Impressum TAO_9_PLM.thy

(*<*)
theory TAO_9_PLM
imports 
  TAO_8_Definitions 
  "HOLopen>\{TAO_PLM_Solver}›d PLM_intro PLM_lim PLM_subst PLM_dest PLM
begin
(*>*)

section‹
text\<subsection\

declare meta_defs[no_atp] meta_aux[no_atp]

locale PLM = Axioms
begin

subsectionopen>A Solver›
text‹\label{TAO_PLM}›Automatic Solver›

  named_theorems PL namd_herem L
 ed_theorems = supo at i A "phi]]" for \<phi> \<open>fact A[axiom_instance)
  named_theorems PLM_elim
  named_theorems 
  named_theorems PLM_subst

  methodtro PLM_elim PLM_substubstst
    = ((assumption    by(simp add: Semantics.T5
          _ bstubsttubstmLM_subst
        

subsection\<open>Modus Ponens\<lose
text\<open>\label{TAO_PLM_ModusPonens}\<close>

  lemma modus_ponens[PLM]:
    "\<lbrakk>[\<phi> in v]; [\<phi> \<^bold>\<rightarrow> \<psi> in v]\<rbrakk> \<Longrightarrow> [\<psitextopen\label{TAO_PLM_ProofsAndDerivations}\<close>\openlabelTAO_PLM_ProofsAndDerivations}\close
    by (simp add: Semantics.T5)

subsection\<open>Axioms\<close>
text\<open>\label{TAO_PLM_Axioms}\<close>

  rpretationioniomsms
  declare axiom[PLM]
  declare conn_defs[PLM]

subsection\\open\label{TAO_PLM_GEN_RN}\<close>
text :.

  mmaAndv .[\psi> inv <>[<>n]Longrightarrow[\^>\<>psi>  ]>\<bold\box\>in])"
    "lbrakk\<phi> in v]; [\<phi> \<^bold>\<rightarrow> \<psi> in v]\<rbrakk> \<Longrightarrow> [\<psi
    using modus_ponens .
  lemma vdash_properties_9\<openNegationsandConditionalslemmaeful_tautologies_5:
    "[\<phi>text{TAO_PLM_NegationsAndConditionalsclose
    using modus_ponens pl_1[axiom_instance]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  lemma vdash_properties_10PLM:
    "[\<phi> \<^bold>private isjILM_intro
    using vdash_properties_6 .

  attribute_setup deduction = \<open>
    .cceedhmule_attribute
      (fn _ => fn thm => thm RS @{thmlemmah_class_taut_5_cjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
\<close>

subsection\<open>GENnd\close
text\usingCP\<bold><equiv>I" by blast

  lemma rule_gen[PLM]:
    "\<lbrakk>\<And>\<alpha> . [\<phi> \<alpha> in \><> [\<^bold>\<forall>\<alpha  in v]"
    by (simp add: Semantics.T8)

  lemma RN_2[PLM]:
    "(\<And> v  emmaoth_class_taut_5_kPLM
    by(simp class_taut_6_aPLM

  lemma RN[PLM]:
    \And emmaa[_ntroo:
    using qml_3[xiom_necessitationxiom_instance_ t

subsection\<"<boldnot>\<phi> \<^bold>& \<psi> ^bold\<equiv> (\<^bold>\not\<phi> \<^bold>\<or> \<^bold>\otpsi in v]"
t\label{TAO_PLM_NegationsAndConditionals}\<close>

  lemma if_p_then_p[PLM]:
    "[\<phi> \<^bold>\<rightarrow> \<phi> in "[\phiolver
    using pl_11pl_2 sh_properties_1010om_instancelast

  < \<^bold>\<rightarrow> (\<psi> <bold><rightarrow> \<chi>)) \<^bold\equiv> (\<psi> \<^bold>\<rightarrow\phi \<^bold><> \chi)) in v]"
    "\<lbrakk>[\<phi> in v] \<Longrightarrow> [\<psi> in v]\<rbrakk> \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<psi> inv
    (simp add: Semantics.T5)
  lemmas CP = deduction_theorem

  lemma ded_thm_cor_3[PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<rightarrow>   lemmaoth_class_taut_5_i
    by (meson pl_2 vdash_properties_10 dash_properties_9perties_9iom_instance
  lemma ded_thm_cor_4PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>) in][> in ]<brakk\<> [\<phi> \<^bold>\<ightarrowhtarrow<>in v]"
    by (meson pl_2 vdash_properties_10 vdash_properties_9 axiom_instance)

  lemma useful_tautologies_1[PLM]:
    "[\<^bold>\<not>\<^bold>\<not>\<phi> \<^boldlemma oth_class_taut_7_a[LM:
    by (meson pl_1    by PLM_solver
  lemma useful_tautologies_2PLM]
    "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<^bold>\<not>\<phi> 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]:
    "[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<psi>) 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]:
    "[(\<^bold>\<not>\<psi> \   h_class_taut_10_c:
    by (meson pl_1 pl_2 pl_3 ded_thm_cor_3 ded_thm_cor_4 axiom_instance)
  lemma useful_tautologies_5[PLM]:
    > \<^bold>\<rightarrow> \<psi>) <\<rightarrow> (\<^bold>\<not>\<psi> \<^bold> \<^bold>\<not>\<phi>) in v]"
by (metis CP useful_tautologies_4 vdash_properties_10)
  lemma useful_tautologies_6[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi>) \
    by (metis CP useful_tautologies_4 vdash_properties_10)
  lemma useful_tautologies_7[PLM]:
    "[(\<^
    using ded_thm_cor_3ogies_4ul_tautologies_5
          useful_tautologies_6 by blast
  lemma useful_tautologies_8[PLM]:
    <> <^bold>\<>(<bold><ot>\<>\<^bold\<> \old><>\^>\<> <psi
    by (meson ded_thm_cor_3 CP useful_tautologies_5)
  useful_tautologies_9[PLM]:
    "[(\<phi>     unfoldingidentity_defsty_defs LM_solver
    by (metis CP useful_tautologies_4 vdash_properties_10)
  lemma3) \<^bold>= G) \<^bold>\<rightarrow> (G \^oldFv"
   >\<^bold>\<rightarrow> \<^bold>\not\psi>) \<^bold>\<rightarrow> ((\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\rightarrow \<^bold>\<not>> in v]"
    by (metis ded_thm_cor_3Puseful_tautologies_6

  lemma modus_tollens_1[PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<rightarrow> \<psi> in v      e\boldexists> \<alpha> . (\<alpha>\<^sup ^= x in v]"
    by (metis ded_thm_cor_3d_thm_cor_4ul_tautologies_3
              useful_tautologies_7 vdash_properties_10)
  lemma modus_tollens_2[PLM]next
    "\<lbrakk>[\<hi^\<rightarrow> \<^bold>\<not>\[> < \<Longrightarrow> [\<^bold>\<not>\<phi> in v]"
    using modus_tollens_1 useful_tautologies_2
          vdash_properties_10 by blast

  lemma contraposition_1[PLM]:
    "[\<phi> \<rightarrow> \<psi> in] = ^><not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> in v]"
    simple_1
          vdash_properties_10
  lemma contraposition_2[PLM]:
    "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi>have"[\bold\<diamond>\<lparr>E!, x\<^sup>P\<rparr>) \<^bold>\<or> (\<^old><ot\<^bold>\<diamond>\<lparr>E!, x\<^sup>P\<rparr>) in v]"
    using contraposition_1 ed_thm_cor_3_
           by blast

  lemma reductio_aa_1[PLM]:
    "\<lbrakkhence\lparr\<^ldlambda>. \<^bold>\<not>\<^bold<>\<E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr> \<
    using CP modus_tollens_2 useful_tautologies_1
          vdash_properties_10 by blast
  lemma reductio_aa_2[PLM]:
    "\<lbrakk>[  To unify the statements propertiesofualityypeassisintroduced
    assumes_ ( :)<bold>)^> (y \<^bold>= z) \<^bold>\<rightarrow> (^= z) in v]"
lemmaPLMjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
    "\<lbrakk>[\<^bold>\<not>\<phi
    using reductio_aa_1 vdash_properties_10 by blast
  lemma   next
    "\<lbrakk>[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> in v]   show "x<bold=\^>\rightarrow> y \<^bold>= x in v]"
    using reductio_aa_2 vdash_properties_10 by blast

  lemma raa_cor_1[PLM]:
    "\<lbrakk>[\<phi> in v]; [\<^bold>\<not>\<psi> in v] \<Longrightarrow> [\<^bold>\<not>\<phi  
    using reductio_aa_1 ash_properties_9perties_9 by blast
  lemma raa_cor_2[PLM]:
    "\<lbrakk>[\<^bold>\<not>\<phi> in vinstanceof
    using reductio_aa_1 vdash_properties_9 by blast
  lemma raa_cor_3[PLM] y) \<^bold>& (y \\<rightarrow> x \<^bold> 
    "\<lbrakk>[\phi in v]; [\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> in v]\<rbrakk> \<Longrightarrow> < in v] \<Longrightarrow> [\<psi> in v])"
    using raa_cor_1 vdash_properties_10 by blast
  lemma raa_cor_4[PLM]:
    "\<lbrakk>[\<^bold>\<not>\<phi> in v]; [\<^bold>\<not>\<psi> \<^bold>hence<\<exists> \<beta> . (\<beta>\<^sup>bold= \<tau> in v]"
    using raa_cor_2 vdash_properties_10 by blast

text\<open>
\begin{remark}
  contrasthessicalduction and elimination rules are proven
  reutologiesgiessTheatementsents oven nfar fficient
  for the proofs and using the derived rules the tautologies can be derived
  automatically.
\end{remark}
\<alpha>\<^sup>P) \^>= (\<^bold>\<iota>x . \<phi> x) in ] and [(\<beta>\<^sup>)bold= (\<^bold>\<iota>x .psi x) in

  lemma intro_elim_1[PLM]:
    "\<lbrakk>[\<phi>       <>([<><exists> \<alpha>. (\<alpha>\<^sup>P) \<^bold>= (\<^bold><a phi x)) in v]and [(\<^bold>\<exists> \<>beta<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<psi> x)) in v])"
    unfolding conj_def\label{TAO_PLM_Identity}\<close>
  lemmas "\<^bold>&I" = intro_elim_1
  lemma intro_elim_2_a[PLM]:
    "[\<phi> \<^bold>& \<psi> in  >[\<phi> in v]"
    unfolding conj_def using CP reductio_aa_1 by blast
  lemma intro_elim_2_b[PLM]:

   conj_def ductio_aa_1instancelast
  lemmas "\<^bold>&E" = intro_elim_2_a intro_elim_2_b
  lemma intro_elim_3_a[PLM]:
    "[\<phi> in v] lemmaprop_prop_7
    unfolding disj_def using ded_thm_cor_4 useful_tautologies_3 by     identity_defssolver
  lemma intro_elim_3_b[PLM]:
    <psi> in v] \<Longrightarrow> [\<phi> \<^bold>\<or> \<psi> in v]"
    bysimpnlysj_defash_properties_9
  lemmas "\<^bold>\<or>I" = intro_elim_3_a intro_elim_3_b
  lemma intro_elim_4_a[PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<or> \<psi> in v >\<^bold>\<rightarrow chi in v]; [ \<^bold>\< \<chi> in     equiv_def \&pan>1intro_elim_4_dbyblast
    unfolding disj_def by (meson reductio_aa_2h_properties_10
  lemmalim_4_bjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    "\<lbrakk>[\<phi> \<^bold>\or <psi> in v]; [\<^bold>\<not>\<phi> in v]\<rbrakk> \<Longrightarrow> [\<psi> in ]
    unfolding disj_def using vdash_properties_10 by blast
  lemma intro_elim_4_c[PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<or> \<siin;^><not>\<psi> in v]\<rbrakk> \<Longrightarrow> [\<phi>  "
    unfolding disj_def using raa_cor_2 vdash_properties_10 by blast
  emmatro_elim_4_d]
    "\<lbrakk>[\<phi> \<^bold"[<bold=<><equiv> ((\<lparr>O!,x\<rparr> \<^bold>& \lparr\>^& \<^bold>\<box>(\<^bold>\<forall>F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>))
    unfolding disj_def using contraposition_1ded_thm_cor_3 by blast
  lemma using<><or>I" by blast
orpsi in v]; [\<phi> \<^ld<>\<chi> in v]; [\<psi> \<^bold>\<equiv> \<Theta> in v]\<rbrakk> \<Longrightarrow> [\<chi> \<^bold>\<or> \<Theta> in
    unfoldingf^>)tro_elim_4_d blast
  lemmas "\<^bold>\<or>E" = intro_elim_4_a intro_elim_4_b intro_elim_4_c intro_elim_4_d
  lemma    insertsmsoldingts_defPLM_solver
    "\<lbrakk>[\<phi> bold\<rightarrow> \<psi> in v]; \ \<^bold>\<rightarrow> in v]\<rbrakk< [\<phi> \<^bold>\<equiv> \<psi> in v]"
     (simplyquiv_def<>I"java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
  lemmas "\<^bold>\<equiv>I" = intro_elim_5
  lemma intro_elim_6_ahowx^bold=  v]
    "\<lbrakk>[\<phi> \<^bold>\<equiv> \<psi> in v]; [\<phi> in v]\<show ^> y \<^bold>\<rightarrow> y \<^bold>= x in v]"
ng_ \^bold>E"(1) vdash_properties_10 by blast
  ntro_elim_6_bbPLM]
    "\<lbrakk>[\<phi> \<^bold>\<equiv> \<psi> in v]; [\<psi> in v]\<rbrakk> \"[\>\^p\>= (\<^bold>\<iota>x . \<phi> x) in v] \and [(\<beta>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<psi> x ]
    unfolding equiv_def using "\<^bold>&E"(2) h_properties_10 lastt
  lemmatro_elim_6_c
    thushesisply yLM_solver
    unfolding equiv_def using "\<^bold>&E"(2
  lemma intro_elim_6_d[PLM]:
    "\<lbrakk>[\<phi> \<^bold>\<equiv> \<  lemmarule_uiLMM_elimM_dest
    unfolding equiv_def using "\<^bold>&E"(1)us_tollens_1t
  lemma intro_elim_6_e[]:
    "\<lbrakk>[\<phi> \<^bold>\<equiv> \"<><forall>\<alpha>. \<phi> \<^bold>rightarrow \<psi> \<alphabold\<rightarrow> (\<phi> \<^bold>\<rightarrow> (\<^bold>forall<alpha>. \<psi> \<alpha>)) in v]"
    by (metis equiv_def ded_thm_cor_3 "\<^bold>&usingmplIy
  lemma intro_elim_6_f[PLM]:
    \\<phi> \<^bold>\<equiv> \<psi>nv<phi \<^bold>\<equiv> <hi in v]\<rbrakk> \<Longrightarrow> [\<chi> \<^bold>\<equiv> \<psi> in v]"
    by (metis uiv_defd_thm_cor_3 "\<^bold&<boldequiv>I")
  lemmas "\^>\<equiv>E" = intro_elim_6_a intro_elim_6_b intro_elim_6_c
                intro_elim_6_d intro_elim_6_ero_elim_6_f
  lemma intro_elim_7[PLM]:
    "[\<phi> inqed
    using if_p_then_p modus_tollens_2 by blast
  s<bold\ot\<^bold>\<not>I" = intro_elim_7
  lemma intro_elim_8[PLM]:
    "[\<^bold>\<not>\<^bold>\<not>\<phi> in v] \<Longrightarrow> [\<phi> in v]"
    usingif_p_then_p_2y ast
  "><not>\<^bold>\<not>E" = intro_elim_8

  ext
  begin
     mmatIntro
      [<phi> in v] \<Longrightarrow> [\<^bold>\<not>(\<^bold>\<not>\<phi>) in v]"
      by ( dd"^bold\not\^bold\<not>I")
    private lemma NotNotD[PLM_dest]:
      "[\<^bold>\<not>(\<^>not\<phi>) in v] \<Longrightarrow in v]"
      using "\<^bold>\<not>\<^bold>\<not>E" by blast

    private lemma ImplI[PLM_intro]:
      "([\<phihence<\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (<>>. \<phi> x) inw
    CP java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    private mmalimM_dest
      " \<^bold>\<rightarrow> psi in v] \<Longrightarrow[phi in v] \<ghtarrow< in v])"
      using modus_ponens .
    private lemma ImplS[PLM_subst]:
      "[\<phi> \<^bold>\<rightarrow> \<psi> in v] = ([\<phi> in v] \<longrightarrow> [\<psi> in v])"
      using ImplI ImplE by blast

    private lemma NotI[PLM_intro]
      "([\<phi> in v] \<Longrightarrow(And\<psi> .[\<psi> in v])) \Longrightarrow><>\<not>\<phi> in v]"
      usingCPodus_tollens_2byblast
    private lemma NotE[PLM_elim,PLM_dest]:
      "<bold\<not>\<phi> in v] \<Longrightarrow> ([\<phi> in v] \<longrightarrow><psi> .[\<psi> in v]))"
      using "\<^bold>\<or>I"(2) "\<^bold>\<or>E"(3) by blast
    private lemma NotS[PLM_subst]:
      "[\<^bold>\<nothence
      using NotI NotE by blast

  aonjI_rojava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      using "\<^bold>&yjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33

      "[\<       obtainere:
      using CP "\<^bold>&E"{ ixa   
    private lemma ConjS[PLM_subst]:
      "[\<phi> \<^bold>& \          expunge_expunge_commutehbconcurrent_def
      using ConjI ConjE by 

    private lemma DisjI[PLM_intro]:
      usingassms prefix_contains_msg apply (metis 
         sing"^bold>\<r" java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  isjElimLM_dest
      "[\<phi> \<^bold>\<orusingi_desc_descA_1y
      using CP "\<^bold>\<or>E"(1) by blast
    private  DisjS[PLM_subst]:
      "[\<phi> \<^boldusing"\<exists>I" by using_st_message
      using DisjI DisjE bybold>\<iota>x. \<phi> x) in v]"

    private lemma EquivIM_intro
">phi> in v] \<Longrightarrow> \>in v];[\<psi> in v Longrightarrow [\<phi> in v]\<rbrakk>< [\<phi> \<^bold>\<equiv> \<psi> in v]"
      using CP "\<^bold>\<equiv>I" by blast
    private lemma EquivE[PLM_elim,PLM_dest]:
      "[\<phi> \<^bold>\<equiv> \<psi> in v] \<Longrightarrow> ((
      using "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2) by blast
    private lemmaEquivSPLM_subst
      "[\phi> \<^bold>\<equiv> \<psi> in v]=< in v] \<longleftrightarrow> [\> n]
      using EquivI EquivE by blast

    private lemma NotOrD[PLM_dest]:
      "\<not>[\<phi> \<^bold>\<or> \<psi> in v] \<Longrightarrow> \<not>[\<phi> in v] \<and> \<not>[\<              have [<bold><A\<phi> z\bold<rightarrow> z \<^bold>= x in v]"
      using "\<^bold>\<or>I" by blast
    private lemma NotAndD[PLM_dest]:
      "\<not>[\<phi> \<& \<psi> in v] \<Longrightarrow> \<not>[\<phi> in v] \<or> \<not>[\<psi> in v]"
              java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    privateassmswhere]
      "\<not>[\<phi><bold><> <psi in v] \<Longrightarrow> [<>in   [\<psi> in v]"
      by (meson NotI hence"\<bold<>z.\^bold\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x)n ule\bold\<forall>I")

     mma[intro
      "(\<And> v . [\<phi> in v]) \<Longrightarrow> [\<^bold>\<box>\<phi> in v]"
      using RN by blast
     lemmaNotBoxD[PLM_dest]:
      "\<not>[\<^bold>\<box>\<phi> in v] \<Longrightarrow> (\<exists> v . \<not>[\<phi                          ]
      using BoxI by blast

    private lemma AllI[PLM_intro]:
      "(\<And> x . [\<phi> 
      using rule_gen by blast
    lemma NotAllD[PLM_dest]:
      "\<not>[\<^bold>\<forall> x . \<phi> x in v] \<Longrightarrowproofjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
      using AllI by 
  end

emmat_1_a:
    "[\<^bold>\<not>(\<phi> \<^bold>& \<^bold>\<not>\<phi>) in v]"quiv_lrt_basic_2[equiv_lr,conj1
    by PLM_solver
  lemma oth_class_taut_1_b[PLM]:
    "[\<^bold>\<not>(\<phi> \<^bold>\<equiv> \<^bold>\<not>\<phi>) in v]"
    by PLM_solver
  lemma oth_class_taut_2[PLM]:
    "[\<phi> \<^bold>\<or> \<^bold>\<not>\<phi> in v]"
    by PLM_solver
  lemma oth_class_taut_3_a[PLM]:
    "[(\<phi> \<^bold>& \<phi>) \<^bold>\<equiv> \<phi> in v]"
    by PLM_solver
  lemma oth_class_taut_3_b[PLM]:
    "[(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> (\<psi> \<^bold>& \<phi>)          >\<^sup><ldbold\<iota>x. \<phi> x)) in v]"
    by PLM_solver
  lemma oth_class_taut_3_cunfoldingts_unique_defrule<old>\<exists>E")
    "[(\<phi> \<^bold>& (\        bold\<forall>z. \<^bold>\<A>\<phi> z \<^bold>equiv> z \<^bold>= x in vjava.lang.StringIndexOutOfBoundsException: Index 94 out of bounds for length 94
    by PLM_solver
  lemma oth_class_taut_3_d:
    "[(\<phi> \<^bold>\<or> \<phi>) \<^bold>\<equiv> \<phi> in v]"
    by PLM_solver
  lemma oth_class_taut_3_e[PLM]:
    "[(\<phi> \<bold>\<or> \<psi>) \<^bold>\<equiv> (\"[<phi \^>rpsi  ]Longrightarrow>[\phi  v <>[<> ]
    by<old\<box>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<^bold>\<box>(\<phi> \<beta>) \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
  lemma oth_class_taut_3_f[PLM]:
    "[(\<phi> \<^bold>or (\<psi> \<^bold>\<or> \<chi>)^<equiv> ((\<phi> \<^bold>\<or> \psi) \<^bold<> \<chi>) in v"
    by PLM_solver
  lemmah_class_taut_3_gtaut_3_gM
    "[(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> (\<psi> \<^bold>\<equiv> \<    \v.[\<phi> \<^bold>\<ightarrowpsi in v]) \Longrightarrow [\<^bold>\<box\ \<^bold>\<w>\<box>\<psi> in v]"
    by PLM_solver
  lemma oth_class_taut_3_i[PLM]:
    "[(\<phi ^>\<equiv> (\<psi> \<^bold>\<equiv> \<chi>)) \<^bold>\<equiv> ((\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> \<chi>) in v]"

