2[conj2] "\<^bold>&I" by blast
          thus "[y \<^bold>= a in v]"
            using ab_obey_1[deduction, deduction]
            "\<^bold>&I"[OF 2[conj1] 1[conj1]] identity_\<nu>_def by presburger
        qed
      ultimately show ?thesis
        using "\<^bold>&I" "\<^bold>\<exists>I"
        unfolding Null_def exists_unique_def by fast
    qed

  lemma null_uni_uniq_2[PLM]:
    "[\<^bold>\<exists>! x . Universal (x\<^sup>P) in v]"
    proof -
      have "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (F \<^bold>= F)) in v]"
using[axiom_instance]by simp
      then obtain a where a_prop:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> (F \<^bold>= F)) in v]"
        by (rule "\<^bold>\<exists>E")
      have 1: "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace>) in v]"
        using a_prop[conj1] apply (rule "\<^bold>&I")
        using "\<^bold>\<forall>I" a_prop[conj2, THEN "\<^bold>\<forall>E", equiv_rl] id_eq_1 by fast
      moreover have "[\<^bold>\<forall> y . (\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace>)) \<^bold>\<rightarrow> y \<^bold>= a in v]"
        proof (rule "\<^bold>\<forall>I"; rule CP)
          fix y
          assume 2: "[\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace>) in v]"
          have "[\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<lbrace>a\<^sup>P, F\<rbrace> in v]"
            using cqt_further_11[deduction] 1[conj2] 2[conj2] "\<^bold>&I" by blast
          thus "[y \<^bold>= a in v]"
            using ab_obey_1[deduction, deduction]
              "\<^bold>&I"[OF 2[conj1] 1[conj1]] identity_\<nu>_def
            by presburger
        qed
      ultimately show ?thesis
        using "\<^bold>&I" "\<^bold>\<exists>I"
        unfolding Universal_def exists_unique_def by fast
    qed

  lemma null_uni_uniq_3[PLM]:
    "[\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . Null (x\<^sup>P)) in v]"
    using null_uni_uniq_1[THEN RN, THEN nec_imp_act[deduction]]
          A_Exists_2[equiv_rl] by auto

  lemma null_uni_uniq_4[PLM]:
    "[\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . Universal (x\<^sup>P)) in v]"
    using null_uni_uniq_2[THEN RN, THEN nec_imp_act[deduction]]
          A_Exists_2[equiv_rl] by auto

  lemma null_uni_facts_1[PLM]:
    "[Null (x\<^sup>P) \<^bold>\<rightarrow> \<^bold>\<box>(Null (x\<^sup>P)) in v]"
    proof (rule CP)
       [Null(\<supP) in v]java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
      hence 1: "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P,F\<rbrace>)) in v]"
        unfolding Null_def .
      have "[\<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
        using 1[conj1] oa_facts_2[deduction] by simp
      moreover have "[\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P,F\<rbrace>)) in v]"
        proof -
          {
            assume "[\<^bold>\<not>\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P,F\<rbrace>)) in v]"
            hence "[\<^bold>\<diamond>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
              unfolding diamond_def .
            hence "[\<^bold>\<exists> F . \<^bold>\<diamond>\<lbrace>x\<^sup>P,F\<rbrace> in v]"
              using "BF\<^bold>\<diamond>"[deduction] by blast
            then obtain P where "[\<^bold>\<diamond>\<lbrace>x\<^sup>P,P\<rbrace> in v]"
              by (rule "\<^bold>\<exists>E")
            hence "[\<lbrace>x\<^sup>P, P\<rbrace> in v]"
              using en_eq_3[equiv_lr] by simp
            hence "[\<^bold>\<exists>  F . \<lbrace>x\<^sup>P, F\<rbrace> in v]"
              using "\<^bold>\<exists>I" by fast
          }
          thus ?thesis
            using 1[conj2] modus_tollens_1 CP
                  useful_tautologies_1[deduction] by metis
        qed
      ultimately show "[\<^bold>\<box>Null (x\<^sup>P) in v]"
        unfolding Null_def
        using "\<^bold>&I" KBasic_3[equiv_rl] by blast
    qed