  lemma oth_class_taut_4_a[PLM]:
    "[\<phi> \<^bold>\<equiv> \<phi      "\<ot>\phi> <bold>\<>in     last
    by PLM_solver
  lemmaoth_class_taut_4_bPLM]:
    "[\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<not>\<phi> in v]"
    by PLM_solver
  lemma th_class_taut_5_aM]:
    "[(\<> <bold>\<rightarrow> \<psi>) \<^bold>\<equiv> \<^bold>\<not>(<i ^>& \<^bold>\<not>\<psi>) in v]"
    by PLM_solver
  lemma oth_class_taut_5_b[PLM]:
    "[\<^bold>\<not>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv \<^bold>& \<^bold\<psi>) in v]"
    by PLM_solver
  lemma oth_class_taut_5_c[PLM]:
   \<> \<^ldrightarrow \<psi>) \<^bold>\<rightarrow> (( \<^bold>\ \<chi>) \<old (\<phi> \<^bold\ \<chi>)) in "
    by PLM_solver
  lemmaath_class_taut_5_d
" \<^bold>\<equiv>\>) \<^bold>\<equiv> (\<^bold>\<not>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<psi>) in v]"
    by PLM_solver
  lemma oth_class_taut_5_e    "[\<^bold\<box>(\<psi> \<^bold><<> in v] \<Longrightarrow> [\<^bold    lemmaNotAllDPLM_dest:
"(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<rightarrow> ((\<phi> \<^bold>\<rightarrow> \<chi>) \<^bold>\<equiv> (\<psi> \<^bold>\<rightarrow> \<chi>)) in v]"
    by PLM_solver
   oth_class_taut_5_fPLM:
 phi \<^bold>\<equiv> \<psi>) \<bold\<rightarrow> ((\<chi> \<^bold>\<rightarrow> \<phi>) \<^bold><quivchi \<^bold>\<rightarrow> \>)) in v]"
    by PLM_solver
  lemma oth_class_taut_5_g[PLM]:
    "[(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\rightarrow> ((\phi \<^bold>\<equiv> \<>\<^bold>\equiv (\<psi> \<^bold>\<equiv> \<chi>"
    by PLM_solver
  lemma oth_class_taut_5_h[PLM]:
    "[(<> \bold<equiv> \<psi>) \<^bold>\<rightarrow> ((\<chi> \<^bold>\<equiv> \<phi>) \<^bold>\<equiv> (\<chi bold\<equiv> \<psi>)) in v]"
    by_r
  lemmath_class_taut_5_i[PLM]:
    "[(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> ((\<phi>\& \<psi>) \<^bold>\<or> (\^>not\<phi> \<^bold>\<not)) in v
    by PLM_solver
  lemma oth_class_taut_5_j[PLM]:
    "[(\<usingbold\<exists>E" by auto          
    
  lemma oth_class_taut_5_k[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> (\<^bold>\<not> dentity<="\<alpha>\^P" and \<alpha>="\<^bold>\<iota>x. \<ndphi>="\<a<>>,
    

  lemma oth_class_taut_6_a[PLM]:
    "[(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv "(\<>v  y [psi x y \<^bold>\ \<chi> x y in v]) \<Longrightarrow>< [\<phi> \<psiin v] >\<Theta> [\<phi> \<chiin
    by PLM_solver
  lemma oth_class_taut_6_b[PLM]:
    "[(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> \<^bold>\<not (\<^bold>x. \<phi> (x\sup"^dexists x . (\<^bold>\<not>^\<diamond>\<lparr>E!,x\<^sup>P\<rparr)in
    by
  apply M_solver
    "[\<^bold>\<not>(\<phi> \<^bold>& \<psi>) \<^bold>apply- PLM_subst_method "\<bold>\<>(\<>\<not><>\<diamond>(\<phi> \          \bold<\phi> z \<^bold>\<rightarrow> z \<^bold>= x in v]" by (rule CP
    ver
  lemma oth_class_taut_6_d[PLM]:
    "[\<^bold>\<not>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> (\<^bold>\<not>\<phi> \<^lemmasic_7M
yjava.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17

  lemma oth_class_taut_7_a[PLM]:
    "[(\<phi> \<^bold>& (\<psi> \<^bold>\<or> \<chi>)) \<^bold>\<equiv> ((\<phi> \<^bold>& \<psi>) \<^bold>\<or> (\<phi> 
    by PLM_solver
  lemma oth_class_taut_7_b[PLM]:
    "[(\<phi> \<^bold>\<or> (\<psi> \<^bold>& \lemma S5Basic_11[PLM]java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    by PLM_solver

  lemma oth_class_taut_8_a
    "[((\<    using  byssumption
    by PLM_solver
   oth_class_taut_8_b[PLM]:
    "[(\<phi> \<^bold>\<rightarrow<><bold<ightarrow> \<chi>)) \<^bold>\<rightarrow>\ \<^bold>& \<psi^\<rightarrow> \<chi>) in v]"
     PLM_solver

  lemma oth_class_taut_9_a[PLM]:
    "[(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> \<phi> in v]"
    by PLM_solver
  lemma oth_class_taut_9_b[PLM]:
    "[(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> \<psi> in v]"
    by PLM_solver

  lemma oth_class_taut_10_a[PLM]:
    "[\<phi> \<^bold>\<rightarrow> (\<psi>
    
  lemma oth_class_taut_10_b[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>)) \<^bold>\<equiv> (\<psi> \<^bold>\<rightarrow> (\<phi> \<^boldqed
    by PLM_solver
  lemma oth_class_taut_10_c[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> ((\<phi> \<^bold>\<rightarrow> \<chi>) \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>& \<chi>))) in v]"
    by PLM_solver
  lemma oth_class_taut_10_d[PLM]:
"(phi \^bold>rightarrow \chi)\^bold\rightarrow((\psi \^>\rightarrow <>\^bold><rightarrow ((\phi> <^><or><si>) \<^bold<rightarrow><>)  ]java.lang.StringIndexOutOfBoundsException: Index 195 out of bounds for length 195
    by PLM_solver
  lemma oth_class_taut_10_e[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> ((\<chi> \<^bold>\<rightarrow> \<Theta>) \<^bold>\<rightarrow> ((\<phi> \<^bold>& \<chi>) \<     PLM_solver
    by PLM_solver
  lemma oth_class_taut_10_f[PLM]:
    "[((\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> (\    by PLM_solver
    by PLM_solver
  lemma oth_class_taut_10_g[PLM]:
    "[((\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> (\<chi> \<^bold>& \<psi>)) \<^bold>\<equiv> (\<psi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<equiv> \<chi>)) in v]"
    by PLM_solver

  attribute_setup equiv_lr = \<open>
    Scan.succeed (Thm.rule_attribute [] 
      (fn _ => fn thm => thm RS @{thm "\<^bold>\<equiv>E"(1)}))
\<close>

  attribute_setup equiv_rl = \<open>
    Scan.succeed (Thm.rule_attribute [] 
      (fn _ => fn thm => thm RS @{thm \^bold<>E"(2)})
<

  attribute_setup equiv_sym = \<open>
     Thmrule_attribute [
      (fn _ => fn thm => thm RS @{thm oth_class_taut_3_g[equiv_lr]}))
\<close>

  attribute_setup conj1 = \<open>
    Scan.succeed (Thm.rule_attribute [] 
      (fn _ => fn thm => thm RS @{thm "\<^bold>&E"(1)}))
\<close>

  attribute_setup conj2 = \<    then caselr of
    Scan.succeed (Thm.rule_attribute [] 
      (fn _ => fn thm => thm RS @{thm "\<^bold>&E"(2)}))
\<close>

  attribute_setup conj_sym = \<open>
    Scan.succeed (Thm.rule_attribute [] 
      (fn _ => fn thm => thm RS @{thm oth_class_taut_3_b[equiv_lr]}))
\<close>

  
subsection\<open>Identity\<close>
text\<open>\label{TAO_PLM_Identity}\<close>

  lemma id_eq_prop_prop_1[PLM]:
    "[(F::\<Pi>\<^sub>1) \<^bold>= F in v]"
    unfolding identity_defs by PLM_solver 
  lemma id_eq_prop_prop_2[PLM]:
    "[((F::\<Pi>                  instanceuctionduction
    by metisl_identityaxiom_instanceded_thm_cor_4 CP"\<>E)
  lemma id_eq_prop_prop_3[PLM]:
    "[(((F::\<Pi>\<^sub>1) \<^bold>= G) \<^bold>& (G \<^bold>= unfoldingtable_def
byl_identityaxiom_instanceded_thm_cor_4 CP \^boldE)
   __rop_prop_4_a]
    "[(F::\<Pi>\<sub)^> F in v]"
    unfolding          a"height mkt_bal_r     =eight  )+"
  lemma id_eq_prop_prop_4_b[PLM]:
    "[(F:Pi\<^sub>3) \<^bold>= F]
    unfolding identity_defs by PLM_solver
  lemma id_eq_prop_prop_5_a[PLM]:
    "[((F::\<Pi>\<^sub>2) \<^bold>= G) \<^bold>\<rightarrow> (G \<^bold>= F) in v]"
    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::\<Pi>\<^sub>3) \<^bold>= G) \<^bold>\<rightarrow> 
    by (meson id_eq_prop_prop_4_b CP ded_thm_cor_3 l_identity[axiom_instance])
  lemma id_eq_prop_prop_6_a[PLM]:
    "[(((F::\<Pi>\<^sub>2) \<^bold>= G) \<^bold>& (G \<^bold>= H)) \<^bold>\<rightarrow> (F \<^bold>= H) in v]"
    by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "\<^bold>&E")
  lemma id_eq_prop_prop_6_b[PLM]:
    "[(((F::\<Pi>\<^sub>3) \      using PLM.id_eq_prop_prop_7 .
    by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "\<^bold>&E")
  lemma id_eq_prop_prop_7[PLM]:
    "[(p::\<Pi>\<^sub>0) \<^bold>= p in v]"
    unfolding identity_defs by PLM_solver
  lemma id_eq_prop_prop_7_b[PLM]:
    "[(p::\<o>) \<^bold>= p in v]"
    unfolding identity_defs by PLM_solver
  lemma id_eq_prop_prop_8[PLM]:
    "[((p::\<Pi>\<^sub>0) \<^bold>= q) \<^bold>\<rightarrow> (q \<^bold>= p) in v]"
    by (meson id_eq_prop_prop_7 CP ded_thm_cor_3 l_identity[axiom_instance])
  lemma id_eq_prop_prop_8_b[PLM]:
    "end
    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::\<Pi>\<^sub>0) \<^bold>= q) \<^bold>& (q \<^bold>= r)) \<^bold>\<rightarrow> (p \<^bold>= r) in v]"
    by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "\<^bold>&E")
  lemma id_eq_prop_prop_9_b[PLM]:
    "[(((p::\<o>) \<^bold>= q) \<^bold>& (q \<^bold>= r)) \<^bold>\<rightarrow> (p \<^bold>= r) in v]"
    by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "\<^bold>&E")

  lemma fix xy  :\Pi><>andjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
    "[(x \<^bold>=\<^sub>E y) \<^bold>\<equiv> (\<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>)) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume 1: "[x \<^bold>=\<^sub>E y in v]"
      have "[\<^bold>\<forall> x y . ((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> (\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr>
              \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>)) in v]"
        unfolding identity\<^sub>E_infix_def identity\<^sub>E_def
        apply (rule lambda_predicates_2_2[axiom_universal, axiom_universal, axiom_instance])
        by show_proper
      moreover have "[\<^bold>\<exists> \<alpha> . (\<alpha>\<^sup>P) \<^bold>= x in v]"
applyrule cqt_5_mod[here \<psi>"<lambda> x.x\^bold>\<subE y",axiom_instance,deduction
        unfolding identity\<^sub>E_infix_def
        apply (rule SimpleExOrEnc.intros)
        using 1 unfolding identity\<^sub>E_infix_def by auto
      moreover have "[\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= y in v]"
        apply (rule cqt_5_mod[where \<psi>="\<lambda> y . x \<^bold>=\<^sub>E y",axiom_instance,deduction])
        unfolding identity\<^sub>E_infix_def
        apply [simp]: "time ( x y) = "
        unfolding identity\<^subp"
      ultimatelyhave "[(x \^bold>\<sub>E y) \<^old\<quiv> (\<lparr>O!x\rparr <bold& \<lparr>O!,\rparr>
                        <bold&\<^bold>\<box>(\<^bold>\<forall>F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>)) in v]"
        using cqt_1_\<kappa>[axiom_instance,deduction, deduction] by meson
      thus"(<lparr>O!,x\<rparr \<bold& \lparrO,y<rparr>\<bold&\^>\<box>(\<bold>\<forall>F . F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>)) in v]"
        using
    next" where
      assume 1: "[\<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) in v]"
    >> xy  (\^>)\<bold=<sub>(\<^upP)\^bold\<>x<>\rparr\><parrO,\^P<>
              
        unfolding identity\<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   ambda_predicates_2_2alxiom_universalom_instance
        
      moreover have "[\<^bold>\<exists>  assumes "
        apply (rule cqt_5_mod[where \<psi>="\<lambda> x . \<lparr>O!,x\<rparr>",axiom_instance,deduction])
        apply (rule SimpleExOrEnc.intros)

       avl_dist2
        apply (rule cqt_5_mod[where \<psi>="\<lambda> y . \<lparr>O!,y\<rparr>",axiom_instance,deduction])
         apply (rule SimpleExOrEnc.intros)
        using 1[conj1,conj2] by auto
   vex\^bold>\^subE y) \<^bold>\<equiv> (\<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr>
                        \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>)) inusingBasic_9lyjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
      using_<>[axiom_instancection eductionbymeson
      thus "[(x \<^bold>=\<^sub>E)"ng <bold\E"(2) by blast
qed
  lemma eq_E_simple_2[PLM]:
    
    unfolding identity_defs by PLM_solver  "<lbrakk>avlMKT  n   ;is_ord n  )r <noteq> ET\<rbrakk> <Longrightarrow>
lemma3M
    ( <bold> )\^bold><equiv> ((<>O!\rparr <bold&\>O!y<rparr <^
               \<^bold>\<or> (\<lparr>A!,\rparr \<^old&<>A!y<rparr>\^old> <^old\box>(\<^bold>\<forall>F. \<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,F\<rbrace>))) in v]"
    using eq_E_simple_1
    apply - unfolding identity_defs
    by PLM_solver

  lemma id_eq_obj_1[PLM]: "[(x\<^sup>P) \<^bold>= (x\<^sup>P) in v]"
    f-
      have "[(\<^bold>\<diamondassume "\^bold>\<box(^><diamond>\<phi> \<^bold>\<rightarrow> \<psi>)  ultimately show ?case
        using PLM.oth_class_taut_2 by simp
      hence "[(\<^        }
        using CP "\<^bold>\<or>E"(1) by blast
      moreover {
        assume "[(\<^bold>\<diamond>\<lparr>E!, x\<^sup>P\<rparr>) in v]"
        hence "[\<lparr       "[\<^(\<^bold>\<diamond>\<phi\^bold\<rightarrow> \<phi>) in v]"
          apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl, rotated]
          by show_proper
        hence "[\<lparr>\<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>         using ded_thm_cor_3 by blast
                \<^bold>& \<^bold>\<box>(\<^bold>\<forall>F. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,x\<^sup>P\<rparr>) in v]"
          apply - by PLM_solver
        hence "[(x\<^sup>P) \<^bold>=\<^sub>E (x\<^sup>P) in     [^bold>\<diamond><>xFrbrace \<^bold>\<equiv> \<^bold>\<box>\<lbrace>x,F\rbracein]
          using q_E_simple_1rlunfoldingdinary_def ast
      }
      moreover {
        assume "[(\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!, x\<^sup>P\<rparr>) in v]    [<^bold>\box(<lbrace>x,F\<rbrace> \<^bold>\<equiv><>yGrbrace) \<^bold>\<equiv> (\<^bold>\<box>\lbracexFrbrace <^bold>\<equiv> \<^bold>\<box>\<lbrace>y,G\<rbrace>) in v]"
       \><^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr> in v]"
          apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl, rotated]                    , THEN qml_act_2axiom_instance, equiv_rljava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
          by show_proper
        hence "[\<lparr>\<^bold>    assumes "IsProperInX <hijava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
                & \<^bold>\<box>(\<^bold>\<forall>F. \<lbrace>x\<^sup>P,\<brace<bold\<equiv> \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
          apply - by PLM_solver
      }
      ultimately show ?thesis unfolding identity_defs Ordinary_def Abstract_def
        using "\<^bold>\<or>I" by blast
    qed
  lemma id_eq_obj_2[PLM]:
    "[((x\^sup>)\^> (\sup>)\^><rightarrow (y<(
    by (meson l_identity[axiom_instanceusing assmsbeta_equiv_eq_1_3byauto
  lemma id_eq_obj_3[PLM]:
    "[((x\<^sup>P) \<^bold>= (y\<^sup>P)) \<^bold>& ((y\<^sup>P) \<^bold>= (z\<^sup>P)) \<^bold>\<rightarrow> ((x\<^sup>P) \<^bold>= (z\<^sup>P)) in v]"
    by (metis l_identity[axiom_instance] ded_thm_cor_4 CP "\<^bold>&E")
end

text\<open>
\begin{remark}
  To unify   oftheproperties ofequalityatypeclassisintroduced.
\end{remark}
\<close>

class id_eq = quantifiable_and_identifiable +
assumes: [(x : ')\bold=x  v]"
  assumes id_eq_2: "[((x :: 'a) \<^bold>= y) \<^bold>\<rightarrow> (y \<^bold>= x) in v]"
  assumes id_eq_3: "[((x :: 'a) \<^bold>= y) \<^bold>& (y \<^bold>= z) \<^bold>\<rightarrow> (x \<^bold>= z) in v]"

instantiation \> : id_eq
begin
  instance proof
    fix x :: \<nu> and v
    show "[x \<^bold>= x in v]"
using.d_eq_obj_1
      by (simp add: identity_\<nu>_def)
  next
    fix x y::\<nu> and v
    show "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in v]"
      using PLM.id_eq_obj_2
      by (simp add: identity_\<nu>_def)
  next
    fix x y z::\<nu> and v
    show "[((x \<^bold>= y) \<^bold>& (y \<^bold>= z)) \<^bold>\<rightarrow> x \<^hence [<bold\>< (\beta\<sup)\<bold= \>in]
usingPLM.id_eq_obj_3
      by (simp add: identity_\<nu>_def)
  qed
end

instantiation \<o> :: id_eq
begin
  instanceproof
    fix x :: \<o> and v
    show "[x \<^bold>= x in v]"
      using PLM.id_eq_prop_prop_7 .
  next
    fix x y :: \<o> and v
    show "[x \<^bold>= y \<^><rightarrowy\^bold> x  ]
      using PLM.id_eq_prop_prop_8 .
  next
    fix x y z :: \<o> and vdefs
    show "[((x \<^bold>= y) \<^bold>& (y \<^bold>= z)) \<^bold>\<rightarrow> x \<^bold>= z in v]"
      using PLM.id_eq_prop_prop_9 .
  qed
end

instantiation \<Pi>\<^sub>1 :: id_eq
begin
  instance proof
    fix x :: \<Pi>\<^sub>1 and v
    show "[x \<^bold>= x in v]"
      usingPLMid_eq_prop_prop_1.
  next
    fix x y :: \<Pi>\<^sub>1 and v
    show "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in v]"
      using PLM.id_eq_prop_prop_2 .
  next
    fix x y z :: \<Pi>\<^sub>1 and v
    show "[((x \<^bold>= y) \<^bold>& (y \<^bold>= z)) \<^bold>\<rightarrow> x \<^bold>= z in v]"
      using PLM.id_eq_prop_prop_3 .
  qed
end

instantiation unfoldingdiamond_def
begin
  instance proof
    fix x :: \<Pi>\<^sub>2 and v
    show "[x \<^bold>= x in v]"
      using PLM.id_eq_prop_prop_4_a .
  next
    fix x y :: \<Pi>\<^sub>2 and v
    show "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in v]"
      using PLM.id_eq_prop_prop_5_a .
  next
    fix x y z :: \<Pi>\<^sub>2 and v
    show "[((x \<^bold>= y) \<^bold>& (y \<^bold>= z)) \<^bold>\<rightarrow> x \<^bold>= z in v]"
      using PLM.id_eq_prop_prop_6_a .
  qed
end

instantiation \<Pi>\<^sub>3 :: id_eq
begin
  instance proof
    fix x :: \<Pi>\<^sub>3 and v
    show "[x \<^bold>= x in v]"
      using PLM.id_eq_prop_prop_4_b .
  next
    fix x y :: \<Pi>\<^sub>3 and v
    show "[x \<^bold>= y \<^bold>\<rightarrow> y \<^bold>= x in]
      using PLM.id_eq_prop_prop_5_b .
  next
    fix x y z :: \<Pi>\<^sub>3 and v
    show "(( \<bold> y) \<^bold>& (y \<^bold>= z)) \<^bold>\<rightarrow> x \<^bold>= z in v]"
      using PLM.id_eq_prop_prop_6_b .
  qed
end

context PLM
begin
  lemma id_eq_1[PLM]:
"(x::'a::id_eq) \<^bold>= x in v]"
    using id_eq_1 .
  lemma id_eq_2[PLM]:
    "[((x::'a::id_eq) \<^bold>= y) \<^bold>\<rightarrow> (y \<^bold>= x) in v]"
    using id_eq_2 .
  lemma id_eq_3[PLM]:
    "[((x::'a::id_eq) \<^bold>= y) \<^bold>& (y \<^bold>= z) \<^bold>\<rightarrow> (x \<^bold>= z) in v]"
    using id_eq_3 .