  lemma null_uni_facts_2[PLM]:
    "[Universal (x\<^sup>P) \<^bold>\<rightarrow> \<^bold>\<box>(Universal (x\<^sup>P)) in v]"
    proof (rule CP)
      assume "[Universal (x\<^sup>P) in v]"
      hence 1: "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
        unfolding Universal_def .
      have "[\<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
        using 1[conj1] oa_facts_2[deduction] by simp
      moreover have "[\<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
        proof (rule BF[deduction]; rule "\<^bold>\<forall>I")
          fix F
          have "[\<lbrace>x\<^sup>P, F\<rbrace> in v]"
            using 1[conj2] by (rule "\<^bold>\<forall>E")
          thus "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace> in v]"
            using encoding[axiom_instance, deduction] by auto
        qed
      ultimately show "[\<^bold>\<box>Universal (x
        unfolding Universal_def
        using "\<^bold>&I" KBasic_3[equiv_rl] by blast
    qed

  lemma null_uni_facts_3[PLM]:
    "[Null (\<^bold>a\<^sub>\<emptyset>) in v]"
    proof -
      let ?\<psi> = "\<lambda> x . Null x"
      have "[((\<^bold>\<exists>! x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> (\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> ?\<psi> (y\<^sup>P))) in v]"
        using unique_box_desc[deduction] null_uni_facts_1[THEN "              apply -byPLM_solver
      have 1: "[(\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> ?\<psi> (y\<^sup>P)) in v]"
        using unique_box_desc[deduction, deduction] null_uni_uniq_1
              null_uni_facts_1[THEN "\<^bold>\<forall>I"] by fast
      have "[\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>a\<^sub>\<emptyset>) in v]"
        unfolding NullObject_def using null_uni_uniq_3 .
      then obtain y where "[y\<^sup>P \<^bold>= (\<^bold>a\<^sub>\<emptyset>) in v]"
        by (rule "\<^bold>\<exists>E")
      moreover hence "[?\<psi> (y\<^sup>P) in v]"
        using 1[THEN "\<^bold>\<forall>E", deduction] unfolding NullObject_def by simp
      ultimately show "[?\<psi> (\<^bold>a\<^sub>\<emptyset>) in v]"
        using l_identity[axiom_instance, deduction, deduction] by blast
    qed

  lemma null_uni_facts_4[PLM]:
    "[Universal (\<^bold>a\<^sub>V) in v]"
    proof -
      let ?\<psi> = "\<lambda> x . Universal x"
      have "[((\<^bold>\<exists>! x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> (\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> ?\<psi> (y\<^sup>P))) in v]"
        using unique_box_desc[deduction] null_uni_facts_2[THEN "\<^bold>\<forall>I"] by fast
      have 1: "[(\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> (x\<^sup>P)) \<^bold>\<rightarrow> ?\<psi> (y\<^sup>P)) in v]"
        using unique_box_desc[deduction, deduction] null_uni_uniq_2
              null_uni_facts_2[THEN "\<^bold>\<forall>I"] by fast
      have "[\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>a\<^sub>V) in v]"
        unfolding UniversalObject_def using null_uni_uniq_4 .
      then obtain y where "[y\<^sup>P \<^bold>= (\<^bold>a\<^sub>V) in v]"
        by (rule "\<^bold>\<exists>E")
      moreover hence "[?\<psi> (y\<^sup>P) in v]"
        using 1[THEN "\<^bold>\<forall>E", deduction]
        unfolding UniversalObject_def by simp
      ultimately show "[?\<psi> (\<^bold>a\<^sub>V) in v]"
        using l_identity[axiom_instance, deduction, deduction] by blast
    qed

  lemma aclassical_1[PLM]:
    "[\<^bold>\<forall> R . \<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (x \<^bold>\<noteq> y)
      \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
    proof (rule "\<^bold>\<forall>I")
      fix R
      obtain a where \<theta>:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists> y          using[][ =]
          \<^bold>& F \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P, F\<rbrace>)) in v]"
        using A_objects[axiom_instance] by (rule "\<^bold>\<exists>E")
      {
        assume "[\<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace> in v]"
        hence "[\<^bold>\<not>(\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)
                \<^bold>& \<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace>) in v]"
          using \<theta>[conj2, THEN "\<^bold>\<forall>E", THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
                cqt_further_4[equiv_lr] "\<^bold>\<forall>E" by fast
        hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)
                <><rightarrow \lbrace>P,(<bold\> z . \<parrRz<supP,\<sup>P<)>inv"
          apply - by PLM_solver
        hence "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace> in v]"
          using \<theta>[conj1] id_eq_1 "\<^bold>&I" vdash_properties_10 by fast
      }
      hence 1: "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace> in v]"
        using reductio_aa_1 CP if_p_then_p by blast
      then obtain b where \<xi>:
        "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>=           using "\<>" blast
          \<^bold>& \<^           z
        using \<theta>[conj2, THEN "\<^bold>\<forall>E", equiv_lr] "\<^bold>\<exists>E" by blast
      have "[a \<^bold>\<noteq> b in v]"
        proof -
          {
            assume "[a \<^bold>= b in v]"
            hence "[\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace> in v]"
              using 1 l_identity[axiom_instance, deduction, deduction] by fast
            hence ?thesis
              using \<xi>[conj2] reductio_aa_1 by blast
          }
          thus ?thesis using reductio_aa_1 by blast
        qed
      hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> b
              \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,b\<^sup>P\<rparr>) in v]"
        using \<theta>[conj1] \<xi>[conj1, conj1] \<xi>[conj1, conj2] "\<^bold>&I" by presburger
      hence "[\<^bold>\<exists> y . \<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> y
              \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
      thus "[\<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y
             \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
    qed

  lemma aclassical_2[PLM]:
    "[\<^bold>\<forall> R . \<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (x \<^bold>\<noteq> y)
      \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,x\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,y\<^sup>P,z\<^sup>P\<rparr>) in v]"
    proof (rule "\<^bold>\<forall>I")
      fix R
      obtain a where \<theta>:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists> y . \<lparr>A!,y\<^sup>P\<rparr>
          \<^bold>& F \<^bold>= (\<^bold>\<lambda> z .      
        using A_objects[axiom_instance] by (rule "\<^bold>\<exists>E")
       with3have"(^><\<>\<alpha <bold& \^><>z.<bold><>z boldrightarrow z <bold=\alpha))<^bold>& \<psi> (\<alpha>\<^sup>)in v"
        assume "[\<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
        hence "[\<^bold>\<not>(\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)
                \<^bold>& \<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace>) in v]"
          using \<theta>[conj2, THEN "\<^bold>\<forall>E", THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
                cqt_further_4[equiv_lr] "\<^bold>\<forall>E" by fast
        hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)
                \<^bold>\<rightarrow> \<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
          apply - by PLM_solver
        hence "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
          using \<theta>[conj1] id_eq_1 "\<^bold>&I" vdash_properties_10 by fast
      }
      hence 1: "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
        using reductio_aa_1 CP if_p_then_p by blast
      then obtain b where \<xi>:
        "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,b\<^sup>P,z\<^sup>P\<rparr>)
          \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
        using \<theta>[conj2, THEN "\<^bold>\<forall>E", equiv_lr] "\<^bold>\<exists>E" by blast
      have "[a \<^bold>\<noteq> b in v]"
        proof -
          {
            assume "[a \<^bold>= b in v]"
            hence "[\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>)\<rbrace> in v]"
              using 1 l_identity[axiom_instance, deduction, deduction] by fast
            hence ?thesis using \<xi>[conj2] reductio_aa_1 by blast
          }
          thus ?thesis using \<xi>[conj2] reductio_aa_1 by blast
        qed
      hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> b
              \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,b\<^sup>P,z\<^sup>P\<rparr>) in v]"
        using \<theta>[conj1] \<xi>[conj1, conj1] \<xi>[conj1, conj2] "\<^bold>&I" by presburger
      hence "[\<^bold>\<exists> y . \<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> y
              \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,y\<^sup>P,z\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
      thus "[\<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y
             \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,x\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,y\<^sup>P,z\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
    qed