  attribute_setup eq_sym = \<open>
    Scan.succeed (Thm.rule_attribute [] 
      (fn   nhm=hm@{hmid_eq_2deduction))
\<close>


  lemma all_self_eq_1[PLM]:
    "[\<^bold>\<box>(\<^bold>\<forall> \<alpha> :: 'a::id_eq . \<alphaoth_class_taut_4_ath_class_taut_1_b
    by PLM_solver
  lemma all_self_eq_2[PLM]:
    "[\<^bold>\<forall>\<alpha> :: 'a::id_eq . \<^bold>\<box>(\<alpha> \<bold>=\<lpha>) in v]"
    by PLM_solver

  lemma t_id_t_proper_1[PLM]:
    "[<tau <bold>=\tau>'\^lddrightarrow> (\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold= \tau) nv"
    proof (rule CP)
      assume "[\<tau> \<^bold>= \<tau>' in v]"
      moreover {
        assume "[\<tau> \<^bold>=\<^sub>E \<tau>' in v]"
        hence "[\<^bold>\<exists> <beta> . (\beta\^sup>P)\^old>= \<tau> in v]"
          apply -
          apply (rule cqt_5_mod[where \<psi>="\<lambda> \<tau> . \<tau> \<^bold>=\<^sub>E \<tau>'", axiom_instance, deduction])
           subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
          by simp
      }
      moreover {
        assume "[\<lparr>A!,\<tau>\<rparr> \<^bold>& \<lparr>A!,\<tau>'\<rparr> \<^bold>& \<^bold>\<box>(\<^bold><forallF lbrace\<au>F\<rbrace> \<^bold>\<equiv> \<lbrace>\<tau>',F\<rbrace>) in v]"
hencebold\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \<tau> in v]"
          apply -
          apply (rule         apply (PLM_subst_method  "\<lambda> x . \<^bold>\<not>\<lparr>F<^up>-x<sup>P\<rparr>" \lambda  \>F,\^up><arr"
           subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
          by PLM_solver
      }
      ultimately show "[\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \<tau> in v]" unfolding identity\<^sub>\<kappa>_def
        using intro_elim_4_b reductio_aa_1 by blast
    qed

  lemma t_id_t_proper_2[PLM]: "[\<tau> \<^bold>= \<tau>' \<^bold>\<rightarrow> (\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \<tau>') in v]"
  proof (rule CP)
    assume "[\<tau> \<^bold>= \<tau>' in v]"
    moreover {
      assume "[\<tau> \<^bold>=\<^sub>E \<tau>' in v]"
      hence "[\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \<tau>' in v]"
         -
        apply (rule cqt_5_mod[where \<psi>="\<lambda> \<tau>' . \<tau> \<^bold>=\<^sub>E \<tau>'", axiom_instance, deduction])
         subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
        by simp
    }
    moreover {
      assume "[\<lparr>A!,\<tau>rparr\^bold&\>A!,\tau>\>\^old& <bold><box>(\<^bold>\<forall>F. \<lbrace>\<tau>,F\<rbrace> \<^bold>\<equiv> \<lbrace>\<tau>',F\<rbrace>) in v]"
      hence "[\<^bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \<tau>' in v]"
        apply -
        apply (rule cqt_5_mod[where \<psi>="\<lambda> \<tau> . \<lparr>A!,\<tau>\<rparr>", axiom_instance, deduction])
         subgoal unfolding identity_defs by (rule SimpleExOrEnc.intros)
        by LM_solver
    }
    ultimately show "\<bold>\<exists> \<beta> . (\<beta>\<^sup>P) \<^bold>= \tau> in v]" unfolding identity\<sub>\<kappa>_def
      using intro_elim_4_b reductio_aa_1 by blast
  qed

  lemma id_nec[PLM]: "[((\<alpha>::'a::id_eq) \<^bold>= (\<beta>)) \<^bold      also have ".. =<bold><exists>x  <bold>diamond(<bold\not\lparrFx<supP<parr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> in v]" 
    apply (rule "\<^bold>\<equiv>I")
     using l_identitywhere \> ="\<lambda><beta>.<bold<box>((\<alpha>) \<^bold>= (\<beta>)))", axiom_instance]
            RN ded_thm_cor_4 unfolding identity_\<nu_def
     apply blast
    using qml_2[axiom_instance] by blast

  lemma id_nec_desc[PLM]:
    "[((\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<^bold>\<iota>x. \<psi> x)) \<^bold>\<equiv> \<^bold>\<box>((\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<^bold>\<iota>x. \<psi> x)) in v]"
roof"\<bold\<exists>\<alpha>. (<><supP)\<bold> \<bold\iota>x  . <>x)inv \and> (\bold\<exists \beta. (<>\^sup>P) \<^bold>=(<bold>\<iota>x . \<psi> x)) in v]"
      assume "[(\<^bold>\<exists> \<alpha>. (\<alpha>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<phi> x)) in v] \<and> [(\<^bold>\<exists> \<beta>. (\<beta>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<psi> x)) in v]"
      then obtain \<alpha> and \<beta> where
        "[(\<alpha>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<phi> x) in v] \<and> [(\<beta>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<psi> x) in v]"
        apply - unfolding conn_defs by PLM_solver
      moreover {
        moreover have "[(\<alpha>) \<^bold>= (\<beta>) \<^bold>\<equiv> \<^bold>\<box>((\<alpha>) \<^bold>= (\<beta>)) in v]" by PLM_solver
        ultimately have "[((\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<beta>\<^sup>P) \<^bold>\<equiv> \<^bold>\<box>((\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<beta>\<^sup>P))) in v]"
          using l_identity[where \<phi>="\<lambda> \<alpha> . (\<alpha>) \<^bold>= (\<beta>\<^sup>P) \<^bold>\<equiv> \<^bold>\<box>((\<alpha>) \<^bold>= (\<beta>\<^sup>P))", axiom_instance]
          modus_ponens unfolding identity_\<nu>_def by metis
      }
      ultimately show ?thesis
        using l_identity[where \<phi>="\<lambda> \<alpha> . (\<^bold>\<iota>x . \<phi> x) \<^bold>= (\<alpha>)
\^><equiv <><>(<^old\iota>x.\phi )\^> \alpha)" ]
        modus_ponens by metis
    next
      assume "<ot((<bold><>\alpha>.(alpha\^>P \> \^><iotax  \> )  ]\and[\^bold>\<exists> \beta (\<beta\<sup>)\<bold=(<bold\iotax.\> x) in ])
      hence "\<not>[\<lparr>A!,(\<^bold>\<iota>x . \<phi> x)\<rparr> in v] \<and> \<not>[(\<^bold>\<iota>x . \<phi> x) \<^bold>=\<^sub>E (\<^bold>\<iota>x . \<psi> x) in v]
           \<or> \<not>[\<lparr>A!,(\<^bold>\<iota>x . \<psi> x)\<rparr> in v] \<and> \<not>[(\<^bold>\<iota>x . \<phi> x) \<^bold>=\<^sub>E (\<^bold>\<iota>x . \<psi> x) in v]"
      unfolding identity\<^sub>E_infix_def
      using cqt_5[axiom_instance] PLM.contraposition_1 SimpleExOrEnc.intros
            vdash_properties_10 by meson
      hence "\<not>[(\<^bold>\<iota>x . \<phi> x) \<^bold>= (\<^bold>\<iota>x . \<psi> x) in v]"
        apply - unfolding identity_defs by PLM_solver
      thus ?thesis apply - apply PLM_solver
        using qml_2[axiom_instance, deduction] by auto
    qed

subsection\<open>Quantification\<close>
text\<open>\label{TAO_PLM_Quantification}\<close>

  lemma rule_ui[PLM,PLM_elim,PLM_dest]:
    "[\<^bold>\<forall>\<alpha> . \<phi> \<alpha> in v] \<Longrightarrow> [\<phi> \<beta> in v]"
    by (meson cqt_1[axiom_instance, deduction])
  lemmas "\<^bold>\<forall>E" = rule_ui

  lemma rule_ui_2[PLM,PLM_elim,PLM_dest]:
    "\<lbrakk>[\<^bold>\<forall>\<alpha> . \<phi> (\<alpha>\<^sup>P) in v]; [\<^bold>\<exists> \<alpha> . (\<alpha>)\<^sup>P \<^bold>= \<beta> in v]\<rbrakk> \<Longrightarrow> [\<phi> \<beta> in v]"
    using cqt_1_\<kappa>[axiom_instance, deduction, deduction] by blast

  lemma cqt_orig_1[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<beta> in v]"
    by PLM_solver
  lemma cqt_orig_2[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \      assume "[onContingent F in v]java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 

  lemma universal[PLM]:
    "(\<And>\<alpha.[<> <> n]<Longrightarrow[<bold\<forall> \<alpha> . \<phi> \<alpha> in v]"
    using rule_gen .
  lemmas "\<^bold>\<forall>I" = universal

  lemma cqt_basic_1[PLM]:
    "[(\<^bold>\<forall>\<alpha>. (\<^bold>\<forall>\<beta> . \<phi> \<alpha> \<beta>)) \<^bold>\<equiv> (\<^bold>\<forall>\<beta>. (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<beta>)) in v]"
    by PLM_solver
  lemma cqt_basic_2[PLM]:
"(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>\<equiv> ((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<phi> \<alpha>)) in v]"
     PLM_solver
  lemma cqt_basic_3[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>\<rightarrow> ((\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<psi> \<alpha[ContingentlyTrue<bold>\<rightarrow> Contingent p in v]"
    by PLM_solver
  lemma cqt_basic_4[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<equiv> ((\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>)) in v]"
    by PLM_solver
  lemma cqt_basic_6[PLM]:
    "[(\<^bold>\<forall>\<alpha>. (\<^bold>\<forall>\<alpha>. \<phi> \<alpha>)) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
    by PLM_solver
  lemma cqt_basic_7[PLM]:
    "[(\<phi> \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha> . \<psi> \<alpha>)) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>.(\<phi> \<^bold>\<rightarrow> \<psi> \<alpha>)) in v]"
    by PLM_solver
  lemma cqt_basic_8[PLM]:
    "[((\<^bold>\<forall>\<alpha>. \<phi> \<>)\bold\<or> (\<^bold>\<forall>\<alpha>. \<psi> \<alpha) \><rightarrow> (\<^bold>\<forall>\<alpha>. (\<phi> \<alpha> \^bold>\<or> \psi <>) v]"
    by PLM_solver
  lemma cqt_basic_9[PLM]:
    "[((\<^bold>\<forall>\<alpha>. \<phi> \[PLM]:
    by PLM_solver
  lemma cqt_basic_10[PLM]:
    "[((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<           "[\^sub>0 \<bold>& \<^bold\<iamond\<^bold>\<not>q\<^sub>0 in v]"
    by PLM_solver
  lemma cqt_basic_11[PLM]:
    "[(\<^bold
    by PLM_solver
  lemma cqt_basic_12[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<      }
    by PLM_solver

  [,LM_intro
    "[\<phi> \<    next
    unfolding exists_def by PLM_solver
  lemmas "\<^bold>\<exists>I" = existential
lemmaPLM_elimt:
    "\<lbrakk>[\<^bold>\<exists>\<alpha> . \<phi> \<alpha> in v]; (\<And>\<alpha>.[\<phi> \<alpha> in v] \<Longrightarrow> [\<psi> in v])\<rbrakk> \<Longrightarrow> [\<psi> in v]"
    unfolding exists_def by PLM_solver

  lemma Instantiate:
    assumes "[\<^bold>
    obtains x where "[\<phi> x in v]"
    show "[ContingentlyTrue q\<^sub>0 \<^bold>\<or> ContingentlyFalse q\<^sub>0 in v]"
  mas<^bold>\<xistsE tantiate

  lemma cqt_further_1[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) in v]"
    by PLM_solver
  lemma cqt_further_2[PLM]:
    "[(\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>)) \<^bold>\<equiv> (\<^bold>\<exists>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 38
  lemma cqt_further_3[PLM]:
    "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
    unfolding exists_def by PLM_solver
  lemma cqt_further_4[PLM]:
    "[(\<^bold>\<not>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>)) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
    unfolding exists_def by PLM_solver
  lemma cqt_further_5[PLM]:
    "[(\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<end
      unfolding exists_def by PLM_solver
  lemma cqt_further_6[PLM]:
    "[(\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>\<or> \<psi> \<alpha>) \<^bold>\<equiv> \<psi> <alpha>) \<^bold>\<equiv> ((\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>\<or> (\<^bold>\exists>\alpha \>\<alpha>)) in v]"
    unfolding exists_defyPLM_solver
  lemma cqt_further_10[PLM]:
<(\<alpha>::'a::id_eq)\bold> <boldd\orall \<beta> . \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>)) \<^bold>\<equiv> (\<^bold>\<forall> \<beta> . \<phi> \<beta> \<^bold>\<equiv> \<beta> \<^bold>= \<alpha>) in v]"
    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]:
    "[((\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>)) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>.           assume"[L \<> E! in v]"
    by PLM_solver
  lemma cqt_further_12[PLM]:
    "[((\<^bold>\<not>(\<^bold>\<exists>\<alpha>.    "\lparr>!<supP<parrbold<equiv> \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr> in v]"
unfoldingefy M_solver
  lemma cqt_further_13[PLM]:
    "[((\<^bold>\<exists>\alpha. \phi><alpha>>) \^old&(<bold\<ot>(\<^bold>\<exists>\<alpha>. \<psi> \<alpha>))) \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>)) in v]"
    unfolding exists_def by PLM_solver
  lemma cqt_further_14[PLM]:
    "[(\<^bold>\<exists>\<alpha>. \<^bold>\<exists>\<beta>. \<phi> \<alpha> \<beta>) \<^bold>\<equiv> (\<^bold>\<exists>\<beta>. \<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<beta>) in v]"
    unfolding exists_def by PLM_solver

  lemma nec_exist_unique[PLM]:
    "[(\<^bold>\<forall> x. \<phi> x \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> x)) \<^bold>\<rightarrow> ((\<^bold>\<exists>!x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<exists>!x. \<^bold>\<box>(\<phi> x))) in v]"
    proof (rule CP)
      assume a: [\<bold\<orallx \<phi>x \<bold\<rightarrow> \^bold>\box\phi x in v]"
      show "[(\<^bold>\<exists>!x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<exists>!x. \<^bold>\<box>\<phi> x) in v]"
      proof (rule CP)
        assume "[(\<^bold>\<exists>!x. \<phi> x) in v]"
        hence "[\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
          by (simp only: exists_unique_def)
        then obtain \<alpha> where 1:
          "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
          by (rule "\<^bold>\<exists>E")
        {
          fix \<beta>
          have "[\<^bold>\<box>\<phi <eta \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha> in v]"
            by (metis "1" Semantics.T5 Semantics.T6 cqt_orig_1 oth_class_taut_9_b)
        }
        hence "[\<^bold>\<forall>\<beta>. \<^bold>\<box>\<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha> in v]" by (rule "\<^bold>\<forall>I")
        moreover have "[\<^bold>\<box>(\<phi> \<alpha>) in v]"
          using 1 "\<^bold>&E"(1) a vdash_properties_10 cqt_orig_1[deduction]
          by fast
        ultimately have "[\<^bold>\<exists>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<beta>. \<^bold>\<box>\<phi> \<beta> \<^}
          using "\<^bold>&I" "\<^bold>\<exists>I" by fast
        thus "[(\<^bold>\<exists>!x. \<^bold>\<box>\<phi> x) in v]"
          unfolding exists_unique_def by assumption
      qed
    qed


subsection\<open>Actuality and Descriptions\<close>
text\<open>\label{TAO_PLM_ActualityAndDescriptions}\<close>

  lemma nec_imp_act[PLM]: "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<phi> in v]"
    apply (rule CP)
    using qml_act_2[axiom_instance, equiv_lr]
          qml_2[axiom_actualization, axiom_instance]
          logic_actual_nec_2[axiom_instance, equiv_lr, deduction]
    by blast
  lemma act_conj_act_1[PLM]:
    "[\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<phi>) in v]"
    using equiv_def logic_actual_nec_2[axiom_instance]
          logic_actual_nec_4[axiom_instance] "\<^bold>&E"(2) "\<^bold>\<equiv>E"(2)
    by metis
  lemma act_conj_act_2[PLM]:
    "[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<phi>) in v]"
    using logic_actual_nec_2[axiom_instance] qml_act_1[axiom_instance]
          ded_thm_cor_3 "\<^bold>\<equiv>E"(2) nec_imp_act
    by blast
  lemma act_conj_act_3[PLM]:
    "[(\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi>) \<^bold>\<rightarrow> \<^bold>\<A>(\<phi> \<^bold>&n> \<psi>) in v]"
    unfolding conn_defs
    by (metis logic_actual_nec_2[axiom_instance]
              logic_actual_nec_1[axiom_instance]
              "\<^bold>\<equiv>E"(2) CP "\<^bold>\<equiv>E"(4) reductio_aa_2
              vdash_properties_10)
  lemma act_conj_act_4[PLM]:
    "[\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
    unfolding equiv_def
    by (PLM_solver PLM_intro: act_conj_act_3[where \<phi>="\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<phi>"
                                and \<psi>="\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<phi>", deduction])
  lemma closure_act_1a[PLM]:
    "[\<^bold>\<A>\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
    using logic_actual_nec_4[axiom_instance]
          act_conj_act_4 "\<^bold>\<equiv>E"(1)
    by blast
  lemma closure_act_1b[PLM]:
    "[\<^bold>\<A>\<^bold>\<A>\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
    using logic_actual_nec_4[axiom_instance]
          act_conj_act_4 "\<^bold>\<equiv>E"(1)
    by blast
  lemma closure_act_1c[PLM]:
    "[\<^bold>\<A>\<^bold>\<A>\<^bold>\<A>\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
    using logic_actual_nec_4[axiom_instance]
          act_conj_act_4 "\<^bold>\<equiv>E"(1)
    by blast
   closure_act_2[PLM]:
    "[\<^bold>\<forall>\<alpha>. \<^bold>\<A>(\<^bold>\<A>(\<phi> \<alpha>) \<^bold>\<equiv>\phi> <alpha> njava.lang.StringIndexOutOfBoundsException: Index 116 out of bounds for length 116
    by PLM_solver

 PLM
    "[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<^bold>\<A>(\<phi> \<alpha>) \<^bold>\<equiv> \<phi> \<alpha>) in v]"
    by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
  lemma closure_act_4[PLM]:
    "[\<^bold>\<A>(\<^bold>\<forall>\<alpha>\<^sub>1 \<alpha>\<^sub>2. \<^bold>\<A>(\<phi> \<alpha>\<^sub>1 \<alpha>\<^sub>2) \<^bold>\<equiv> \<phi> \<alpha>\<^sub>1 \<alpha>\<^sub>2) in v]"
    by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
lemma[LM]:
    "[\<^bold>\<A>(\<^bold>\<forall>\<alpha>\<^sub>1 \<alpha>\<^sub>2 \<alpha>\<^sub>3.         apply (PLM_subst_method"""\\^><>"
    by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])
  lemma closure_act_4_c[PLM]:
    "[\<^bold>\<A>(\<^bold>\<forall>\<alpha>\<^sub>1 \<alpha>\<^sub>2 \<alpha>\<^sub>3 \<alpha>\<^sub>4. \<^bold>\<A>(\<phi> \<alpha>\<^sub>1 \<alpha>\<^sub>2 \<alpha>\<^sub>3 \<alpha>\<^sub>4) \<^bold>\<equiv> \<phi> \<alpha>\        using KBasic2_2[]"\bold& <bold&  
    by (PLM_solver PLM_intro: logic_actual_nec_3[axiom_instance, equiv_rl])

  lemma RA[PLM,PLM_intro]:
    "([\<phi> in dw]) \<Longrightarrow> [\<^bold>\<A>\<phi> in dw]"
    using logic_actual[necessitation_averse_axiom_instance, equiv_rl] .

  lemma RA_2[PLM,PLM_intro]:
    "([\<psi> in dw] \<Longrightarrow> [\<phi> in dw]) \<Longrightarrow> ([\<^bold>\<A>\<psi> in dw] \<Longrightarrow> [\<^bold>\<A>\<phi> in dw])"
    

  context
  begin
    private lemma ActualE[PLM,PLM_elim,PLM_dest]:
      "[\<^bold>\<A>\<phi> in dw] \<Longrightarrow> [\<phi> in dw]"
      using logic_actual[necessitation_averse_axiom_instance, equiv_lr] .
    