  lemma aclassical_3[PLM]:
    "[\<^bold>\<forall> F . \<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (x \<^bold>\<noteq> y)
      \<^bold>& ((\<^bold>\<lambda>\<^sup>0 \<lparr>F,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>\<^sup>0 \<lparr>F,y\<^sup>P\<rparr>)) in v]"
    proof (rule "\<^bold>\<forall>I")
      fix R
      obtain a where \<theta>:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists> y . \<lparr>A!,y\<^sup>P\<rparr>
          \<^bold>& F \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,y\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P, F\<rbrace>)) in v]"
        using A_objects[axiom_instance] by (rule "\<^bold>\<exists>E")
      {
        assume "[\<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
        hence "[\<^bold>\<not>(\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)
                \<^bold>& \<^bold>\<not>\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace>) in v]"
          using \<theta>[conj2, THEN "\<^bold>\<forall>E", THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
                cqt_further_4[equiv_lr] "\<^bold>\<forall>E" by fast
        hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)
                \<^bold>\<rightarrow> \<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
          apply - by PLM_solver
        hence "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
          using \<theta>[conj1] id_eq_1 "\<^bold>&I" vdash_properties_10 by fast
      }
      hence 1: "[\<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
        using reductio_aa_1 CP if_p_then_p by blast
      then obtain b where \<xi>:
        "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda> z . \<lparr>R,b\<^sup>P\<rparr>)
          \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
        using \<theta>[conj2, THEN "\<^bold>\<forall>E", equiv_lr] "\<^bold>\<exists>E" by blast
      have "[a \<^bold>\<noteq> b in v]"
        proof -
          {
            assume "[a \<^bold>= b in v]"
            hence "[\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,a\<^sup>P\<rparr>)\<rbrace> in v]"
              using 1 l_identity[axiom_instance, deduction, deduction] by fast
            hence ?thesis
              using \<xi>[conj2] reductio_aa_1 by blast
          }
          thus ?thesis using reductio_aa_1 by blast
        qed
      moreover {
        have "[\<lparr>R,a\<^sup>P\<rparr> \<^bold>= \<lparr>R,b\<^sup>P\<rparr> in v]"
          unfolding identity\<^sub>\<o>_def
          usingxi>[conj1, conj2] by auto
        \>\<lambda>\<^sup>0 \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>\<^sup>0 \<>R,b\<^sup>P\<rparr ]
          using lambda_p_q_p_eq_q[equiv_rl] bymp
      }
      ultimately have "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>&<A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\> b
                \<^bold>& ((\<^bold><lambda\<^sup>0 \<lparr>R,a\<^>\<rparr>) \<^bold>=(\<^bold>\<lambda>\<^sup>0 \<lparr>R,b\<^sup>P\<rparr>)) in v]"
    >conj1] \<xi>[conj1, conj1 <>[conj1, j2"\^bold&I"
        by presburgerhave\star UNIV \<subseteq> X \<star> Inv X \<star> UNIV"
      hence "[\<^bold>\<exists> y . \<lparr>A!,a\<^sup>P<> \bold& <>!\^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> y
              \<^bold>& (\<^bold>\<lambda>\<sup0\<lparrR,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda<sup0\<lparr>R,y\<^sup>P\<rparr>) in v]"
        using "\<^bold>\< groupoid_:z>(inv y) \<odot> x) = (x \<in> y \<odot> z)"
      thus "[\<^bold>\<exists> x y <>!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<arr\bold& x \<^bold>\<noteq> y
             \<^bold>& (\<^bold>\<lambda>\<^sup>0 \<lparr>R,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\lambda>\<^sup>0 \<lparr>R,y\<^sup>P> in v]"
        using "\<^bold>\<exists>also"dots = (\<Union>x \<in> X. \<Union>y \<in> Y. inv y \<odot> inv x)"
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

  lemma aclassical2[PLM]:
    "[\<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>&</span> x \<^bold>\<noteq> y \<^bold>& (\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) in v]"
    proof -
      let ?R\<^sub>1 = "\<^bold>\<lambda>\<^sup         \^>equivI CPbyauto
      have "[\<^bold>\<exists> x y . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y
             \<^bold>& (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
        using aclassical_1 by (rule "\<^bold>\<forall>E")
      then obtain a where
        "[\<^bold>\<exists> y . \<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>&span> a \<^bold>\<noteq> y
          \<^bold>& (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
        by (rule "\<^bold>\<exists>E")
      then obtain b where ab_prop:
        "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> b
          \<^bold>& (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>?R\<^sub>1,z\<^sup>P,b\<^sup>P\<rparr>) in v]"
        by (rule "\<^bold>\<exists>E")
      have "[\<lparr>?R\<^sub>1, a\<^sup>P, a\<^sup>P\<rparr> in v]"
        apply (rule beta_C_meta_2[equiv_rl])
         apply show_proper
       oth_class_taut_4_a[HEN "<>\forall>I"]by fast
      hence "[\<lparr>\<^bold>\<lambda> z . \<lparr>?R\<^sub>1, z\<^sup>P, a\<^sup>P\<rparr>, a\<^sup>P\<rparr> in v]"
        apply - apply (rule beta_C_meta_1[equiv_rl])
         apply show_proper
        by auto
      hence "[\<lparr>\<^bold>\<lambda> z . \<lparr>?R\<^sub>1, z\<^sup>P, b\<^sup>P\<rparr>, a\<^sup>P\<rparr> in v]"
        using ab_prop[conj2] l_identity[axiom_instance, deduction, deduction]
        by fast
      hence "[\<lparr>?R\<^sub>1, a\<^sup>P, b\<^sup>P\<rparr> in v]"
        apply (safe intro!: beta_C_meta_1[where \<phi>=
               "\<lambda>z . \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>),z,b\<^sup>P\<rparr>", equiv_lr])
        by show_proper
      moreover have "IsProperInXY (\<lambda>x y. \<^bold>\<forall>F. \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>)"
        by show_proper
      ultimately have "[\<^bold>\<forall>F. \<lparr>F,a\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,b\<^sup>P\<rparr> in v]"
        using beta_C_meta_2[equiv_lr] by blast
      hence "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> b \<^bold>& (\<^bold>\<forall>F. \<lparr>F,a\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,b\<^sup>P\<rparr>) in v]"
        using ab_prop[conj1] "\<^bold>&I" by presburger
      hence "[\<^bold>\<exists> y . \<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& a \<^bold>\<noteq> y \<^bold>& (\<^bold>\<forall>F. \<lparr>F,a\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
      thus ?thesis using "\<^bold>\<exists>I" by fast
    qed