    private lemma NotActualD[PLM_dest]:
      "\<not>[\<^bold>\<A>\<phi> in dw] \<Longrightarrow> \<not>[\<phi> in dw]"
      using RA by metis
    
    private lemma ActualImplI[PLM_intro[NonContingentp<^sub>0) in v]"
      "[\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v] \<Longrightarrow> [\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
      using logic_actual_nec_2[axiom_instance, equiv_rl] .
    private lemma ActualImplE[PLM_dest, PLM_elim]:
      "[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v] \<Longrightarrow> [\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v]"
      using       "\^bold\exists>p:\o) .  \^\noteq q \^old&NonContingentp\^>   inv]java.lang.StringIndexOutOfBoundsException: Index 119 out of bounds for length 119
    private lemma NotActualImplD[PLM_dest]:
      "\<not>[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v] \<Longrightarrow> \<not>[\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v]"
      using ActualImplI by blast
    
    private lemma ActualNotI[PLM_intro]:
      "[\<^bold>\<not>\<^bold>\<A>\<phi> in v] \<Longrightarrow> [\<^bold>\<A>\<^bold>\<not>\<phi> in v]"
      using logic_actual_nec_1[axiom_instance, equiv_rl] .
    lemma ActualNotE[PLM_elim,PLM_dest]:
      "[\<^bold>\<A>\<^
      using logic_actual_nec_1[axiom_instance, equiv_lr] .
    lemma NotActualNotD[PLM_dest]:
      "\<not>[\<^bold>\<A>\<^bold>\<not>\<phi> in v] \<Longrightarrow> \<not>[\<^bold>\<not>\<^bold>\<A>\<phi> in v]"
      using ActualNotI by blast
    
    private  lemma ActualConjI[PLM_intro]:
      "[\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> in v] \<Longrightarrow> [\<^bold>\<A>(\<phi> \<^bold>&an> \<psi>) in v]"
      unfolding equiv_def
      by (PLM_solver PLM_intro: act_conj_act_3[deduction])
    private lemma ActualConjE[PLM_elim,PLM_dest]:
      "[\<^bold>\<A>(\<phi> \<^bold>& \<psi>) in v] \<Longrightarrow> [\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi> in v]"
      unfolding conj_def by PLM_solver
    
    private lemma ActualEquivI[PLM_intro]:
      "[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi> in v] \<Longrightarrow> [\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
      unfolding equiv_def
      by (PLM_solver PLM_intro: act_conj_act_3[deduction])
ualEquivE,PLM_dest]
      "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) in v] \<Longrightarrow> [\<^bold (rule CP)
      unfolding equiv_def by PLM_solver

    private lemma ActualBoxI[PLM_intro]:
      "[\<^bold>\<box>\<phi> in v] \<Longrightarrow> [\<^bold>\<A>(\<^bold>\<box>\<phi>) in v]"
      using qml_act_2[axiom_instance, equiv_lr] .
    private lemma ActualBoxE[PLM_elim, PLM_dest]:
      "[\<^bold>\<A>(\<^bold>\<box>\<phi>) in v] \<Longrightarrow> [\<^bold>\<box>\<phi> in v]"
      using qml_act_2[axiom_instance equiv_rl .
    private lemma NotActualBoxD[PLM_dest]:
"not\<bold\A>\^><box><>  ]\<Longrightarrow> \<>\^bold><box>\<> in v]              blast
      using ActualBoxI by blast

    private lemma ActualDisjI[PLM_intro]:
[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v] \<Longrightarrow> [\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) in v]"
      unfolding disj_def by PLM_solver
    privatemma ctualDisjE_,LM_dest
      "[\<^bold>\<A>(<hi\<>\<or> \<psi>) in v] \<Longrightarrow> [\<^bold>\< \<^bold>\<or> \old<>\<psi> in v]"
      unfolding disj_def by PLM_solver
    private lemma NotActualDisjD[PLM_dest]:
      "\<not>[\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) in v] \<Longrightarrow> \<not>[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]"
      using ActualDisjI by blast

    private lemma ActualForallI[PLM_intro]:
      "[\<^bold>\<forall> x . \<^bold>\<A>(\<phi> x) in v] \<Longrightarrow> [\<^bold>\<A>(\<^bold>\<forall> x . \<phi> x) in v]"
      using logic_actual_nec_3[axiom_instance, equiv_rl] .
    lemma ActualForallE[PLM_elim,PLM_dest]:
      "[\<^bold>\<A>(\<^bold>\<forall> x . \<phi> x) in v] \<Longrightarrow> [      finallyshow hesis
      using logic_actual_nec_3[axiom_instance, equiv_lr] .
    orallDstjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
      "\<not>[\<^bold>\<A>(\<^bold>\<forall> x . \<phi> x) in v] \<Longrightarrow> \<not>[\<^bold>\<forall> x . \<^bold>\<A>(\<phi> x) in v]"
      using ActualForallI yblast

    lemma ActualActualI[PLM_intro]:
      "[\<^bold>\<A>\<phi> in v] \<Longrightarrow> [\<^bold>\<A>\<^bold>\<A>\<phi> in v]"
      using logic_actual_nec_4[axiom_instance, equiv_lr] .
    lemma ActualActualE[PLM_elim,PLM_dest]:
      
      using logic_actual_nec_4[axiom_instance, equiv_rl] .
    lemma NotActualActualD[PLM_dest]:
      "\<not
      using ActualActualI by blast
  end

  lemma ANeg_1[PLM]:
    "[\<^bold>\<not>\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<phi> in dw       hence "("\<^old>exists q.NonContingent q\^bold>  \bold> in]
    by PLM_solver
              using thm_noncont_propos_4 "\<bold&I <\exists[where<>"\^sub\^>"] by auto
    "[\<^bold>\<not>\<^bold>\A\^>\<not>\<phi> \<^bold>\<equiv> \<phi> in dw]"
    by PLM_solver
  lemma Act_Basic_1[PLM]:
    "[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<^bold>\<not>\<phi> in v]"
    by PLM_solver
  lemma Act_Basic_2[PLM]:
    "[\<^bold>\<A>(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> (\<^bold>\<A>\<phi> \<^bold>& \<^bold>\<A>\<psi>) in v]"
    by PLM_solver
  lemma Act_Basic_3[PLM]:
    "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> ((\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>)) \<^bold>& (\<^bold>\<A>(\<psi> \<^bold>\<rightarrow> \<phi>))) in v]"
    by PLM_solver
  lemma Act_Basic_4[PLM]:
    "[(\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>& \<^bold>\<A>(\<psi> \<^bold>\<rightarrow> \<phi>)) \<^bold>\<equiv> (\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi>) in v]"
    by PLM_solver
  lemma Act_Basic_5[PLM]:
    "[\<^bold>\<A>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> (\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<A>\<psi>) in v]"
    by PLM_solver
  lemma Act_Basic_6[PLM]:
    "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<diamond>\<phi>) in v]"
    unfolding diamond_def by PLM_solver
  lemma Act_Basic_7[PLM]:
    lemma cont_true_cont_1[PLM]:
    by (simp add: qml_2[axiom_instance] qml_act_1[axiom_instance] "\<^bold>\<equiv>I")
  lemma Act_Basic_8[PLM]:
    "[\<^bold>\<A>(\<^bold>\<box>\<phi>) \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<A>\<phi> in v]"
    by (metis qml_act_2[axiom_instance] CP Act_Basic_7 "\<^bold>\<equiv>E"(1)
              "\<^bold>\<equiv>E"(2) nec_imp_act vdash_properties_10)
  lemma Act_Basic_9[PLM]:
    "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^boldunfolding
    using qml_act_1[axiom_instance] ded_thm_cor_3 nec_imp_act by blast
  lemma Act_Basic_10[PLM]:
    "[\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> \<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]"
    by PLM_solver

  lemma Act_Basic_11[PLM]:
    "[\<^bold>\<A>(\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<exists>\<alpha>.\<^bold>\<A>(\<phi> \<alpha>)) in v]"
    proof -
      have "[\<^bold>\<A>(\<^bold>\<forall> \<alpha> . \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall> \<alpha> . \<^bold>\<A>\<^bold>\<not>\<phi> \<alpha>) in v]"
        using logic_actual_nec_3[axiom_instance] by blast
      hence "[\<^bold>\<not>\<^bold>\<A>(\<^bold>\<forall> \<alpha> . \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<forall> \<    "[(x\\<sup> <bold> \^>\iota> <>) ^\ \bold\forall   <       usingthm_relation_negation_3simp
        using oth_class_taut_5_d[equiv_lr] by blast
      moreover have "[\<^bold>\<A>\<^bold>\<not>(\<^bold>\<forall> \<alpha> . \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<equiv> \<^by 
tual_nec_1
      ultimately have "[\<^bold>\<A>\<^bold>\<not>(\<^bold>\<forall> \<alpha> . \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<forall> \<alpha> . \<^bold>\<A>\<^bold>\<not>\<phi> \<alpha>) in v]"
        using "\<^bold>\<equiv>E"(5) by blast
      moreover {
        have "[\<^bold><orall \<alpha> . \<^bold>\<A>\<^bold>\<not>\<phi> \<alpha> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<A>\<phi> \<alpha> in v]"
          using logic_actual_nec_1[axiom_universal, axiom_instance] by blast 
        hence "[(\<^bold>\<forall> \<alpha> . \<^bold>\<A>\<^bold>\<not>\<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall> \<alpha> . \<^bold>\<not>\<^bold>\<A>\<phi> \<alpha>) in v]"
          using cqt_basic_3[deduction] by fast
         [\^><>\^><^><ot<<) <^bold\<equiv><bold\<>\^><>\<>.\^bold\not\^bold>\A>\<phi \<alpha
using[equiv_lr by blast
      }
      ultimately show ?thesis
        by (metis "\<^bold>\<exists>E" MetaSolver.EquivI Semantics.T7 existential)
    qed

  t_quant_uniq
    "[(\<^bold>\<forall> z . \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x) \<^bold>\<equiv> (\<^bold>\<forall> z . \<phi> z \<^bold>\<equiv> z \<^bold>= x) in dw]"
    by PLM_solver

  lemma fund_cont_desc[PLM]:
    "[(x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<forall> z . \<phi> z \<^bold>\<equiv> (z \<^bold>= x)) in dw]"
    using descriptions[axiom_instance] act_quant_uniq "\<^bold>\<equiv>E"(5) by fast

  lemma hintikka[PLM]:
    "[(x           cont_true_cont_4]by simp
    proof -
      have       assumeenc_equiv<\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P, F\<rbrace> in v]"
unfolding>def apply PLM_solver using id_eq_obj_1 apply imp
        hence><box>(\<^bold>\<forall> F. lbracex\<^sup>P, F\<rbrace> \<^< \<lbrace>y\<show[ q^>\^bold\> ContingentlyFalse^>  java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90
                          deduction, deduction]
        using id_eq_obj_2[deduction] unfolding identity_\<nu>_def by fastforce
      thus ?thesis using "\<^bold>\<equiv>E"(5) fund_cont_desc by blast
    qed

  lemma russell_axiom_a[PLM]:
    "[(\<lparr>F, \<^bold>\<iota>x. \<phi> x\<rparr>) \<^bold>\<equiv> (\<^bold>\<exists> x . \<phi> x \<^bold>&> (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= x) \<^bold>& \<lparr>F, x\<^sup>P\<rparr>) in dw]"
    (is "[?lhs \<^bold>\<equiv> ?rhs in dw]")
    proof -
      {
        assume 1: "[?lhs in dw]"
        hence "[\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        using cqt_5[axiom_instance, deduction]
              SimpleExOrEnc.intros
        by blast
        then obtain \<alpha> where 2:
          "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
          using "\<^bold>\<exists>E" by auto          
        hence 3: "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>) in dw]"
          using hintikka[equiv_lr] by simp
        from 2 have "[(\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<alpha>\<^sup>P)  in dw]"
=alpha^>"and <eta=\^><>x <>x  \" x \bold=<\<sup>"
                axiom_instance, deduction, deduction]
id_eq_obj_1 =alpha]by 
        hence "[\<lparr>F, \<alpha>\<^sup>P\<rparr> in dw]"
        using 1 l_identity[where \<beta>="\<alpha>\<^sup>P" and \<alpha>="\<^bold>\<iota>x. \<phi> x" and \<phi>="\<lambda> x . \<lparr>F,x\<rparr>",
                           axiom_instance, deduction, deduction] by auto
        with 3 have "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \             l_identity[axiom_instance, deduction deduction]1conj1]
        hence "[?rhs in dw]" using "\<^bold>\<exists>I"[where \<alpha>=\<alpha>] by simp
      }
      moreover {
        assume "[?rhs in dw]"
        then obtain \<alpha> where 4:
          "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>) \<^bold>& \<lparr>F, \<alpha>\<^sup>P\<rparr> in dw]"
          using "\<^bold>\<exists>E" by auto
        hence "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x) in dw] \<and> [\<lparr>F, \<alpha>\<^sup>P\<rparr> in dw]"
          using hintikka[equiv_rl] "\<^bold>&E" by blast
        hence "[?lhs in dw]"
          using l_identity[axiom_instance, deduction, deduction]
          by blast
      }
      ultimately show ?thesis by PLM_solver
    qed

  lemma russell_axiom_g[PLM]:
    "[\<lbrace>\<^bold>\<iota>x. \<phi> x,F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists> x . \<phi> x \<^bold>&> (\<^bold>\<forall> z . \<phi> z \<^bold>apply  _v_sym
    (is "[?lhs \<^bold>\<equiv> ?rhs in dw]")
    proof -
      {
        assume 1: "[?lhs in dw]          using A_objectsaxiom_instance]bysimp
        hence "[\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        using cqt_5[axiom_instance, deduction] SimpleExOrEnc.intros by blast
        then obtain \<alpha> where 2: "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]" by (rule "\<^bold>\<exists>E")
        hence 3: "[(\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>)) in dw]"
          using hintikka[equiv_lr] by simp
        from 2 have "[(\<^bold>\<iota>x. \<phi> x) \<^bold>= \<alpha>\<^sup>P  in dw]"
          using l_identity[where \<alpha>="\<alpha>\<^sup>P" and \<beta>="\<^bold>\<iota>x. \<phi> x" and \<phi>="\<lambda> x . x \<^bold>= \<alpha>\<^sup>P",
                axiom_instance, deduction, deduction]
                id_eq_obj_1[where x=\<alpha>] by auto
        hence "[\<lbrace>\<pha<sup>, F\<rbrace> in dw]"
        using 1 l_identity[where \<beta>="\<alpha>\<^sup>P" and \<alpha>="\<^bold>\<iota>x. \<phi> x" and \<phi>="\<lambda> x . \<lbrace>x,F\<rbrace>",
                           axiom_instance, deduction, deduction] by auto
        with 3 have "[(\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>)) \<^bold>& \<lbrace>\<alpha>\<^sup>P, F\<rbrace> in dw]"
          using "\<^bold>&I" by auto
        hence "[?rhs in dw]" using "\<^bold>\<exists>I                    CPdash_properties_10bold\<forall>E" by metis
      }
      moreover {
        assume "[?rhs in dw]"
        then obtain \<alpha> where 4:
          "[\<phi> 
          using "\<^bold>\<exists>E" by auto
        hence "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x) in dw] \<and> [\<lbrace>\<alpha>\<^sup>P, F\<rbrace> in dw]"
          using hintikka[equiv_rl] "\<^bold>&E" by blast
        hence "[?lhs in dw]"
          "<bold<exists! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>(<bold><forall> y . \<lparr>G, y\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>F, y\<^sup>\<rparrarrn
      
      }
      ultimately show ?thesis by PLM_solver
    qed

  lemma russell_axiom[PLM]:
            using 1[conj1"<^bold>I "\^bold<exists>"byfast
    showspsi (\<^bold>\<iota>\ x) \<^bold>\<equiv><\<exists> x . \<phi> x \<^\<forall> z . \<phi> z \<^boldrightarrow z \<^bold>= x) \<^bold>\psi (x\<^sup>P)w
    (is "[?lhs \<^bold>\<equiv> ?rhs in dw]")
    proof -
      {
        assume 1: "[?lhs in dw]"
        hence "[\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        using cqt_5[axiom_instance, deduction] assms by blast
        then obtain \<alpha> where 2: "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]" by (rule "\<^bold>\<exists>E")
        hence 3: "[(\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>)) in dw]"
          using hintikka[equiv_lr] by simp
        from 2 have "[(\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<alpha>\<^sup>P)  in dw]"
          using l_identity[where \<alpha>="\<alpha>\<^sup>P" and \<beta>="\<^bold>\<iota>x. \<phi> x" and \<phi>="\<lambda> x . x \<^bold>= \<alpha>\<^sup>P",
                axiom_instance, deduction, deduction]
                id_eq_obj_1[where x=\<alpha>] by auto
        hence "[\<psi> (\<alpha>\<^sup>P) in dw]"
          using 1 l_identity[where \<beta>="\<alpha>\<^sup>P" and \<alpha>="\<^bold>\<iota>x. \<phi> x" and \<phi>="\<lambda> x . \<psi> x",
                             axiom_instance, deduction, deduction] by auto        ultimately show ?thesis
        with 3 have "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>) \<^bold>& \<psi> (\<alpha>\<^sup>P) in dw]"
          using "\<^bold>&I" by auto
        hence "?hsin ]  \^><>I"where\alpha=\alpha]by (simpadd identity_defsjava.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 109
      }
      moreover {
        assume "[?rhs in dw]"
        then obtain \<alpha> where 4:
          
          contingent_5PLM:
        hence "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x) in dw] \<and> [\<psi> (\<alpha>\<^sup>P) in dw]"
          using hintikka[equiv_rl] "\<^bold>&E" by blast
        hence "[?lhs in dw]"
          using l_identity[axiom_instance, deduction, deduction]
          by fast
  lemma A_Exists_2[PLM]:
      ultimately show ?thesis by PLM_solver
    qed

  lemma unique_exists[PLM]:
    "[(\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<exists>!x . \<phi> x) in dw]"     
    proof((rule "\<^bold>\<equiv>I", rule CP, rule_tac[2] CP))
      assume "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
      then obtain \<alpha> where
        "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        by (rule "\<^bold>\<exists>E")
      hence "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in dw]"
        using hintikka[equiv_lr] by auto
      thus "[\<^bold>\<exists>!x . \<phi> x in dw]"
        unfolding exists_unique_def using "\<^bold>\<exists>I" by fast
    next
      assume "[\<^bold>\<exists>!x . \<phi> x in dw]"
      then obtain \<alpha> where
        "[\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in dw]"
        unfolding exists_unique_def by (rule "\<^bold>\<exists>E")
      hence "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        using hintikka[equiv_rl] by auto
      thus "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in dw]"
        using "\<^bold>\<exists>I" by fast
    qed

  lemma y_in_1[PLM]:
    "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi>) \<^bold>\<rightarrow> \<phi> in dw]"
    using hintikka[equiv_lr, conj1] by (rule CP)

  lemma y_in_2[PLM]:
    "[z\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x) \<^bold>\<rightarrow> \<phi> z in dw]"
    using hintikka[equiv_lr, conj1] by (rule CP)

  lemma y_in_3[PLM]:
    "[(\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> (x\<^sup>P))) \<^bold>\<rightarrow> \<phi> (\<^bold>\<iota>x . \<phi> (x\<^sup>P)) in dw]"
    proof (rule CP)
      assume "[(\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> (x\<^sup>P))) in dw]"
      then obtain y where 1:
        "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> (x\<^sup>P)) in dw]"
        by (rule "\<^bold>\<exists>E")
ljava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
        
      thus "[\<phi> (\<^bold>\<iota>x. \<phi> (x\<^sup>P)) in dw]"
        using l_identity[axiom_instance, deduction,
deduction] 1 by fast
    qed

 act_quant_nec[PLM]:
    "[(\<^bold>\<forall>z . (\<bold\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x)) \<^bold>\<equiv> (\<^bold>\<forall>z. \<^bold>\<A>\^bold><A><phi> z \<^bold\ AOT_urrel_equiv sAOT_model_valid_in v (Rep_urrel r (\<omega> u)) 
    by PLM_solver

  lemma equi_desc_descA_1[PLM]:
    "[(x\<^sup>P \<^bold>= (\<\open\not>AOT_model_regular x \<Longrightarrow> fix_irregular \<phi> x = AOT_model_irregular \<phi
    using descriptions[axiom_instance] apply \<
    using act_quant_nec apply (rule \<boldequivE"(5))
    using descriptions[axiom_instance]
    by (meson "\<^bold>\<equiv>E"(6) oth_class_taut_4_a)

  lemma equi_desc_descA_2[PLM]:
    "[(\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<moreoverave<>Rep_rel <>(\omega\kappa> x)  Rep_rel <>SOME\>\psilony<><> x)\<close> for x
    proof (rule CP)
      assume "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)]
      then obtain y where
        "AOT_model_valid_in>x)\<close> for v x
        by (rule "\<^bold>\<exists>E")
      moreover hence "[y\<^sup>P^bold> (\<^bold>\<iota>x. \<^bold>\AOT_model_denotes \<kappa>'\<close>
        using equi_desc_descA_1[equiv_lr] by auto
      ultimately show "[(\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x) in v]"
        using l_identity[axiom_instance, deduction, deduction]
java.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 15
    qed

  lemma equi_desc_descA_3[PLM]:
    assumes "SimpleExOrEnc \<psi>"
    shows "[\<psi> (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x)) in v]"
    proof (rule CP)
      assume "[\<psi> (\<^bold>\<iota>x. \<phi> x) in v]"
      hence "[\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
        using cqt_5[OF assms, axiom_instance, deduction] by auto
      then obtain \<alpha> where "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]" by (rule "\<^bold>\<exists>E")
      hence "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<^bold>\<A>\<phi> x) in v]"
        using equi_desc_descA_1[equiv_lr] by auto
      thus "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x) in v]"
        using "\<^bold>\<exists>I" by fast
    qed

  lemma equi_desc_descA_4[PLM]:
    assumes "SimpleExOrEnc \<psi>"
    shows "[\<psi> (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> ((\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<^bold>\<iota>x. \<^bold>\<A>\<phi> x)) in v]"
    proof (rule CP)
      assume "[\<psi> (\<^bold>\<iota>x. \<phi> x) in v]"
      hence "[\<^bold>\<exists>\<alpha>. \<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
        using cqt_5[OF assms, axiom_instance, deduction] by auto
      then obtain \<alpha> where "[\<alpha>\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]" by (rule "\<^bold>\<exists>E")
      moreover hence "[\<alpha>\<^sup>P  \<^bold>= (\<^bold>\<iota>x . \<^bold>\<A>\<phi> x) in v]"
        using equi_desc_descA_1[equiv_lr] by auto
      ultimately show "[(\<^bold>\<iota>x. \<phi> x)  \<^bold>= (\<^bold>\<iota>x . \<^bold>\<A>\<phi> x) in v]"
        using l_identity[axiom_instance, deduction, deduction] by fast
    qed

  lemma nec_hintikka_scheme[PLM]:
    "[(x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x)) in v]"
    using descriptions[axiom_instance]
    apply (rule "\<^bold>\<equiv>E"(5))
    apply PLM_solver
     using id_eq_obj_1 apply simp
     using id_eq_obj_2[deduction]
           l_identity[where \<alpha>="x", axiom_instance, deduction, deduction]
     identity_nu_def
     apply blast
    using l_identity[where \<alpha>="x", axiom_instance, deduction, deduction]
    id_eq_2[where 'a=\<nu>, deduction] unfolding identity_\<nu>_def by meson

  lemma equiv_desc_eq[PLM]:
    assumes "\<And>x.[\<^bold>\<A>(\<phi> x \<^bold>\<equiv> \<psi> x) in v]"
    hows"(\bold\<orall> x . ((x<sup>P\<>=\^><>x. <> )) <bold\equiv (\^> <bold=\^old\iotax  <>x))inv]"
    proof(rule "\<^bold>\<forall>I")
      fix x
      {
        assume "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x) in v]"
        hence 1: "[\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
          using nec_hintikka_scheme[equiv_lr] by auto
        hence 2: "[\<^bold>\<A>\<phi> x in v] \<and> [(\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
          using "\<^bold>&E" by blast
        {
           fix z
           {
             assume "[\<^bold>\<A>\<psi> z in v]"
             hence "[\<^bold>\<A>\<phi> z in v]"
              using assms[where x=z] apply - by PLM_solver
             moreover have "[\<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x in v]"
               using 2 cqt_1[axiom_instance,deduction] by auto
             ultimately have "[z \<^bold>= x in v]"
              using vdash_properties_10 by auto
           }
           hence "[\<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x in v]" by (rule CP)
        }
        hence "[(\<^bold>\<forall> z . \<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]" by (rule "\<^bold>\<forall>I")
        moreover have "[\<^bold>\<A>\<psi> x in v]"
          using 1[conj1] assms[where x=x]
          apply - by PLM_solver
        ultimately have "[\<^bold>\<A>\<psi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
          by PLM_solver
        hence "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<psi> x) in v]"
         using nec_hintikka_scheme[where \<phi>="\<psi>", equiv_rl] by auto
      }
      moreover {
        assume "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<psi> x) in v]"
        hence 1: "[\<^bold>\<A>\<psi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
          using nec_hintikka_scheme[equiv_lr] by auto
        hence 2: "[\<^bold>\<A>\<psi> x in v] \<and> [(\<^bold>\<forall>z. \<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"