subsection\<open>Propositional Properties\<close>
text\<open>\label{TAO_PLM_PropositionalProperties}\<close>

  lemma prop_prop2_1:
    "[\<^bold>\<forall> p . \<^bolds unfolding identity\<^subE_infix_def
    proof (rule "\<^bold>\<forall>I")
      fix p
      have "[(\<^bold>\<lambda> x . p) \<^bold>= (\<^bold>\<lambda> x . p) in v]"
        using id_eq_prop_prop_1 by auto
      thus "[\<^bold>\<exists>  F . F \<^bold>= (\<^bold>\<lambda> x . p) in v]"
        by PLM_solver
    qed

  lemma prop_prop2_2:
    "[F \<^bold>= (\<^bold>\<lambda> x . p) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall> x . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> p) in v]"
    proof (rule CP)
      assume 1: "[F \<^bold>= (\<^bold>\<lambda> x . p) in v]"
      {
        fix v
        {
          fix x
          have "[\<lparr>(\<^bold>\<lambda> x . p), x\<^sup>P\<rparr> \<^bold>\<equiv> p in v]"
            apply (rule beta_C_meta_1)
            by show_proper
        }
        hence "[\<^bold>\<forall> x . \<lparr>(\<^bold>\<lambda> x . p), x\<^sup>P\<rparr> \<^bold>\<equiv> p in v]"
          by (rule "\<^bold>\<forall>I")
      }
      hence "[\<^bold>\<box>(\<^bold>\<forall> x . \<lparr>(\<^bold>\<lambda> x . p), x\<^sup>P\<rparr> \<^bold>\<equiv> p) in v]"
        by (rule RN)
      thus "[\<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> p) in v]"
        using l_identity[axiom_instance,deduction,deduction,
              OF 1[THEN id_eq_prop_prop_2[deduction]]] by fast
    qed

  lemma prop_prop2_3:
    "[Propositional F \<^bold>\<rightarrow> \<^bold>\<box>(Propositional F) in v]"
    proof (rule CP)
      assume "[Propositional F in v]"
      hence "[\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p) in v]"
        unfolding Propositional_def .
      then obtain q where "[F \<^bold>= (\<^bold>\<lambda> x . q) in v]"
        by (rule "\<^bold>\<exists>E")
      hence "[\<^bold>\<box>(F \<^bold>= (\<^bold>\<lambda> x . q)) in v]"
        using id_nec[equiv_lr] by auto
      hence "[\<^bold>\<exists> p . \<^bold>\<box>(F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
        using "\<^bold>\<exists>I" by fast
      thus "[\<^bold>\<box>(Propositional F) in v]"
        unfolding Propositional_def
        using sign_S5_thm_1[deduction] by fast
    qed