        {
          fix z
          {
            assume "[\<^bold>\<A>\<phi> z in v]"
            hence "[\<^bold>\<A>\<psi> z in v]"
              using assms[where x=z]
              - java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
            moreover have "[\<^bold>\<A>\<psi> z \<^bold>\<rightarrow> z \<^bold>= x in v]"
              using 2 cqt_1[axiom_instance,deduction] by auto
            ultimately have "[z \<^bold>= x in v]"
              using vdash_properties_10 by auto
          }
          hence "[\<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x in v]" by (rule CP)
        }
        hence "[(\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]" by (id_eq_2[ a\nu> deduction  identity_<> byjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
        moreover have "[\<^bold>\<A>\<phi> x in v]"
          using 1[conj1] assms[where x=x]
          apply - by PLM_solver
        ultimately have "[\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) in v]"
          by PLM_solver
        hence "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
          using nec_hintikka_scheme[where \<phi>="\<phi>",equiv_rl]
          by auto
      }
      ultimately show "[x\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> (x\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<psi> x) in v]"
        using "\<^bold>\<equiv>I" CP by auto
    qed

  lemma UniqueAux:
    assumes "[(\<^bold>\<A>\<phi> (\<alpha>::\<nu>) \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> z \<^bold>= \<alpha>)) in v]"
    shows "[(\<^bold>\<forall> z . (\<^bold>\<A>(\<phi> z) \<^bold>\<equiv> (z \<^bold>= \<alpha>))) in v]"
    proof -
      {
        fix z
                      unfoldingpropnot_defs .
          assume "[\<^bold>\<A>(\<phi> z) in v]"
          hence "[z \<^bold>= \<alpha> in v]"
            using assms[conj2, THEN cqt_1[where \<alpha>=z,
                          axiom_instance, deduction],
                        deduction] by auto
        }
        moreover {
          assume "[z \<^bold>= \<alpha> in v]"
          hence "[\<alpha> \<^bold>= z in v]"
            unfolding identity_\<nu>_def
            using id_eq_obj_2[deduction] by fast
          hence "[\<^bold>\<A>(\<phi> z) in v]" using assms[conj1]
            using l_identity[axiom_instance, deduction,
                             deduction] by fast
        }
        ultimately have "[(\<^bold>\<A>(\<phi> z) \<^bold>\<equiv> (z \<^bold>= \<alpha>)) in v]"
          using "\<^bold>\<equiv>I" CP by auto
      }
      thus "[(\<^bold>\<forall> z . (\<^bold>\<A>(\<phi> z) \<^bold>\<equiv> (z \<^bold>= \<alpha>))) in v]"
      by (rule "\<^bold>\<forall>I")
    qed

  lemma nec_russell_axiom[PLM]:
    assumes "SimpleExOrEnc \<psi>"
    shows "[(\<psi> (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<exists> x . (\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> z \<^bold>= x))
                              \<^bold>& \<psi> (x\<^sup>P)) in v]"
    (is "[?lhs \<^bold>\<equiv> ?rhs in v]")
    proof -
      {
        assume 1: "[?lhs in v]"
        hence "[\<^bold>\<exists>\<alpha>. (\<alpha>\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
          using cqt_5[axiom_instance, deduction] assms by blast
        then obtain \<alpha> where 2: "[(\<alpha>\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]" by (rule "\<^bold>\<exists>E")
        hence "[(\<^bold>\<forall> z . (\<^bold>\<A>(\<phi> z) \<^bold>\<equiv> (z \<^bold>= \<alpha>))) in v]"
          using descriptions[axiom_instance, equiv_lr] by auto
        hence 3: "[(\<^bold>\<A>\<phi> \<alpha>) \<^bold>& (\<^bold>\<forall> z . (\<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> (z \<^bold>= \<alpha>))) in v]"
          using cqt_1[where \<alpha>=\<alpha> and \<phi>="\<lambda> z . (\<^bold>\<A>(\<phi> z) \<^bold>\<equiv> (z \<^bold>= \<alpha>))",
                      axiom_instance, deduction, equiv_rl]
          using id_eq_obj_1[where x=\<alpha>] unfolding identity_\<nu>_def
         hintikkaequiv_lr]cqt_basic_2[equiv_lrconj1]
          "\<^bold>&I" by fast
        from 2 have "[(\<^bold>\<iota>x. \<phi> x) \<^bold>= (\<alpha>\<^sup>P)  in v]"
          using l_identity[where \<beta>="(\<^bold>\<iota>x. \<phi> x)" and \<phi>="\<lambda> x . x \<^bold>= (\<alpha>\<^sup>P)",
                axiom_instance, deduction, deduction]
                id_eq_obj_1[where x=\<alpha>] by auto
        hence "[\<psi> (\<alpha>\<^sup>P) in v]"
          using 1 l_identity[where \<alpha>="(\<^bold>\<iota>x. \<phi> x)" and \<phi>="\<lambda> x . \<psi> x",
                             axiom_instance, deduction,
                             deduction] by auto
        with 3 have "(<bold\A>\<hi \alpha> \<bold>& (<bold>\forall z .\<^bold>\<A(\phi z) \<^>\<rightarrow> (\^> \alpha>)))\>  ]java.lang.StringIndexOutOfBoundsException: Index 196 out of bounds for length 196
          using "\<^bold>&I" by simp
        hence "[?rhs in v]"
          using "\<^bold>\<exists>I"[where \<alpha>=\<alpha>]
bysimp :identity_defs)
      }
      moreover {
        assume "[?rhs in v]"
        then obtain \<alpha> where 4:
          "[(\<^bold>\<A>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> z \<^bold>= \<alpha>)) \<^bold>& \<psi> (\<alpha>\<^sup>P) in v]"
          using "\<^bold>\<exists>E" by auto
        hence "[(\<^bold>\<forall> z . (\<^bold>\<A>(\<phi                KBasic2_2[equiv_rl blast
          using UniqueAux "\<^bold>&E"(1) by auto
        hence "[(\<alpha>\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<phi> x) in v] \<and> [\<psi> (\<alpha>\<^sup>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

  actual_desc_1]
    "[(\<^bold>\<exists> y . (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<equiv> (\<^bold>\<exists>! x . \<^bold>\<A>(\<phi> x)) in v]" (is "[?lhs \<^bold>\<equiv> ?rhs in v]")
    proof -
      {
        assume "[?lhs in v]"
        then obtain \<alpha> where
                {
          by (rule "\<^bold>\<exists>E")
        hence "[\<lparr>A!,(\<^bold>\<iota>x. \<phi> x)\<rparr> in v] \<or> [(\<alpha>\<^sup>P) \<^bold>=\<^sub>E (\<^bold>\<iota>x. \<phi> x) in v]"
          apply - unfolding identity_defs by PLM_solver
        then obtain x where
          "[((\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> z \<^bold>= x))) in v]"
          using nec_russell_axiom[where \<psi>="\<lambda>x . \<lparr>A!,x\<rparr>", equiv_lr, THEN "\<^bold>\<exists>E"]
          using nec_russell_axiom[where \<psi>="\<lambda>x . (\<alpha>\<^sup>P) \<^bold>=\<^sub>E x", equiv_lr, THEN "\<^bold>\<exists>E"]
          using SimpleExOrEnc.intros unfolding identity\<^sub>E_infix_def
          by (meson "\<^bold>&E")
        hence "[?rhs in v]" unfolding exists_unique_def by (rule "\<^bold>\<exists>I")
      }
      moreover {
        assume "[?rhs in v]"
        then obtain x where
          "[((\<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall> z . \<^bold>\<A>(\<phi> z) \<^bold>\<rightarrow> z \<^bold>= x))) in v]"
          unfolding exists_unique_def by (rule "\<^bold>\<exists>E")
        hence "[\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= x in v]"
          using UniqueAux by auto
        hence "[(x\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
          using descriptions[axiom_instance, equiv_rl] by auto
        hence "[?lhs in v]" by (rule "\<^bold>\<exists>I")
      }
      ultimately show ?thesis
        using"\<bold>\<equiv>I" CP by auto
    qed

  lemma actual_desc_2[PLM]:
    "[(x\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi>) \<^bold>\<rightarrow> \<^bold>\<A>\<phi> in v]"
    using nec_hintikka_scheme[equiv_lr, conj1]
    by (rule CP)

  lemma actual_desc_3[PLM]:
    "[(z\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<^bold>\<A>(\<phi> z) in v]"
    using nec_hintikka_scheme[equiv_lr, conj1]
    by (rule CP)

  lemma actual_desc_4[PLM]:
    "[(\<^bold>\<exists> y . ((y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> (x\<^sup>P)))) \<^bold>\<rightarrow> \<^bold>\<A>(\<phi> (\<^bold>\<iota>x. \<phi> (x\<^sup>P))) in v]"
    proof (rule CP)
      assume "[(\<^bold>\<exists> y . (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x . \<phi> (x\<^sup>P))) in v]"
      then obtain y where 1:
        "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> (x\<^sup>P)) in v]"
        by (rule "\<^bold>\<exists>E")
      hence "[\<^bold>\<A>(\<phi> (y\<^sup>P))       using [HEN "^bold>\forall>I] fast
      thus "[\<^bold>\<A>(\<phi> (\<^bold>\<iota>x. \<phi> (x\<^sup>P))) in v]"
        using l_identity[axiom_instance, deduction,
                         deduction] 1 by fast
    qed

  lemma unique_box_desc_1[PLM]:
    "[(\<^bold>\<exists>!x . \<^bold>\<box>(\<phi> x)) \<^bold>\<rightarrow> (\<^bold>\<forall> y . (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> y) in v]"
    proof (rule CP)
      assume "[(\<^bold>\<exists>!x . \<^bold>\<box>(\<phi> x)) in v]"
      then obtain \<alpha> where 1:
        "[\<^bold>\<box>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<^bold>\<box>(\<phi> \<beta>) \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
        unfolding exists_unique_def by (rule "\<^bold>\<exists>E")
      {
        fix y
        {
          assume "[(y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
          hence "[\<^bold>\<A>\<phi> \<alpha> \<^bold>\<rightarrow> \<alpha> \<^bold>= y in v]"
            using nec_hintikka_scheme[where x="y" and \<phi>="\<phi>", equiv_lr, conj2,
                          THEN cqt_1[where \<alpha>=\<alpha>,axiom_instance, deduction]] by simp
          hence "[\<alpha> \<^bold>= y in v]"
            using 1[conj1] nec_imp_act vdash_properties_10 by blast
          hence "[\<phi> y in v]"
             1[conj1]qml_2[xiom_instance deduction]
                  l_identity[axiom_instance, deduction, deduction]
by
        }
        hence "[(y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> y in v]"
          by (rule CP)
      }
      thus "[\<^bold>\<forall> y . (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> y in v]"
        by (rule "\<^bold>\<forall        using cont_nec_fact1_1[HENoth_class_taut_5_dequiv_lr] equiv_lr]
    qed

  lemma unique_box_desc[PLM]:
    "[(\<^bold>\<forall> x . (\<phi> x \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> x))) \<^bold>\<rightarrow> ((\<^bold>\<exists>!x . \<phi> x)
      \<^bold>\<rightarrow> (\<^bold>\<forall> y . (y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x)) \<^bold>\<rightarrow> \<phi> y)) in v]"
    apply (rule CP, rule CP)
    using nec_exist_unique[deduction, deduction]
          unique_box_desc_1[deduction] by blast

subsection\<open>Necessity\<close>
text\<open>\label{TAO_PLM_Necessity}\<close>

  lemma RM_1[PLM]:
    "(\<And>v.[\<phi> \<^bold>\<rightarrow> \<psi> in v]) \<Longrightarrow> [\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
    using RN qml_1

  lemma RM_1_b[PLM]:
    "(\<And>v.[\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<psi> in v]) \<Longrightarrow> ([\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v])"
    using RN_2 qml_1[axiom_instance] vdash_properties_10 by blast

  lemma RM_2[PLM]:
    "(\<And>v.[\<phi> \<^bold>\<rightarrow> \<psi> in v]) \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi> in v]"
    unfolding diamond_def
    using RM_1 contraposition_1 by auto

  lemma RM_2_b[PLM]:
    "(\<And>v.[\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<psi> in v]) \<Longrightarrow> ([\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi> in v])"
    unfolding diamond_def
    using RM_1_b contraposition_1 by blast

  KBasic_1PLM]java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
    "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>) in v]"
    by (simp only: pl_1[axiom_instance] RM_1)
  lemma KBasic_2[PLM]:
    "[\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
    by (simp only: RM_1 useful_tautologies_3)
  lemma KBasic_3[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>& \<psi>) \<^bold>\<equiv> \<^bold>\<box>\<phi> \<^bold>& \           descriptionsaxiom_instanceequiv_rlauto
    apply (rule "\<^bold>\<equiv>I")
     apply (rule CP)
     apply (rule "\<^bold>&I")
      using RM_1 oth_class_taut_9_a vdash_properties_6 apply 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]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<equiv> (\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>)) in v]"
    apply (rule "\<^bold>\<equiv>I")
     unfolding equiv_def using KBasic_3 PLM.CP "\<^bold>\<equiv>E"(1)
     apply blast
    using KBasic_3 PLM.CP "\<^bold>\<equiv>E"(2)
    by blast
  lemma KBasic_5[PLM]:
    "[(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>)) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) in v]"
    by (metis qml_1[axiom_instance] CP "\<^bold>&E" "\<^bold>\<equiv>I" vdash_properties_10)
  lemma KBasic_6[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) in v]"
    using KBasic_4 KBasic_5 by (metis equiv_def ded_thm_cor_3 "\<^bold>&E"(1))
  lemma "[(\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
    nitpick[expect=genuine, user_axioms, card = 1, card i = 2]
    oops \<comment> \<open>countermodel as desired\<close>
  lemma KBasic_7[PLM]:
    "[(\<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
    proof (rule CP)
      assume "[\<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi> in v]"
      hence "[\<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>) in v] \<and> [\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
        using "\    qed
      thus "[\<^bold>\<box>(\<phi    "[\<bold<forall>x \phix <bold><ightarrow^>\box(\phi>x) ^><>(bold<>x \<>xjava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
        using KBasic_4 "\<^bold>\<equiv>E"(2) intro_elim_1 by blast
    qed

   KBasic_8PLM]
    "[\<^bold>\<box>(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
    using KBasic_7 KBasic_3
    by (metis equiv_def PLM.ded_thm_cor_3 "\<^bold>&E"(1))
lemmaKBasic_9[LM]:
    "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>)) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
    proof (rule CP)
      assume "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>)) in v]"
      hence "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>\<equiv> (\<^bold>\<not>\<psi>)) in v]"
        using KBasic_8 vdash_properties_10 by blast
      moreover have "\<And>v.[((\<^bold>\<not>\<phi>) \<^bold>\<equiv> (\  lemma prop_in_f_2:
        using CP "\<^bold>\<equiv>E"(2) oth_class_taut_5_d by blast
      ultimately show "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
        using RM_1 PLM.vdash_properties_10 by blast
    qed

  lemma rule_sub_lem_1_a[PLM]:
    "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v] \<Longrightarrow> [(\<^bold>\<not>\<psi>) \<^bold>\<equiv> (\<^bold>\<not>\<chi>) in v]"
    using qml_2[axiom_instance] "\<^bold>\<equiv>E"(1) oth_class_taut_5_d
          vdash_properties_10
    by blast
  lemma rule_sub_lem_1_b[    proof rulereductio_aa_2)
    "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v] \<Longrightarrow> [(\<psi> \<^bold>\<rightarrow> \<Theta>) \<^bold>\<equiv> (\<chi> \<^bold>\<rightarrow> \<Theta>) in v]"
    by (metis equiv_def contraposition_1 CP "\<^bold>&E"(2) "\<^bold>\<equiv>I"
              "\<^bold>\<equiv>E"(1) rule_sub_lem_1_a)
  lemma rule_sub_lem_1_c[PLM]:
    "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v] \<Longrightarrow> [(\<Theta> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> (\<Theta> \<^bold>\<rightarrow> \<chi>) in v]"
    by (metis CP "\<^bold>\<equiv>I" "\<^bold>\<equiv>E"(3) "\<^bold>\<equiv>E"(4) "\<^bold>\<not>\<^bold>\<not>I"
              "\<^bold>\<not>\<^bold>\<not>E" rule_sub_lem_1_a)
   rule_sub_lem_1_d[PLM]:
    "(\<And>x.[\<^bold>\<box>(\<psi> x \<^bold>\<equiv> \<chi> x) in v]) \<Longrightarrow> [(\<^bold>\<forall>\<alpha>. \<psi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<chi> \<alpha>) in v]" 
    by (metis equiv_def "\<^bold>\<forall>I" CP "\<^bold>&E" "\<^bold>\<equiv>I" raa_cor_1
              vdash_properties_10 rule_sub_lem_1_a "\<^bold>\<forall>E")
  lemma rule_sub_lem_1_e[PLM]:
    "[\<^               (\<lparr>!x<sup>< <> \<>!,y\^>\rparr>\^bold>\^bold\>\bold>forall>F.\>,\^>P\<rparr \bold<equiv>\lparr>,^>\>))
    using Act_Basic_5 "\<^bold>\<equiv>E"(1) nec_imp_act
          vdash_properties_10
    by blast
  lemma rule_sub_lem_1_f[PLM]:
    "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v] \<Longrightarrow> [\<^bold>\<box>\<psi> \<^bold>\<equiv> \<^bold>\<box>\<chi> in v]" 
    using KBasic_6 "\<^bold>\<equiv>I" "\<^bold>\<equiv>E"(1) vdash_properties_9
    by blast


  named_theorems Substable_intros
  
  definition Substable :: "('a\<Rightarrow>'a\<Rightarrow>bool)\<Rightarrow>('a\<Rightarrow>\<o>) \<Rightarrow> bool"
    where "Substable \<equiv> (\<lambda> cond \<phi> . \<forall> \<psi> \<chi> v . (cond \<psi> \<chi>) \<longrightarrow> [\<phi> \<psi>classid_act =id_eq+
  
  lemma Substable_intro_const[Substable_intros]:
    "Substable cond (\<lambda> \<phi> . \<Theta>)"
    unfolding Substable_def using oth_class_taut_4_a by blast

  lemma Substable_intro_not[Substable_intros]:
    assumes "Substable cond \<psi>"
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<not>(\<psi> \<phi>))"
    using assms unfolding Substable_def
using  "<^><>E" oth_class_taut_5_dmetis
  lemma Substable_intro_impl[Substable_intros]:
    assumes "Substable cond \<psi>"
        and "Substable cond \<chi>"
    shows "Substable cond (\<lambda> \<phi> . \<psi> \<phi> \<^bold>\<rightarrow> \<chi> \<phi>)"
    using assms unfolding Substable_def
    by (metis "\<^bold>\<equiv>I" CP intro_elim_6_a intro_elim_6_b)
  lemma Substable_intro_box[Substable_intros]:
    assumes "Substable cond \<psi>"
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<box>(\<psi> \<phi>))"
    using assms unfolding Substable_def
    using rule_sub_lem_1_f RN by meson
  lemma Substable_intro_actual[Substable_intros]:
    assumes "Substable cond \<psi>"
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<A>(\<psi> \<phi>))"
    using assms unfolding Substable_def
    using rule_sub_lem_1_e RN by meson
  lemma Substable_intro_all[Substable_intros]:
    assumes "\<forall> x . Substable cond (\<psi> x)"
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<forall> x . \<psi> x \<phi>)"
    using assms unfolding Substable_def
    by (simp add: RN rule_sub_lem_1_d)

  named_theorems Substable_Cond_defs
end

class Substable =
  fixes Substable_Cond :: "'a\<Rightarrow>'a\<Rightarrow>bool"
  assumes rule_sub_nec:
    "\<And> \<phi> \<psi> \<chi> \<Theta> v . \<lbrakk>PLM.Substable Substable_Cond \<phi>; Substable_Cond \<psi> \<chi>\<rbrakk>
      \<Longrightarrow> \<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]"

instantiation \<o> :: Substable
begin
  definition Substable_Cond_\<o> where [PLM.Substable_Cond_defs]:
    "Substable_Cond_\<o> \<equiv> \<lambda> \<phi> \<psi> . \<forall> v . [\<phi> \<^bold>\<equiv> \<psi> in v]"
  instance proof
    interpret java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
    fix \<phi> :: "\<o> \<Rightarrow> \<o>" and  \<psi> \<chi> :: \<o> and \<Theta> :: "bool \<Rightarrow> bool" and v::i
    assume "Substable Substable_Cond \<phi>"
    moreover assume "Substable_Cond \<psi> \<chi>"
    ultimately have "[\<phi> \<psi> \<^bold>\<equiv> \<phi> \<chi> in v]"
    unfolding Substable_def by blast
    hence "[\<phi> \<psi> in v] = [\<phi> \<chi> in v]" using "\<^bold>\<equiv>E" by blast
    moreover assume "\<Theta> [\<phi> \<psi> in v]"
    ultimately show "\<Theta> [\<phi> \<chi> in v]" by simp
  qed
end

instantiation "fun" :: (type, Substable) Substable
begin
  definition Substable_Cond_fun where [PLM.Substable_Cond_defs]:
    "Substable_Cond_fun \<equiv> \<lambda> \<phi> \<psi> . \<forall> x . Substable_Cond (\<phi> x) (\<psi> x)"
  instance proof
    interpret PLM .
    fix \<phi>:: "('a \<Rightarrow> 'b) \<Rightarrow> \<o>" and  \<psi> \<chi> :: "'a \<Rightarrow> 'b" and \<Theta> v
    assume "Substable Substable_Cond \<phi>"
      lemma prop_prop_nec_1:
    ultimately have "[\<phi> \<psi> \<^bold>\<equiv> \<phi> \<chi> in v]"
      unfolding Substable_def by blast
    hence "[\<phi> \<psi> in v] = [\<phi> \<chi> in v]" using "\<^bold>\<equiv>E" by blast
    moreover assume "\<Theta> [\<phi> \<psi> in v]"
ultimately<>[<phi> \chi inv] bysimp
  qed
end

context PLM


  lemma Substable_intro_equiv[Substable_intros]:
    assumes "Substable cond \<psi>"
        and "Substable cond \<chi>"
    shows "Substable cond (\<lambda> \<phi> . \<psi> \<phi> \<^bold>\<equiv> \<chi> \<phi>)"
    unfolding conn_defs by (simp add: assms Substable_intros)
  lemma Substable_intro_conj[Substable_intros]:
    assumes "Substable cond \<psi>"
        and "Substable cond \<chi>"
    shows "Substable cond (\<lambda> \<phi> . \<psi> \<phi> \<^bold>& \<chi> \<phi>)"
    unfolding conn_defs by (simp add: assms Substable_intros)
  lemma Substable_intro_disj[Substable_intros]:
    assumes "Substable cond \<psi>"
        and "Substable cond \<chi>"
    shows "Substable cond (\<lambda> \<phi> . \<psi> \<phi> \<^bold>\<or> \<chi> \<phi>)"
    unfolding conn_defs by (simp add: assms Substable_intros)
  lemma Substable_intro_diamond[Substable_intros]:
   \>
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<diamond>(\<psi> \<phi>))"
    unfolding conn_defs by (simp add: assms Substable_intros)
  lemma Substable_intro_exist[Substable_intros]:
    assumes "\<forall> x . Substable cond (\<psi> x)"
    shows "Substable cond (\<lambda> \<phi> . \<^bold>\<exists> x . \<psi> x \<phi>)"
    unfolding

  lemma Substable_intro_id_\<o>[Substable_intros]:
    "Substable Substable_Cond \<phi> . \<phi>)
    unfolding Substable_def Substable_Cond_      ( TimesR  [ c
  lemma Substable_intro_id_fun[Substable_intros]:
    assumes "Substable Substable_Cond \<psi>"
    shows "Substable Substable_Cond (\<lambda> \<phi> . \<psi> (\<phi> x))"
    using assms unfolding Substable_def Substable_Cond_fun_def
    by blast

  method PLM_subst_method for \<psi>::"'a::Substable" and \<chi>::"'a::Substable" =
    (match conclusion in "\<Theta> [\<phi> \<chi> in v]" for \<Theta> and \<phi> and v \<Rightarrow>
      \<open>(rule rule_sub_nec[where \<Theta>=\<Theta> and \<chi>=\<chi> and \<psi>=\<psi> and \<phi>=\<phi> and v=v],
        ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
        unfold Substable_Cond_defs)\<close>)

  method PLM_autosubst =
    (match premises in "\<And>v . [\<psi> \<^bold>\<equiv> \<chi> in v]" for \<psi> and \<chi> \<Rightarrow>
      \<open> match conclusion in "\<Theta> [\<phi> \<chi> in v]" for \<Theta> \<phi> and v \<Rightarrow>
        \<open>(rule rule_sub_nec[where \<Theta>=\<Theta> and \<chi>=\<chi> and \<psi>=\<psi> and \<phi>=\<phi> and v=v],
          ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
          unfold Substable_Cond_defs)\<close> \<close>)

  method PLM_autosubst1 =
    (match premises in "\<And>v x . [\<psi> x \<^bold>\<equiv> \<chi> x in v]"
      for \<psi>::"'a::type\<Rightarrow>\<o>" and \<chi>::"'a\<Rightarrow>\<o>" \<Rightarrow>
      \<open> match conclusion in "\<Theta> [\<phi> \<chi> in v]" for \<Theta> \<phi> and v \<Rightarrow>
        \<open>(rule rule_sub_nec[where \<Theta>=\<Theta> and \<chi>=\<chi> and \<psi>=\<psi> and \<phi>=\<phi> and v=v],
          ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
          unfold Substable_Cond_defs)\<close> \<close>)

  method PLM_autosubst2 =
    (match premises in "\<And>v x y . [\<psi> x y \<^bold>\<equiv> \<chi> x y in v]"
      for \<psi>::"'a::type\<Rightarrow>'a\<Rightarrow>\<o>" and \<chi>::"'a::type\<Rightarrow>'a\<Rightarrow>\<o>" \<Rightarrow>
      \<open> match conclusion in "\<Theta> [\<phi> \<chi> in v]" for \<
        <open(rule rule_sub_nec[  <>\psi  <phi=<phi andv=v],
          ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
          unfold Substable_Cond_defs)\<close> \<close>)

  method PLM_subst_goal_method for \<phi>::"'a::Substable\<Rightarrow>\<o>" and \<psi>::"'a" =
    (match conclusion in "\<Theta> [\<phi> \<chi> in v]" for \<Theta> and \<chi> and v \<Rightarrow>
      \<open>(rule rule_sub_nec[where \<Theta>=\<Theta> and \<chi>=\<chi> and \<psi>=\<psi> and \<phi>=\<phi> and v=v],
        ((fast intro: Substable_intros, ((assumption)+)?)+; fail),
        unfold Substable_Cond_defs)\<close>)

(*
  text{* \begin{TODO}
            This can only be proven using the Semantics of the Box operator.
            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 \<phi>"
    shows "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v] \<Longrightarrow> [\<phi> \<psi> \<^bold>\<equiv> \<phi> \<chi> in v]"
    using assms unfolding Substable_def Substable_Cond_defs
    using Semantics.T6 by fast
*)

  lemma rule_sub_nec[PLM]:
    assumes "Substable Substable_Cond \<phi>"
    shows "(\<And>v.[(\<psi> \<^bold>\<equiv> \<chi>) in v]) \<Longrightarrow> \<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]"
    proof -
      assume "(\<And>v.[(\<psi> \<^bold>\<equiv> \<chi>) in v])"
      hence "[\<phi> \<psi> in v] = [\<phi> \<chi> in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<^bold>\<equiv>I" CP "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2) by meson
      thus "\<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]" by auto
    qed

  lemma rule_sub_nec1[PLM]:
    assumes "Substable Substable_Cond \<phi>"
    shows "(\<And>v x .[(\<psi> x \<^bold>\<equiv> \<chi> x) in v]) \<Longrightarrow> \<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]"
    proof -
      assume "(\<And>v x.[(\<psi> x \<^bold>\<equiv> \<chi> x) in v])"
      hence "[\<phi> \<psi> in v] = [\<phi> \<chi> in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<^bold>\<equiv>I" CP "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2) by metis
      thus "\<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]" by auto
    qed

  lemma rule_sub_nec2[PLM]:
    assumes "Substable Substable_Cond \<phi>"
> x y in v]) \<Longrightarrow> \<Theta> [\<phi> \<psi> in v] \<Longrightarrow<Theta [\<phi> \<chi> in v]"
    proof -
      assume "(\<And>v x y .[\<psi> x y \<^bold>\<equiv> \<chi> x y in v])"
      hence "[\<phi> \<psi> in v] = [\<phi> \<chi> in v]"
        using assms RN unfolding Substable_def Substable_Cond_defs
        using "\<^bold>\<equiv>I" CP "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2) by metis
      thus "\<Theta> [\<phi> \<psi> in v] \<Longrightarrow> \<Theta> [\<phi> \<chi> in v]" by auto
    qed

  lemmale_sub_remark_1_autosubst
sumes\v.[\<lparr>A!,x\<rparr> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<rparr>)) in v])"
        and "[\<^bold>\<not>\<lparr>A!,x\<rparr> in v]"
    shows"[\<^bold>\<not>\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<rparr> in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_1:
    assumes "(\<And>v.[\<lparr>A!,x\<rparr> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<rparr>)) in v])"
        and "[\<^bold>\<not>\<lparr>A!,x\<rparr> in v]"
      shows"[\<^bold>\<not>bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<rparr> in v]"
    apply (PLM_subst_method "\<lparr>A!,x\<rparr>" "(\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<rparr>))")
     apply (simp add: assms(1))
    by (simp add: assms(2))

  lemma rule_sub_remark_2:
    assumes "(\<And>v.[\<lparr>R,x,y\<rparr> \<^bold>\<equiv> (\<lparr>R,x,y\<rparr> \<^bold>& (\<lparr>Q,a\<rparr> \<^bold>\<or> (\<^bold>\<not>\<lparr>Q,a\<rparr>))) in v])"
        and "[p \<^bold>\<rightarrow> \<lparr>R,x,y\<rparr> in v]"
    shows"[p \<^bold>\<rightarrow> (\<lparr>R,x,y\<rparr> \<^bold>& (\<lparr>Q,a\<rparr> \<^bold>\<or> (\<^bold>\<not>\<lparr>Q,a\<rparr>)))  in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_3_autosubst:
    assumes "(\<And>v x.[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>)) in v])"
        and "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> in v]"
    shows"[\<^bold>\<exists> x . (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>))  in v]"
    apply (insert assms) apply PLM_autosubst1 by auto

  lemma rule_sub_remark_3:
    assumes "(\<And>v x.[\<lparr>A!,x\<^sup>P\<rparr> \<^
        and "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> in  "ll_deduct_wf(ill_deduct_identity_compact G"
    shows "[\<^bold>\<exists> x . (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>"ll_conclusion ill_deduct_identity_compact  Sequent  )
    apply (PLM_subst_method "\<lambda>x . \<lparr>A!,x\<^sup>P\<rparr>" "\<lambda>x . (\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>))")
     apply (simp add: assms(1))
    by (simp add: assms(2))

  lemma rule_sub_remark_4:
    assumes "\<And>v x.[(\<^bold>\<not>(\<^bold>\<not>\<lparr>P,x\<^sup>P\<rparr>)) \<^bold>\<equiv> \<lparr>P,x\<^sup>P\<rparr> in v]"
        and "[\<^bold>\<A>(\<^bold>\<not>(\<^bold>\<not>\<lparr>P,x\<^sup>P\<rparr>)) in v]"
    shows "[\<^bold>\<A>\<lparr>P,x\<^sup>P\<rparr> in v]"
    apply (insert assms) apply PLM_autosubst1 by auto

  lemma rule_sub_remark_5:
    assumes "\<And>v.[(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> ((\<^bold>\<not>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<not>\<phi>)) in v]"
        and "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
    shows "[\<^bold>\<box>((\<^bold>\<not>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<not>\<phi>)) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_6:
    assumes "\<And>v.[\<psi> \<^bold>\<equiv> \<chi> in v]"
        and "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
    shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<chi>) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_7:
    assumes "\<And>v.[\<phi> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<not>\<phi>)) in v]"
        and "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<phi>) in v]"
    shows "[\<^bold>\<box>((\<^bold>\<not>(\<^bold>\<not>\<phi>)) \<^bold>\<rightarrow> \<phi>) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  lemma rule_sub_remark_8:
    assumes "\<And>v.[\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi> in v]"
        and "[\<^bold>\<box>(\<^bold>\<A>\<phi>) in v]"
    shows "[\<^bold>\<box>(\<phi>) in v]"
    apply (insert assms) apply PLM_autosubst by auto

  rule_sub_remark_9
    assumes "\<And>v.[\<lparr>P,a\<rparr> \<^bold>\<equiv> (\<lparr>P,a\<rparr> \<^bold>& (\<lparr>Q,b\<rparr> \<^bold>\<or> (\<^bold>\<not>\<lparr>Q,b\<rparr>))) in v]"
        and "[\<lparr>P,a\<rparr> \<^bold>= \<lparr>P,a\<rparr> in v]"
    shows "[\<lparr>P,a\<rparr> \<^bold>= (\<lparr>P,a\<rparr> \<^bold>& (\<lparr>Q,b\<rparr> \<^bold>\<or> (\<^bold>\<not>\<lparr>Q,b\<rparr>))) in v]"
      unfolding identity_defs apply (insert assms)
      apply PLM_autosubst oops \<comment> \<open>no match as desired\<close>

  \<comment> \<open>@{text "dr_alphabetic_rules"} implicitly holds\<close>
  \<comment> \<open>@{text "dr_alphabetic_thm"} implicitly holds\<close>

  " (ll_deduct_exchange_listA B  c P  Sequent (G @ B @ A @ D) c"
    "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<not>(\<^bold>\<not>\<phi>)) in v]"
    apply (PLM_subst_method "\<phi>" "(\<^bold>\<not>(\<^bold>\<not>\<phi>))")
     by PLM_solver+

  lemma KBasic2_2[PLM]:
    "[(\<^bold>\<not>(\<^bold>\<box>\<phi>)) \<^bold>\<equiv> \<^bold>\<diamond>(\<^bold>\<not>\<phi>) in v]"
    unfolding diamond_def
    apply (PLM_subst_method "\<phi>" "\<^bold>\<not>(\<^bold>\<not>\<phi>)")
     by PLM_solver+

  lemma KBasic2_3[PLM]:
    "[\<^bold>\<box>\<phi> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<diamond>(\<^bold>\<not>\<phi>))) in v]"
    unfolding diamond_def
    apply (PLM_subst_method "\<phi>" "\<^bold>\<not>(\<^bold>\<not>\<phi>)")
     apply PLM_solver
    by (simp add: oth_class_taut_4_b)
  lemmas "Df\<^bold>\<box>" = KBasic2_3

  lemma KBasic2_4[PLM]:
    "[\<^bold>\<box>(\<^bold>\<not>(\<phi>)) \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) in v]"
    unfolding diamond_def
    by (simp add: oth_class_taut_4_b)

  lemma KBasic2_5[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>) in v]"
    by (simp only: CP RM_2_b)
  lemmas "K\<^bold>\<diamond>" = KBasic2_5

  lemma KBasic2_6[PLM]:
    "[\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> (\<^bold>\<diamond>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>) in v]"
    proof -
      have "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>)) \<^bold>\<equiv> (\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>& \<^bold>\<box>(\<^bold>\<not>\<psi>)) in v]"
        using KBasic_3 by blast
      hence "[(\<^bold>\<not>(\<^bold>\<diamond>(\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>))))) \<^bold>\<equiv> (\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>& \<^bold>\<box>(\<^bold>\<not>\<psi>)) in v]"
        using "Df\<^bold>\<box>" by (rule "\<^bold>\<equiv>E"(6))
      hence "[(\<^bold>\<not>(\<^bold>\<diamond>(\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>))))) \<^bold>\<equiv> ((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>& (\<^bold>\<not>(\<^bold>\<diamond>\<psi>))) in v]"
        apply - apply (PLM_subst_method "\<^bold>\<box>(\<^bold>\<not>\<phi>)" "\<^bold>\<not>(\<^bold>\<diamond>\<phi>)")
         apply (simp add: KBasic2_4)
        apply (PLM_subst_method "\<^bold>\<box>(\<^bold>\<not>\<psi>)" "\<^bold>\<not>(\<^bold>\<diamond>\<psi>)")
         apply (simp add: KBasic2_4)
        unfolding diamond_def by assumption
      hence "[(\<^bold>\<not>(\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>))) \<^bold>\<equiv> ((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>& (\<^bold>\<not>(\<^bold>\<diamond>\<psi>))) in v]"
        apply - apply (PLM_subst_method "\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>))" "\<phi> \<^bold>\<or> \<psi>")
  \bold\forallE"
      hence "[(\<^bold>\<not>(\<^bold>\<not>(\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>)))) \<^bold>\<equiv> (\<^bold>\<not>((\<^bold>\<not>(\<^bold>\<diamond>\<phi>))\<^bold>&(\<^bold>\<not>(\<^bold>\<diamond>\<psi>)))) in v]"
        by (rule oth_class_taut_5_d[equiv_lr])
      hence "[\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<equiv> (\<^bold>\<not>((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>& (\<^bold>\<not>(\<^bold>\<diamond       <>  \bold>phi\\< ]
        apply - apply (PLM_subst_method "\<^bold>\<not>(\<^bold>\<not>(\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>)))" "\<^bold>\<diamond>(\<phi> \<^bold>\<or> \<psi>)")
        using oth_class_taut_4_b[equiv_sym] by auto
      thus ?thesis
        apply - apply (PLM_subst_method "\<^bold>\<not>((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>&pan> (\<^bold>\<not>(\<^bold>\<diamond>\<psi>)))" "(\<^bold>\<diamond>\<phi>) \<^bold>\<or> (\<^bold>\<diamond>\<psi>)")
        using oth_class_taut_6_b[equiv_sym] by auto
    qed

  lemma KBasic2_7[PLM]:
    "[(\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) in v]"
    proof -
      have "\<And> v . [\<phi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<or> \<psi>) in v]"
        by (metis contraposition_1 contraposition_2 useful_tautologies_3 disj_def)
      hence "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) in v]" using RM_1 by auto
      moreover {
          have "\<And> v . [\<psi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<or> \<psi>) in v]"
            by (simp only: pl_1[axiom_instance] disj_def)
          hence "[\<^bold>\<box>\<psi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) in v]"
            using RM_1 by auto
      }
      ultimately show ?thesis
        using oth_class_taut_10_d vdash_properties_10 by blast
    qed

  lemma KBasic2_8[PLM]:
    "[\<^bold>\<diamond>(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>&</span> \<^bold>\<diamond>\<psi>) in v]"
    by (metis CP RM_2 "\<^bold>&I" oth_class_taut_9_a
              oth_class_taut_9_b vdash_properties_10)

  lemma KBasic2_9[PLM]:
    "[\<^bold>\<diamond>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<equiv> (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>) in v]"
    apply (PLM_subst_method "(\<^bold>\<not>(\<^bold>\<box>\<phi>)) \<^bold>\<or> (\<^bold>\<diamond>\<psi>)" "\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi>")
     using oth_class_taut_5_k[equiv_sym] apply simp
    apply (PLM_subst_method "(\<^bold>\<not>\<phi>) \<^bold>\<or> \<psi>" "\<phi> \<^bold>\<rightarrow> \<psi>")
     using oth_class_taut_5_k[equiv_sym] apply simp
    apply (PLM_subst_method "\<^bold>\<diamond>(\<^bold>\<not>\<phi>  notepad
     using KBasic2_2[equiv_sym] apply simp
    using KBasic2_6 .