  lemma prop_indis:
    "[Indiscriminate F \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<exists> x y . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr>))) in v]"
    proof (rule CP)
      assume "[Indiscriminate F in v]"
      hence 1: "[\<^bold>\<box>((\<^bold>\<exists>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr>)) in v]"
        unfolding Indiscriminate_def .
      {
        assume "[\<^bold>\<exists> x y . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr> in v]"
        then obtain x where "[\<^bold>\<exists> y . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr> in v]"
          by (rule "\<^bold>\<exists>E")
        then obtain y where 2: "[\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr> in v]"
          by (rule "\<^bold>\<exists>E")
        hence "[\<^bold>\<exists> x . \<lparr>F, x\<^sup>P\<rparr> in v]"
          using "\<^bold>&E"(1) "\<^bold>\<exists>I" by fast
        hence "[\<^bold>\<forall> x . \<lparr>F,x\<^sup>P\<rparr> in v]"
          using 1[THEN qml_2[axiom_instance, deduction], deduction] by fast
        hence "[\<lparr>F,y\<^sup>P\<rparr> in v]"
          using cqt_orig_1[deduction] by fast
        hence "[\<lparr>F,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr>) in v]"
          using 2 "\<^bold>&I" "\<^bold>&E" by fast
        hence "[\<^bold>\<not>(\<^bold>\<exists> x y . \<lparr>F,x\<^sup>P\<rparr> \<^bold>             fast
          using pl_1[axiom_instance, deduction, THEN modus_tollens_1]
                oth_class_taut_1_a by blast
      }
      thus "[\<^bold>\<not>(\<^bold>\<exists> x y . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr>) in v]"
        using reductio_aa_2 if_p_then_p deduction_theorem by blast
    qed


  lemma prop_in_thm:
    "[Propositional F \<^bold>\<rightarrow> Indiscriminate F in v]"
    proof (rule CP)
      assume "[Propositional F in v]"
      hence "[\<^bold>\<box>(Propositional F) in v]"
        using prop_prop2_3[deduction] by auto
      moreover {
        fix w
        assume "[\<^bold>\<exists> p . (F \<^bold>= (\<^bold>\<lambda>        apply (PLM_subst_method
        then obtain q where q_prop: "[F \<^bold>= (\<^bold>\<lambda> y . q) in w]"
          by (rule "\<^bold>\<exists>E")
        {
          assume "[\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr> in w]"
          then obtain a where "[\<lparr>F,a\<^sup>P\<rparr> in w]"
            by (rule "\<^bold>\<exists>E")
          hence "[\<lparr>\<^bold>\<lambda> y . q, a\<^sup>P\<rparr> in w]"
            using q_prop l_identity[axiom_instance,deduction,deduction] by fast
          hence q: "[q in w]"
            apply (safe intro!: beta_C_meta_1[where \<phi>="\<lambda>y. q", equiv_lr])
             apply show_proper
            by simp
          {
            fix x
            have "[\<lparr>\<^bold>\<lambda> y . q, x\<^sup>P\<rparr> in w]"
              apply (safe intro!: q beta_C_meta_1[equiv_rl])
              by show_proper
            hence "[\<lparr>F,x\<^sup>P\<rparr> in w]"
              using q_prop[eq_sym] l_identity[axiom_instance, deduction, deduction]
              by fast
          }
          hence "[\<^bold>\<forall> x . \<lparr>F,x\<^sup>P\<rparr> in w]"
            by (rule "\<^bold>\<forall>I")
        }
        hence "[(\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall> x . \<lparr>F, x\<^sup>P\<rparr>) in w]"
          by (rule CP)
      }
      ultimately show "[Indiscriminate F in v]"
        unfolding Propositional_def Indiscriminate_def
        using RM_1[deduction] deduction_theorem by blast
    qed

  lemma prop_in_f_1:
    "[Necessary F \<^bold>\<rightarrow> Indiscriminate F in v]"
    unfolding Necessary_defs Indiscriminate_def
    using pl_1[axiom_instance, THEN RM_1] by simp

  java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
    "[Impossible F \<^bold>\<rightarrow> Indiscriminate F in v]"
    proof -
      {
        fix w
        have "[(\<^bold>\<not>(\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr>)) \<^bold>\<rightarrow> ((\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall> x . \<lparr>F,x\<^sup>P\<rparr>)) in w]"
          using useful_tautologies_3 by auto
        hence "[(\<^bold>\<forall> x . \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> ((\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<rightarrow> (\<^bold>\<forall> x . \<lparr>F,x\<^sup>P\<rparr>)) in w]"
          apply - apply (PLM_subst_method "\<^bold>\<not>(\<^bold>\<exists> x. \<lparr>F,x\<^sup>P\<rparr>)" "(\<^bold>\<forall> x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)")
          using cqt_further_4 unfolding exists_def by fast+
      }
      thus ?thesis
        unfolding Impossible_defs Indiscriminate_def using RM_1 CP by blast
    qed