  lemma KBasic2_10[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<box>\<phi>) \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<box>\<^bold>\<diamond>(\<^bold>\<not>\<phi>))) in v]"
    unfolding diamond_def apply (PLM_subst_method "\<phi>" "\<^bold>\<not>\<^bold>\<not>\<phi>")
    using oth_class_taut_4_b oth_class_taut_4_a by auto

  lemma KBasic2_11[PLM]:
    "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<equiv> (\<^bold>\<not>(\<^bold>\<box>\<^bold>\<box>(\<^bold>\<not>\<phi>))) in v]"
    unfolding diamond_def
    apply (PLM_subst_method "\<^bold>\<box>(\<^bold>\<not>\<phi>)" "\<^bold>\<not>(\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>\<phi>)))")
    using oth_class_taut_4_b oth_class_taut_4_a by auto

  lemma KBasic2_12[PLM]: "[\<^bold>\<box>(\<phi> \<^bold>\<or> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>) in v]"
    proof -
      have "[\<^bold>\<box>(\<psi> \<^bold>\<or> \<phi>) \<^bold>\<rightarrow> (\<^bold>\<box>(\<^bold>\<not>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) in v]"
        using CP RM_1_b "\<^bold>\<or>E"(2) by blast
      hence "[\<^bold>\<box>(\<psi> \<^bold>\<or> \<phi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<psi> \<^bold>\<or> \<^bold>\<box>\<phi>) in v]"
        unfolding diamond_def disj_def
        by (meson CP "\<^bold>\<not>\<^bold>\<not>E" vdash_properties_6)
      thus ?thesis apply -
        apply (PLM_subst_method "(\<^bold>\<diamond>\<psi> \<^bold>\<or> \<^bold>\<box>\<phi>)" "(\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>)")
         apply (simp add: PLM.oth_class_taut_3_e)
        apply (PLM_subst_method "(\<psi> \<^bold>\<or> \<phi>)" "(\<phi> \<^bold>\<or> \<psi>)")
         apply (simp add: PLM.oth_class_taut_3_e)
        by assumption
    qed

  lemma TBasic[PLM]:
    "[\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> in v]"
    unfolding diamond_def
    apply (subst contraposition_1)
    apply (PLM_subst_method "\<^bold>\<box>\<^bold>\<not>\<phi>" "\<^bold>\<not>\<^bold>\<not>\<^bold>\<box>\<^bold>\<not>\<phi>")
     apply (simp add: PLM.oth_class_taut_4_b)
   [here<phi>="\^bold\not>\<hi>", axiom_instance]
    by simp
masbold>\<diamond>" = TBasic

  lemma S5Basic_1[PLM]:
    "[\<^bold>\<diamond>\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi> in v]"
    proof (rule CP)
 "\<diamond>\<^bold>\<box>\<phi> in v]"
      hence "[\<^bold>\<not>\<^bold>\<box>\<^bold>\<diamond>\<^bold>\<not>\<phi> in v]"
        using KBasic2_10[equiv_lr] by simp
      moreover have "[\<^bold>\<diamond>(\<^bold>\<not>\<phi>) \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<diamond>(\<^bold>\<not>\<phi>) in v]"
        by (simp add: qml_3[axiom_instance])
      ultimately have "[\<^bold>\<not>\<^bold>\<diamond>\<^bold>\<not>\<phi> in v]"
        by (simp add: PLM.modus_tollens_1)
      thus "[\<^bold>\<box>\<phi> in v]"
        unfolding diamond_def apply -
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<not>\<phi>" "\<phi>")
         using oth_class_taut_4_b[equiv_sym] apply simp
        unfolding diamond_def using oth_class_taut_4_b[equiv_rl]
        by simp
    qed
  lemmas "5\<^bold>\<diamond>" = S5Basic_1

  lemma S5Basic_2[PLM]:
    "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<diamond>\<^bold>\<box>\<phi> in v]"
    using "5\<^bold>\<diamond>" "T\<^bold>\<diamond>" "\<^bold>\<equiv>I" by blast

  lemma S5Basic_3[PLM]:
    "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<diamond>\<phi> in v]"
    using qml_3[axiom_instance] qml_2[axiom_instance] "\<^bold>\<equiv>I" by blast

  lemma S5Basic_4[PLM]:
    "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<diamond>\<phi> in v]"
    using "T\<^bold>\<diamond>"[deduction THEN S5Basic_3[uiv_lr
    by (rule CP)

  lemma S5Basic_5[PLM]:
    "[\<^bold>\<diamond>\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<phi> in v]"
    using S5Basic_2[equiv_rl, THEN qml_2[axiom_instance, deduction]]
    by (rule CP)
  lemmas "B\<^bold>\<diamond>" = S5Basic_5

  lemma S5Basic_6[PLM]:
    "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<box>\<phi> in v]"
    using S5Basic_4[deduction] RM_1[OF S5Basic_1, deduction] CP by auto
  lemmas "4\<^bold>\<box>" = S5Basic_6

  lemmaS5Basic_7
    "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<box>\<phi> in v]"
    using "4\<^bold>\<box>" qml_2[axiom_instance] by (rule "\<^bold>\<equiv>I")

  lemma S5Basic_8[PLM]:
    "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> in v]"
    using S5Basic_6[where \<phi>="\<^bold>\<not>\<phi>", THEN contraposition_1[THEN iffD1], deduction]
          KBasic2_11[equiv_lr] CP unfolding diamond_def by auto
  lemmas "4\<^bold>\<diamond>" = S5Basic_8

  lemma S5Basic_9[PLM]:
    "[\<^bold>\<diamond>\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<diamond>\<phi> in v]"
    using "4\<^bold>\<diamond>" "T\<^bold>\<diamond>" by (rule "\<^bold>\<equiv>I")

  lemma S5Basic_10[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>) \<^bold>\<equiv> (\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>) in v]"
    apply (rule "\<^bold>\<equiv>I")
     apply (PLM_subst_goal_method "\<lambda> \<chi> . \<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<or> \<chi>)" "\<^bold>\<diamond>\<^bold>\<box>\<psi>")
      using S5Basic_2[equiv_sym] apply simp
     using KBasic2_12 apply assumption
    apply (PLM_subst_goal_method "\<lambda> \<chi> .(\<^bold>\<box>\<phi> \<^bold>\<or> \<chi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<box>\<psi>)" "\<^bold>\<box>\<^bold>\<box>\<psi>")
     using S5Basic_7[equiv_sym] apply simp
    using KBasic2_7 by auto

  lemma S5Basic_11[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>) \<^bold>\<equiv> (\<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>) in v]"
    apply (rule "\<^bold>\<equiv>I")
     apply (PLM_subst_goal_method "\<lambda> \<chi> . \<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<or> \<chi>)" "\<^bold>\<diamond>\<^bold>\<diamond>\<psi>")
      using S5Basic_9 apply simp
     using KBasic2_12 apply assumption
    apply (PLM_subst_goal_method "\<lambda> \<chi> .(\<^bold>\<box>\<phi> \<^bold>\<or> \<chi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<or> \<^bold>\<diamond>\<psi>)" "\<^bold>\<box>\<^bold>\<diamond>\<psi>")
     using S5Basic_3[equiv_sym] apply simp
using_7byassumption

  lemma S5Basic_12[LM
    "[\<^bold>\<diamond>(\<phi> \<^bold>& \<^bold>\<diamond>\<psi>) \<^bold>\<equiv> (\<^bold>\<diamond>\<phi> \<^bold>& \<^bold>\<diamond>\<psi>) in v]"
    proof -
      have "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>\<psi>)) \<^bold>\<equiv> (\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>\<psi>)) in v]"
        using S5Basic_10 by auto
      hence 1: "[(\<^bold>\<not>\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>\<psi>))) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>\<psi>)) in v]"
        using oth_class_taut_5_d[equiv_lr] by auto
      have 2: "[(\<^bold>\<diamond>(\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>\<or> (\<^bold>\<not>(\<^bold>\<diamond>\<psi>))))) \<^bold>\<equiv> (\<^bold>\<not>((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>\<or> (\<^bold>\<not>(\<^bold>\<diamond>\<psi>)))) in v]"
        apply (PLM_subst_method "\<^bold>\<box>\<^bold>\<not>\<psi>" "\<^bold>\<not>\<^bold>\<diamond>\<psi>")
         using KBasic2_4 apply simp
        apply (PLM_subst_method "\<^bold>\<box>\<^bold>\<not>\<phi>" "\<^bold>\<not>\<^bold>\<diamond>\<phi>")
         using KBasic2_4 apply simp
        apply (PLM_subst_method "(\<^bold>\<not>\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>\<psi>)))" "(\<^bold>\<diamond>(\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>\<or> (\<^bold>\<box>(\<^bold>\<not>\<psi>)))))")
         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
      show ?thesis
        apply (PLM_subst_method "\<^bold>\<not>((\<^bold>\<not>\<phi>) \<^bold>\<or> (\<^bold>\<not>\<^bold>\<diamond>\<psi>))" "\<phi> \<^bold>& \<^bold>\<diamond>\<psi>")
         using oth_class_taut_6_a[equiv_sym] apply simp
        apply (PLM_subst_method "\<^bold>\<not>((\<^bold>\<not>(\<^bold>\<diamond>\<phi>)) \<^bold>\<or> (\<^bold>\<not>\<^bold>\<diamond>\<psi>))" "\<^bold>\<diamond>\<phi> \<^bold>& \<^bold>\<diamond>\<psi>")
         using oth_class_taut_6_a[equiv_sym] apply simp
        using 2 by assumption
    qed

  lemma S5Basic_13[PLM]:
    "[\<^bold>\<diamond>(\<phi> \<^bold>& (\<^bold>\<box>\<psi>)) \<^bold>\<equiv> (\<^bold>\<diamond>\<phi> \<^bold>& (\<^bold>\<box>\<psi>)) in v]"
    apply (PLM_subst_method "\<^bold>\<diamond>\<^bold>\<box>\<psi>" "\<^bold>\<box>\<psi>")
     using S5Basic_2[equiv_sym] apply simp
    using S5Basic_12 by simp

  lemma S5Basic_14[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> (\<^bold>\<box>\<psi>)) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) in v]"
      moreoverS5Basic_4 vdash_properties_10CPby tis
        have "\<And>v.[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
          proof (rule CP)
            fix v
            assume "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box) in
        <><diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<^bold>\<box>\<psi> in v]"
              using "K\<^bold>\<diamond>"[deduction] by auto
            thus "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
              using "B\<^bold>\<diamond>" ded_thm_cor_3 by blast
          qed
        hence "[\<^bold>\<box>(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>)) in v]"
          by (rule RN)
        hence "[\<^bold>\<box>(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)) \<^bold>\<rightarrow> \<^bold>\<box>((\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>)) in v]    proof (rule derived_S5_rules_1_b
          using qml_1[axiom_instance, deduction] by auto
      }
      ultimately show "[\<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
        using S5Basic_6 CP vdash_properties_10 by meson
    next
       have<bold>diamond>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> (\<phi> \<alpha>) in v]"
      moreover {
        fix v
        {
          assume "[\<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
          hence 1: "[\<^bold>\<box>\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
            using qml_1[axiom_instance, deduction] by auto
          assume "[\<phi> in v]"
          hence "[\<^bold>\<box>\<^bold>\<diamond>\<phi> in v]"
            using S5Basic_4[deduction] by auto
          hence "[\<^bold>\<box>\<psi> in v]"
            using 1[deduction] by auto
        }
        hence "[\<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi>) in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
          using CP by auto
      }
      ultimately show "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>) in v]"
        using S5Basic_6 RN_2 vdash_properties_10 by blast
    qed

  lemma sc_eq_box_box_1[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>\<rightarrow> (\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<phi>) in v]"
    proof(rule CP)
assume\bold\box\<>\^><><^><><>inv]
      hence "[\<^bold>\<box>(\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<phi>) in v]"
        using S5Basic_14[equiv_lr] by auto
      hence "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<phi> in v]"
        using qml_2[axiom_instance, deduction] by auto
      moreover from 1 have "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi> in v]"
        using qml_2[axiom_instance, deduction] by auto
      ultimately have "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi> in v]"
        using ded_thm_cor_3 by blast
      moreover have "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> in v]"
        using qml_2[axiom_instance] "T\<^bold>\<diamond>"
        by (rule ded_thm_cor_3)
      ultimately show "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<phi> in v]"
        by (rule "\<^bold>\<equiv>I")
    qed

  lemma sc_eq_box_box_2[PLM]:
    "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>\<rightarrow> ((\<^bold>\<not>\<^bold>\<box>\<phi>) \<^bold>\<equiv> (\<^bold>\<box>(\<^bold>\<not>\<phi>))) in v]"
    proof (rule CP)
      assume "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) in v]"
      hence "[(\<^bold>\<not>\<^bold>\<box>(\<^bold>\<not>\<phi>)) \<^bold>\<equiv> \<^bold>\<box>\<phi> in v]"
        using sc_eq_box_box_1[deduction] unfolding diamond_def by auto
      thus "[((\<^bold>\<not>\<^bold>\<box>\<phi>) \<^bold>\<equiv> (\<^bold>\<box>(\<^bold>\<not>\<phi>))) in v]"
        by (meson CP "\<^bold>\<equiv>I" "\<^bold>\<equiv>E"(3)
                  "\<^bold>\<equiv>E"(4) "\<^bold>\<not>\<^bold>\<not>I" "\<^bold>\<not>\<^bold>\<not>E")
    qed

  lemma sc_eq_box_box_3[PLM]:
    "[(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)) \<^bold>\<rightarrow> ((\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>)) in v]"
    proof (rule CP)
      assume 1: "[(\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>)) in v]"
      {
        assume "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi> in v]"
        hence "[(\<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi>) \<^bold>\<or> ((\<^bold>\<not>(\<^bold>\<box>\<phi>)) \<^bold>& (\<^bold>\<not>(\<^bold>\<box>\<psi>))) in v]"
          using oth_class_taut_5_i[equiv_lr] by auto
        moreover {
          assume "[\<^bold>\<box>\<phi> \<^bold>& \<^bold>\<box>\<psi> in v]"
          hence "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
            using KBasic_7[deduction] by auto
        }
        moreover {
          assume "[(\<^bold>\<not>(\<^bold>\<box>\<phi>)) \<^bold>& (\<^bold>\<not>(\<^bold>\<box>\<psi>)) in v]"
          hence "[\<^bold>\<box>(\<^bold>\<not>\<phi>) \<^bold>& \<^bold>\<box>(\<^bold>\<not>\<psi>) in v]"
             using 1 "\<^bold>&E" "\<^bold>&I" sc_eq_box_box_2[deduction, equiv_lr]
             by metis
          hence "[\<^bold>\<box>((\<^bold>\<not>\<phi>) \<^bold>& (\<^bold>\<not>\<psi>)) in v]"
            using KBasic_3[equiv_rl] by auto
          hence "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
            using KBasic_9[deduction] by auto
        }
        ultimately have "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
          using CP "\<^bold>\<or>E"(1) by blast
      }
      thus "[\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi> \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
        using CP by auto
    qed

  lemma derived_S5_rules_1_a[PLM]:
    assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
    shows "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
    proof -
      have "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<box>\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
        using assms RM_1_b by metis
      thus "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
        using S5Basic_4 vdash_properties_10 CP by metis
    qed

  lemma derived_S5_rules_1_b[PLM]:
    assumes "\<And>v. [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
    shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
    using derived_S5_rules_1_a all_self_eq_1 assms by blast

  lemma derived_S5_rules_2_a[PLM]:
    assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
    shows "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
    proof -
      have "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<^bold>\<box>\<psi> in v]"
        using RM_2_b assms by metis
      thus "[\<^bold>\<box>\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
        using "B\<^bold>\<diamond>" vdash_properties_10 CP by metis
    qed

  lemma derived_S5_rules_2_b[PLM]:
    assumes "\<And>v. [\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
    shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"
    using assms derived_S5_rules_2_a all_self_eq_1 by blast

  lemma BFs_1[PLM]: "[(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
    proof (rule derived_S5_rules_1_b)
      fix v
      {
        fix \<alpha>
        have "\<And>v.[(\<^bold>\<forall>\<alpha> . \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<alpha>) in v]"
          using cqt_orig_1 by metis
        hence "[\<^bold>\<diamond>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> \<^bold>\<diamond>\<^bold>\<box>(\<phi> \<alpha>) in v]"
          using RM_2 by metis
        moreover have "[\<^bold>\<diamond>\<^bold>\<box>(\<phi> \<alpha>) \<^bold>\<rightarrow> (\<phi> \<alpha>) in v]"
          using "B\<^bold>\<diamond>" by auto
        ultimately have "[\<^bold>\<diamond>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> (\<phi> \<alpha>) in v]"
          using ded_thm_cor_3 by blast
      }
      hence "[\<^bold>\<forall> \<alpha> . \<^bold>\<diamond>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> (\<phi> \<alpha>) in v]"
        using "\<^bold>\<forall>I" by metis
      thus "[\<^bold>\<diamond>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
        using cqt_orig_2[deduction] by auto
    qed
  lemmas BF = BFs_1

  lemma BFs_2[PLM]:
    "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) in v]"
    proof -
      {
        fix \<alpha>
        {
           fix v
           have "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<alpha> in v]" using cqt_orig_1 by metis
        }
        hence "[\<^bold>\<box>(\<^bold>\<forall>\<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<alpha>) in v]" using RM_1 by auto
      }
      hence "[\<^bold>\<forall>\<alpha> . \<^bold>\<box>(\<^bold>\<forall>\<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<alpha>) in v]" using "\<^bold>\<forall>I" by metis
      thus ?thesis using cqt_orig_2[deduction] by metis
    qed
  lemmas CBF = BFs_2

  lemma BFs_3[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<exists> \<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists> \<alpha> . \<^bold>\<diamond>(\<phi> \<alpha>)) in v]"
    proof -
      have "[(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>))) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)) in v]"
        using BF by metis
      hence 1: "[(\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)))) \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))) in v]"
        using contraposition_1 by simp
      have 2: "[\<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>))) \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))) in v]"
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>\<alpha> . \<^bold>\<not>(\<phi> \<alpha>))" "\<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)))")
        using KBasic2_2 1 by simp+
      have "[\<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>))) \<^bold>\<rightarrow> (\<^bold>\<exists> \<alpha> . \<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))) in v]"
        apply (PLM_subst_method "\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))" "\<^bold>\<exists> \<alpha>. \<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))")
         using cqt_further_2 apply metis
        using 2 by metis
      thus ?thesis
        unfolding exists_def diamond_def by auto
    qed
  lemmas "BF\<^bold>\<diamond>" = BFs_3

  lemma BFs_4[PLM]:
    "[(\<^bold>\<exists> \<alpha> . \<^bold>\<diamond>(\<phi> \<alpha>)) \<^bold>\<rightarrow> \<^bold>\<diamond>(\<^bold>\<exists> \<alpha>. \<phi> \<alpha>) in v]"
    proof -
      have 1: "[\<^bold>\<box>(\<^bold>\<forall>\<alpha> . \<^bold>\<not>(\<phi> \<alpha>)) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>))) in v]"
        using CBF by auto
      have 2: "[(\<^bold>\<exists> \<alpha> . (\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>))))) \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)))) in v]"
        apply (PLM_subst_method "\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))" "(\<^bold>\<exists> \<alpha> . (\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>)))))")
         using cqt_further_2 apply blast
        using 1 using contraposition_1 by metis
      have "[(\<^bold>\<exists> \<alpha> . (\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<not>(\<phi> \<alpha>))))) \<^bold>\<rightarrow> \<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall> \<alpha> . \<^bold>\<not>(\<phi> \<alpha>))) in v]"
        apply (PLM_subst_method "\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)))" "\<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>(\<phi> \<alpha>)))")
         using KBasic2_2 apply blast
        using 2 by assumption
      thus ?thesis
        unfolding diamond_def exists_def by auto
    qed
  lemmas "CBF\<^bold>\<diamond>" = BFs_4

  lemma sign_S5_thm_1[PLM]:
    "[(\<^bold>\<exists> \<alpha>. \<^bold>\<box>(\<phi> \<alpha>)) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<exists> \<alpha>. \<phi> \<alpha>) in v]"
    proof (rule CP)
      assume "[\<^bold>\<exists>  \<alpha> . \<^bold>\<box>(\<phi> \<alpha>) in v]"
      then obtain \<tau> where "[\<^bold>\<box>(\<phi> \<tau>) in v]"
        by (rule "\<^bold>\<exists>E")
      moreover {
        fix v
        assume "[\<phi> \<tau> in v]"
        hence "[\<^bold>\<exists> \<alpha> . \<phi> \<alpha> in v]"
          by (rule "\<^bold>\<exists>I")
      }
      ultimately show "[\<^bold>\<box>(\<^bold>\<exists>  \<alpha> . \<phi> \<alpha>) in v]"
        using RN_2 by blast
    qed
  lemmas Buridan = sign_S5_thm_1

  lemma sign_S5_thm_2[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<forall> \<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall> \<alpha> . \<^bold>\<diamond>(\<phi> \<alpha>)) in v]"
    proof -
      {
        fix \<alpha>
        {
          fix v
          have "[(\<^bold>\<forall> \<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<alpha> in v]"
            using cqt_orig_1 by metis
        }
        hence "[\<^bold>\<diamond>(\<^bold>\<forall> \<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<diamond>(\<phi> \<alpha>) in v]"
          using RM_2 by metis
      }
      hence "[\<^bold>\<forall> \<alpha> . \<^bold>\<diamond>(\<^bold>\<forall> \<alpha> . \<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<diamond>(\<phi> \<alpha>) in v]"
        using "\<^bold>\<forall>I" by metis
      thus ?thesis
        using cqt_orig_2[deduction] by metis
    qed
  lemmas "Buridan\<^bold>\<diamond>" = sign_S5_thm_2

  lemma sign_S5_thm_3[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<exists> \<alpha> . \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<diamond>((\<^bold>\<exists> \<alpha> . \<phi> \<alpha>) \<^bold>& (\<^bold>\<exists> \<alpha> . \<psi> \<alpha>)) in v]"
    by (simp only: RM_2 cqt_further_5)

  lemma sign_S5_thm_4[PLM]:
    "[((\<^bold>\<box>(\<^bold>\<forall> \<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>)) \<^bold>&</span> (\<^bold>\<box>(\<^bold>\<forall> \<alpha> . \<psi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>))) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) in v]"
    proof (rule CP)
      assume "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) in v]"
      hence "[\<^bold>\<box>((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>)) in v]"
        using KBasic_3[equiv_rl] by blast
      moreover {
        fix v
        assume "[((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>)) in v]"
        hence "[(\<^bold>\<forall> \<alpha> . \<phi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) in v]"
          using cqt_basic_9[deduction] by blast
      }
      ultimately show "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<rightarrow> \<chi> \<alpha>) in v]"
        using RN_2 by blast
    qed

  lemma sign_S5_thm_5[PLM]:
    "[((\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>)) \<^bold>&> (\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>))) \<^bold>\<rightarrow> (\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>)) in v]"
    proof (rule CP)
      assume "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>&pan> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>) in v]"
      hence "[\<^bold>\<box>((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>&pan> (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>)) in v]"
        using KBasic_3[equiv_rl] by blast
      moreover {
        fix v
        assume "[((\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>)) in v]"
        hence "[(\<^bold>\<forall> \<alpha> . \<phi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>) in v]"
          using cqt_basic_10[deduction] by blast
      }
      ultimately show "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<chi> \<alpha>) in v]"
        using RN_2 by blast
    qed
  
  lemma id_nec2_1[PLM]:
    "      \<> \^old><ambda z .<>R,<sup>,<supPrparr\^> \<>\lambda   <>R,\^sup>,\^>P\rparr>  ]"
    apply (rule "\<^bold>\<equiv>I"; rule CP)
     using id_nec[equiv_lr] derived_S5_rules_2_b CP modus_ponens apply blast
    using "T\<^bold>\<diamond>"[deduction] by auto

  lemma id_nec2_2_Aux:
    "[(\<^bold>\<diamond>\<phi>) \<^bold>\<equiv> \<psi> in v] \<Longrightarrow> [(\<^bold>\<not>\<psi>) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<not>\<phi>) in v]"
    proof -
      assume "[(\<^bold>\<diamond>\<phi>) \<^bold>\<equiv> \<psi> in v]"
        "<nd>\phi>\<>.[\bold><>phi)<bold\equiv \psi  ]\Longrightarrow[\^bold>not>\psi)\^bold>\<equiv> \<phi> in v]"
        by PLM_solver
      ultimately show ?thesis
        unfolding diamond_def by blast
    qed