  lemma prop_in_f_3_a:
    "[\<^bold>\<not>(Indiscriminate (E!)) in v]"
    proof( reductio_aa_2
      show "[\<^bold>\<box>\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>E!,x\<^sup>P\<rparr>) in v]"
        using a_objects_exist_3 .
    next
      assume "[Indiscriminate E! in v]"
      thus "[\<^bold>\<not>\<^bold>\<box>\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>E!,x\<^sup>P\<rparr>) in v]"
        unfolding Indiscriminate_def
        using o_objects_exist_1 KBasic2_5[deduction,deduction]
        unfolding diamond_def by blast
    qed

  lemma prop_in_f_3_b:
    "[\<^bold>\<not>(Indiscriminate (E!\<^sup>-)) in v]"
    proof (rule reductio_aa_2)
      assume "[Indiscriminate (E!\<^sup>-) in v]"
      moreover have "[\<^bold>\<box>(\<^bold>\<exists> x . \<lparr>E!\<^sup>-, x\<^sup>P\<rparr>) in v]"
        apply (PLM_subst_method "\<lambda> x . \<^bold>\<not>\<lparr>E!, x\<^sup>P\<rparr>" "\<lambda> x . \<lparr>E!\<^sup>-, x\<^sup>P\<rparr>")
         using thm_relation_negation_1_1[equiv_sym] apply simp
        unfolding exists_def
        apply (PLM_subst_method "\<lambda> x . \<lparr>E!, x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<^bold>\<not>\<lparr>E!, x\<^sup>P\<rparr>")
         using oth_class_taut_4_b apply simp
        using a_objects_exist_3 by auto
      ultimately have "[\<^bold>\<box>(\<^bold>\<forall>x. \<lparr>E!\<^sup>-,x\<^sup>P\<rparr>) in v]"
        unfolding Indiscriminate_def
        using qml_1[axiom_instance, deduction, deduction] by blast
      thus "[\<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>) in v]"
        apply -
        apply (PLM_subst_method "\<lambda> x . \<lparr>E!\<^sup>-, x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<lparr>E!, x\<^sup>P\<rparr>")
        using thm_relation_negation_1_1 by auto
    next
      show "[\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall> x . \<^bold>\<not>\<lparr>E!, x\<^sup>P\<rparr>) in v]"
        using o_objects_exist_1
        unfolding diamond_def exists_def
        apply -
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<not>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>)" "\<^bold>\<forall>x. \<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>")
        using oth_class_taut_4_b[equiv_sym] by auto
    qed

  lemma prop_in_f_3_c:
    "[\<^bold>\<not>(Indiscriminate (O!)) in v]"
    proof (rule reductio_aa_2)
      show "[\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>O!,x\<^sup>P\<rparr>) in v]"
        using a_objects_exist_2[THEN qml_2[axiom_instance, deduction]]
              by blast
    next
      assume "[Indiscriminate O! in v]"
      thus "[(\<^bold>\<forall> x . \<lparr>O!,x\<^sup>P\<rparr>) in v]"
        unfolding Indiscriminate_def
        using o_objects_exist_2 qml_1[axiom_instance, deduction, deduction]
              qml_2[axiom_instance, deduction] by blast
    qed

  lemma prop_in_f_3_d:
    "[\<^bold>\<not>(Indiscriminate (A!)) in v]"
    proof (rule reductio_aa_2)
      show "[\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>A!,x\<^sup>P\<rparr>) in v]"
        using o_objects_exist_3[THEN qml_2[axiom_instance, deduction]]
              interpret PLM .
    next
      assume "[Indiscriminate A! in v]"
      thus "[(\<^bold>\<forall> x . \<lparr>A!,x\<^sup>P\<rparr>) in v]"
        unfolding Indiscriminate_def
        using a_objects_exist_1 qml_1[axiom_instance, deduction, deduction]
              qml_2[axiom_instance, deduction] by blast
    qed

  lemma prop_in_f_4_a:
    "[\<^bold>\<not>(Propositional E!) in v]"
    using prop_in_thm[deduction] prop_in_f_3_a modus_tollens_1 CP
    by meson

  lemma prop_in_f_4_b:
    "[\<^bold>\<not>(Propositional (E!\<^sup>-)) in v]"
    using prop_in_thm[deduction] prop_in_f_3_b modus_tollens_1 CP
    by meson

  lemma prop_in_f_4_c:
    "[\<^bold>\<not>(Propositional (O!)) in v]"
    using prop_in_thm[deduction] prop_in_f_3_c modus_tollens_1 CP
    by meson

  lemma prop_in_f_4_d:
    "[\<^bold>\<not>(Propositional (A!)) in v]"
    using prop_in_thm[deduction] prop_in_f_3_d modus_tollens_1 CP
    by meson

lemma:
    "[\<^bold>\<diamond>(\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p)) \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
    proof (rule CP)
      assume "[\<^bold>\<diamond>(\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
      hence "[\<^bold>\<exists> p . \<^bold>\<diamond>(F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
        using "BF\<^bold>\<diamond>"[deduction] by auto
      then obtain    ultimately  show "\<heta \phi> \>in v" simp
        by (rule "\<^bold>\<exists>E")
      hence "[\<^bold>\<diamond>\<^bold>\<box>(\<^bold>\<forall>x. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>x\<^sup>P,\<^bold>\<lambda>x. p\<rbrace>) in v]"
        unfolding identity_defs .
      hence "[\<^bold>\<box>(\<^bold>\<forall>x. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>x\<^sup>P,\<^bold>\<lambda>x. p\<rbrace
        using "5\<^bold>\<diamond>"[deduction] by auto
      hence "[(F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
        unfolding identity_defs .
      thus "[\<^bold>\<exists> p . (F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
        by PLM_solver
    qed

  lemma prop_prop_nec_2:
    "[(\<^bold>\<forall> p . F \<^bold>\<noteq> (\<^bold>\<lambda> x . p)) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall> p . F \<^bold>\<noteq> (\<^bold>\<lambda> x . p)) in v]"
    apply (PLM_subst_method
           "\<^bold>\<not>(\<^bold>\<exists>  p . (F \<^bold>= (\<^bold>\<lambda> x . p)))"
           "(\<^bold>\<forall> p . \<^bold>\<not>(F \<^bold>= (\<^bold>\<lambda> x . p)))")
     using cqt_further_4 apply blast
    apply (PLM_subst_method
           "\<^bold>\<not>\<^bold>\<diamond>(\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))"
           "\<^bold>\<box>\<^bold>\<not>(\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))")
     using KBasic2_4[equiv_sym] prop_prop_nec_1
           contraposition_1 by auto

  lemma prop_prop_nec_3:
    "[(\<^bold>\<exists> p . F \<^bold>=    assumes"Substable cond\<si"
    using prop_prop_nec_1 derived_S5_rules_1_b by simp

  lemma prop_prop_nec_4:
    "[\<^bold>\<diamond>(\<^bold>\<forall> p . F \<^bold>\<noteq> (\<^bold>\<lambda> x . p)) \<^bold>\<rightarrow> (\<^bold>\<forall> p . F \<^bold>\<noteq> (\<^bold>\<lambda> x . p)) in v]"
    using prop_prop_nec_2 derived_S5_rules_2_b by simp

  lemma enc_prop_nec_1:
    "[\<^bold>\<diamond>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p)))
      \<^bold>\<rightarrow> (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p))) in v]"
    proof (rule CP)
      assume "[\<^bold>\<diamond>(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<  lemma  partition_1[PLM:
      hence 1: "[(\<^bold>\<forall>F. \<^bold>\<diamond>(\<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p)))) in v]"
        using "Buridan\<^bold>\<diamond>"[deduction] by auto
      {
        fix Q
        assume "[\<lbrace>x\<^sup>P,Q\<rbrace> in v]"
        hence "[\<^bold>\<box>\<lbrace>x\<^sup>P,Q\<rbrace> in v]"
          using encoding[axiom_instance, deduction] by auto
        moreover have "[\<^bold>\<diamond>(\<lbrace>x\<^sup>P,Q\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. Q \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
          using cqt_1[axiom_instance, deduction] 1 by fast
        ultimately have "[\<^bold>\<diamond>(\<^bold>\<exists>p. Q \<^bold>= (\<^bold>\<lambda>x. p)) in v]"
          using KBasic2_9[equiv_lr,deduction] by auto
        hence "[(\<^bold>\<exists>p. Q \<^bold>= (\<^bold>\<lambda>x. p)) in v]"
          using prop_prop_nec_1[deduction] by auto
      }
      thus "[(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p))) in v]"
        apply - by PLM_solver
    qed

  lemma enc_prop_nec_2:
    "[(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p))) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace>
      \<^boldjava.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 111
    using derived_S5_rules_1_b enc_prop_nec_1 by blast
end
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3