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

  lemmaid_nec2_3[]:
    "[(\<^bold>\<diamond>((\<alpha>::'a::id_eq) \<^bold>\<noteq> \<beta>)) \<^bold>\<equiv> (\<alpha> \<^bold>\<noteq> \<beta>) in v]"
    using "T\<^bold>\<diamond>" "\<^bold>\<equiv>I" id_nec2_2[equiv_lr]
          CP derived_S5_rules_2_b by metis

  lemma exists_desc_box_1[PLM]:
    "[(\<^bold>\<exists> y . (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x)) \<^bold>\<rightarrow> (\<^bold>\<exists> y . \<^bold>\<box>((y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x))) in v]"
    proof (rule CP)
      assume "[\<^bold>\<exists>y. (y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
      then obtain y where "[(y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x) in v]"
        by (rule "\<^bold>\<exists>E")
      hence "[\<^bold>\<box>(y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x)) in v]"
        using l_identity[axiom_instance, deduction, deduction]
              cqt_1[axiom_instance] all_self_eq_2[where 'a=\<nu>]
              modus_ponens unfolding identity_\<nu>_def by fast
      thus "[\<^bold>\<exists>y. \<^bold>\<box>((y\<^sup>P) \<^bold>= (\<^bold>\<iota>x. \<phi> x)) in v]"
        by (rule "\<^bold>\<exists>I")
    qed

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

  lemma en_eq_1[PLM]:
    "[\<^bold>\<diamond>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>x,F\<rbrace> in v]"
    using encoding[axiom_instance] RN
          sc_eq_box_box_1 modus_ponens by blast
  lemma en_eq_2[PLM]:
    "[\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>x,F\<rbrace> in v]"
    using encoding[axiom_instance] qml_2[axiom_instance] by (rule "\<^bold>\<equiv>I")
  lemma en_eq_3[PLM]:
    "[\<^bold>\<diamond>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>x,F\<rbrace> in v]"
    using encoding[axiom_instance] derived_S5_rules_2_b "\<^bold>\<equiv>I" "T\<^bold>\<diamond>" by auto
  lemma en_eq_4[PLM]:
    "[(\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace>) \<^bold>\<equiv> (\<^bold>\<box>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>y,G\<rbrace>) in v]"
    by (metis CP en_eq_2 "\<^bold>\<equiv>I" "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2))
  lemma en_eq_5[PLM]:
    "[\<^bold>\<box>(\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace>) \<^bold>\<equiv> (\<^bold>\<box>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<lbrace>y,G\<rbrace>) in v]"
    using "\<^bold>\<equiv>I" KBasic_6 encoding[axiom_necessitation, axiom_instance]
    sc_eq_box_box_3[deduction] "\<^bold>&I" by simp
  lemma en_eq_6[PLM          using \<heta>[conj2, THEN "\^bold>\<orall>E" THENoth_class_taut_5_d[equiv_lr,equiv_lr
    "[(\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>y,G\<rbrace>) \<^bold>\<equiv> \<^bold>
    en_eq_4 oth_class_taut_4_a "<\equivE() by meson
  lemma en_eq_7[PLM]:
    "[(\<^bold>\<not>\<lbrace>x,F\<rbrace>) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<not>\<lbrace>x,F\<rbrace>) in v]"
    using en_eq_3[THEN id_nec2_2_Aux] by blast
  lemma en_eq_8[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<not>\<lbrace>x,F\<rbrace>) \<^bold>\<equiv> (\<^bold>\<not>\<lbrace>x,F\<rbrace>) in v]"
     unfolding diamond_def apply (PLM_subst_method "\<lbrace>x,F\<rbrace>" "\<^bold>\<not>\<^bold>\<not>\<lbrace>x,F\<rbrace>")
      using oth_class_taut_4_b apply simp
     _thod"<lbrace>x,F<rbrace>""\<^>\<box>\<lbrace>x,F\<rbrace>")
      using en_eq_2 apply simp
     using oth_class_taut_4_a by assumption
  lemma en_eq_9[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<not>\<lbrace>x,F\<rbrace>) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<not>\<lbrace>x,F\<rbrace>) in v]"
    using en_eq_8 en_eq_7 "\<^bold>\<equiv>E"(5) by blast
  lemma en_eq_10[PLM]:
    "[\<^bold>\<A>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>x,F\<rbrace> in v]"
    apply (rule "\<^bold>\<equiv>I")
     using encoding[axiom_actualization, axiom_instance,
                    THEN logic_actual_nec_2[axiom_instance, equiv_lr],
                    deduction, THEN qml_act_2[axiom_instance, equiv_rl],
                    THEN en_eq_2[equiv_rl]] CP
     apply simp
    using encoding[axiom_instance] nec_imp_act ded_thm_cor_3 by blast

subsection\<open>The Theory of Relations\<close> 
text\<open>\label{TAO_PLM_Relations}\<close>

  lemma beta_equiv_eq_1_1[PLM]:
    assumes "IsProperInX \<phi>"
        and "IsProperInX \<psi>"
        and "\<And>x.[\<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) in v]"
    shows "[\<lparr>\<^bold>\<lambda> y. \<phi> (y\<^sup>P), x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda> y. \<psi> (y\<^sup>P), x\<^sup>P\<rparr> in v]"
    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 "\<^bold>\<equiv>E"(6) oth_class_taut_4_a)

  lemma beta_equiv_eq_1_2[PLM]:
    assumes "IsProperInXY \<phi>"
        and "IsProperInXY \<psi>"
        and "\<And>x y.[\<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) in v]"
    shows "[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<phi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr>
            \<^bold>\<
    using lambda_predicates_2_2[OF assms(x\<in>U. eventually (<>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
    using lambda_predicates_2_2[OF assms(2), axiom_instance]
    using assms(3) by (meson "\<^bold>\<equiv>E"(6) oth_class_taut_4_a)

  lemma beta_equiv_eq_1_3[PLM]:
    assumes"perInXYZphi
        and "IsProperInXYZ \<psi>"
        and "\<And
showslparr>\<^bold>\<lambda>\<^sup>3 (<> x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr>
            \<^ add_hyperdual_parts:
    using [F(,iom_instance
    using lambda_predicates_2_3[OF assms(2), axiom_instance]
    using assms(3) by (meson "\<^E"(6) oth_class_taut_4_a)

  lemma beta_equiv_eq_2_1[PLM]:
  sIsProperInXInX <"
        and "IsProperInX \<psi>"
    shows "[(\<^bold>\<box>(\<^bold>\<forall> x . \<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>Ptext\open
            (\<^bold>\<box>(\<^bold>\<forall> x . \<lparr>\<^bold>\<lambda> y. \<phi> (y\<^sup>P), x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda> y. \<psi> (y\<^sup>P
     apply (rule qml_1[axiom_instance, deduction])
     apply (rule RN)
     proof (rule CP, rule "\<^bold>\<forall>I")
      fix v x
      assume "[\<^bold>\<forall>x. \<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) in v]"
      hence "\<And>x.[\<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) in v]"
        by PLM_solver
      thus "[\<lparr>\<^bold>\<lambda> y. \<phi> (y\<^sup>P), x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda> y. \<psi> (y\<^sup>P), x\<^sup>P\<rparr> in v]"
        using assms beta_equiv_eq_1_1 by auto
     qed

  lemma beta_equiv_eq_2_2[PLM]:
    assumes "IsProperInXY \<phi>"
        and "IsProperInXY \<psi>"
    shows "[(\<^bold>\<box>(\<^bold>\<forall> x y . \<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P))) \<^bold>\<rightarrow>
            (\<^bold>\<box>(\<^bold>\<forall> x y . \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<phi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr>
              \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<psi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr>)) in v]"
    apply (rule qml_1[axiom_instance, deduction])
apply )
    proof (rule CP, rule "\<^bold>\<forall>I", rule "\<^bold>\<forall>I")
      fix v x y
      assume "[\<^bold>\<forall>x y. \<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) in v]"
      hence "(\<And>x y.[\<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) in v])"
        by (meson "\<^bold>\<forall>E")
      thus "[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<phi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr>
            \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<psi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr> in v]"
        using assms beta_equiv_eq_1_2 by auto
    

  lemma beta_equiv_eq_2_3[PLM]:
    assumes "IsProperInXYZ \<phi>"
        and "IsProperInXYZ \<psi>"
    shows "[(\<^bold>\<box>(\<^bold>\<forall> x y z . \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P))) \<^bold>\<rightarrow>
            (\<^bold>\<box>(\<^bold>\<forall> x y z . \<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda> x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr>
                \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda> x y z. \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr>)) in v]"
    apply (rule qml_1[axiom_instance, deduction])
    apply (rule RN)
    proof (rule CP, rule "\<^bold>\<forall>I", rule "\<^bold>\<forall>I", rule "\<^bold>\<forall>I")
      fix v x y z
      assume "[\<^bold>\<forall>x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) in v]"
      hence "(\<And>x y z.[\<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) in v])"
        by (meson "\<^bold>\<forall>E")
      thus "[\<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda> x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr>
              \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda> x y z. \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr> in v]"
        using assms beta_equiv_eq_1_3 by auto
    qed

  lemma beta_C_meta_1[PLM]:
ssumes \<phi>
    shows "[\<lparr>\<^>\<lambda> .\phi (\^P), x\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) in v]"
    using lambda_predicates_2_1[OF assms, axiom_instance] by auto

  lemma beta_C_meta_2[PLM]:
    assumes "IsProperInXY \<phi>"
    shows "[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda> x y. \<phi> (x\<^sup>P) (y\<^sup>P)), x\<^sup>P, y\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P) in v]"
    using lambda_predicates_2_2[OF assms, axiom_instance] by auto

  lemma beta_C_meta_3[PLM]:
    assumes "IsProperInXYZ \<phi>"
    shows "[\<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda> x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)), x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) in v]"
    using lambda_predicates_2_3[OF assms, axiom_instance] by auto

  lemma relations_1[PLM]:
    assumes "IsProperInX \<phi>"
    shows "[\<^bold>\<exists> F. \<^bold>\<box>(\<^bold>\<forall> x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P)) in v]"
    using assms apply - by PLM_solver

  lemma relations_2[PLM]:
    assumes "IsProperInXY \<phi>"
    shows "[\<^bold>\<exists> F. \<^bold>\<box>(\<^bold>\<forall> x y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P)) in v]"
    using assms apply - by PLM_solver

  lemma relations_3[PLM]:
    assumes "IsProperInXYZ \<phi>"
    shows "[\<^bold>\<exists> F. \<^bold>\<box>(\<^bold>\<forall> x y z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)) in v]"
    using assms apply - by PLM_solver

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

  lemma propositions_lemma_1[PLM]:
    "[\<^bold>\<lambda>\<^sup>0 \<phi> \<^bold>= \<phi> in v]"
    using lambda_predicates_3_0[axiom_instance] .

  lemma propositions_lemma_2[PLM]:
    "[\<^bold>\<lambda>\<^sup>0 \<phi> \<^bold>\<equiv> \<phi> in v]"
    using lambda_predicates_3_0[axiom_instance, THEN id_eq_prop_prop_8_b[deduction]]
    apply (rule l_identity[axiom_instance, deduction, deduction])
    by PLM_solver

   propositions_lemma_4[PLM]:
assume\> t etpds
    shows "[(\<chi>::\<kappa>\<Rightarrow>\<o>) (\<^bold>\<iota>x. \<phi> x) \<^bold>= \<chi> (\<^bold>\<iota>x. \<psi> x) in v]"
     -
      have "\^bold><\<^sup>0 (\<chi \^>\<iota>.\> x))\<bold= \<^bold>\<lambda>\<^sup>0 (\<chi> (\<^bold>\<iota>x. \<psi> x)) in v]"
        using assms lambda_predicates_4_0[axiom_instance]
        by blast
      [>(\<^bold>\<iota>x. \<phi> x)) \<^bold>= \<^bold>\<lambda>\<^sup>0 (\<chi> (\<^bold>\<iota>x. \<psi> x)) in v]"
        usingpropositions_lemma_1[THEN id_eq_prop_prop_8_bdeduction]]
              id_eq_prop_prop_9_b[deduction] "\<^bold>&I"
        by blast
thus ?thesis
        using propositions_lemma_1 id_eq_prop_prop_9_b[deduction] "\<^bold>&I"
        byjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
    qed

  tions
    "[\<^bold>\<exists> pjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    by PLM_solver

  lemma pos_not_equiv_then_not_eq[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>G,x\<^sup>P\<rparr>)) \<^bold>\<rightarrow> F \<^bold>\<noteq> G in v]"
    unfolding diamond_def
    proof (subst contraposition_1[symmetric], rule CP)
      assume "[F \<^bold>= G in v]"
      thus "[\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>G,x\<^sup>P\<rparr>))) in v]"
        apply (rule l_identity[axiom_instance, deduction, deduction])
        by PLM_solver
    qed

  lemma thm_relation_negation_1_1[PLM]:
    "[\<lparr>F\<^sup>-, x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F, x\<^sup>P\<rparr> in v]"
    unfolding propnot_defs
    apply (rule lambda_predicates_2_1[axiom_instance])
    by show_proper

  lemma thm_relation_negation_1_2[PLM]:
    "[\<lparr>F\<^sup>-, x\<^sup>P, y\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F, x\<^sup>P, y\<^sup>P\<rparr> in v]"
    unfolding propnot_defs
    apply (rule lambda_predicates_2_2[axiom_instance])
    by show_proper

  lemma thm_relation_negation_1_3[PLM]:
    "[\<lparr>F\<^sup>-, x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F, x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr> in v]"
    unfolding propnot_defs
    apply (rule lambda_predicates_2_3[axiom_instance])
    by show_proper

  lemma thm_relation_negation_2_1[PLM]:
    "[(\<^bold>\<not>\<lparr>F\<^sup>-, x\<^sup>P\<rparr>) \<^bold>\<equiv> \<lparr>F, x\<^sup>P\<rparr> in v]"
    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]:
    "[(\<^bold>\<not>\<lparr>F\<^sup>-, x\<^sup>P, y\<^sup>P\<rparr>) \<^bold>\<equiv> \<lparr>F, x\<^sup>P, y\<^sup>P\<rparr> in v]"
    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]:
    "[(\<^bold>\<not>\<lparr>F\<^sup>-, x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr>) \<^bold>\<equiv> \<lparr>F, x\<^sup>P, y\<^sup>P, z\<^sup>P\<rparr> in v]"
    using thm_relation_negation_1_3[THEN oth_class_taut_5_d[equiv_lr]]
    apply - by PLM_solver

  lemma thm_relation_negation_3[PLM]:
    "[(p)\<^sup>- \<^bold>\<equiv> \<^bold>\<not>p in v]"
    unfolding propnot_defs
    using propositions_lemma_2 by simp

  lemma thm_relation_negation_4[PLM]:
    "[(\<^bold>\<not>((p::\<o>)\<^sup>-)) \<^bold>\<equiv> p in v]"
    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::\<Pi>\<^sub>1) \<^bold>\<noteq> (F\<^sup>-) in v]"
    using id_eq_prop_prop_2[deduction]
          l_identity[where \<phi>="\<lambda> G . \<lparr>G,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F\<^sup>-^>\<rparr>", axiom_instance,
                      deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_1 "\<^bold>\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_5_2[PLM]:
    "[(F::\<Pi>\<^sub>2) \<^bold>\<noteq> (F\<^sup>-) in v]"
    using id_eq_prop_prop_5_a[deduction]
          l_identity[where \<phi>="\<lambda> G . \<lparr>G,x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F\<^sup>-,x\<^sup>P,y\<^sup>P\<rparr>", axiom_instance,
                      deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_2 "\<^bold>\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_5_3[PLM]:
    "[(F::\<Pi>\<^sub>3) \<^bold>\<noteq> (F\<^sup>-) in v]"
    using id_eq_prop_prop_5_b[deduction]
          l_identity[where \<phi>="\<lambda> G . \<lparr>G,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F\<^sup>-,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>",
                     axiom_instance, deduction, deduction]
          oth_class_taut_4_a thm_relation_negation_1_3 "\<^bold>\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_6[PLM]:
    "[(p::\<o>) \<^bold>\<noteq> (p\<^sup>-) in v]"
    using id_eq_prop_prop_8_b[deduction]
          l_identity[where \<phi>="\<lambda> G . G \<^bold>\<equiv> (p\<^sup>-)", axiom_instance,
                      deduction      show ""[\<^old\<not>\bold>>\<forall>x. \\lparr>A!x<^up>P<parr> in v"
          oth_class_taut_4_a thm_relation_negation_3 "\<^bold>\<equiv>E"(5)
          oth_class_taut_1_b modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_7[PLM]:
    "[((p::\<o>)\<^sup>-) \<^bold>= \<^bold>\<not>p in v]"
    unfolding propnot_defs using propositions_lemma_1 by simp

  lemma thm_relation_negation_8[PLM]:
    "[(p::\<o>) \<^bold>\<noteq> \<^bold>\<not>p in v]"
    unfolding propnot_defs 
    using id_eq_prop_prop_8_b[deduction]
          l_identity[where \<phi>="\<lambda> G . G \<^bold>\<equiv> \<^bold>\<not>(p)", axiom_instance,
                      deduction, deduction]
          oth_class_taut_4_a oth_class_taut_1_b
          modus_tollens_1 CP
    by meson

  lemma thm_relation_negation_9[PLM]:
    "[((p::\<o>) \<^bold>= q) \<^bold>\<rightarrow> ((\<^bold>\<not>p) \<^bold>= (\<^bold>\<not>q)) in v]"
    using l_identity[where \<alpha>="p" and \<beta>="q" and \<phi>="\<lambda> x . (\<^bold>\<not>p) \<^bold>= (\<^bold>\<not>x)",
            , deduction]
          id_eq_prop_prop_7_b using CP modus_ponens by blast

  lemma thm_relation_negation_10[PLM]:
    "[((p::\<o>) \<^bold>= q) \<^bold>\<rightarrow> ((p\<^sup>-) \<^bold>= (q\<^sup>-)) in v]"
    here>"p" and \<beta>="q" and \phi>\<lambda x . p<sup>-\^bold>= (\^>-)",
                      axiom_instance, deduction]
          id_eq_prop_prop_7_b using CP modus_ponens by blast

  lemma thm_cont_prop_1[PLM]:
    "[NonContingent (F::\<Pi>\<^sub>1) \<^bold>\<equiv> NonContingent (F\<^sup>-) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[NonContingent F in v]"
      hence "[\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
      hence "[\<^    fix f java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
        apply  have "\<> \lesssim>\<^bsup>M
        apply (PLM_subst_method "\<lambda> x . \<lparr>F,x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>")
        using thm_relation_negation_2_1[equiv_sym] by auto
      hence "[\<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F\<^sup>-,x\<^sup>P\<rparr>) in v]"
        apply -
        apply (PLM_subst_goal_method
               "\<lambda> \<phi> . \<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x. \<phi> x)" "\<lambda> x . \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>")
        using thm_relation_negation_1_1[equiv_sym] by auto
      hence "[\<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F\<^sup>-,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) in v]"
        by (rule oth_class_taut_3_e[equiv_lr])
      thus "[NonContingent (F\<^sup>-) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    next
      assume "[NonContingent (F\<^sup>-) in v]"
      hence "[\<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F\<^sup>-,x\<^sup>P\<rparr>) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs
        by (rule oth_class_taut_3_e[equiv_lr])
      hence "[\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F\<^sup>-,x\<^sup>P\<rparr>) in v]"
        apply -
        apply (PLM_subst_method  "\<lambda> x . \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>" "\<lambda> x . \<lparr>F,x\<^sup>P\<rparr>")
        using thm_relation_negation_2_1 by auto
      hence "[\<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) in v]"
     
        applyLM_subst_method\lambda>x .\lparrF\<^sup>-,x\<^sup>P\<rparr>" "\<lambda> x . \<>\<not>\<lparr>F,<sup>P\<rparr>)
using thm_relation_negation_1_1 by auto
      thus "[NonContingent F in v]"
         NonContingent_def Necessary_defs Impossible_defs .
    

  ( A)
    "[Contingent F \^bold>\<quiv\<bold<diamond>(<bold>\<xistsx.\<parrF,x\<^sup>P\<rparr>) \<^bold>&n> \<^bold>\<diamond>(\<^bold>\<exists> x . \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule case_nat  ) f"
      assume "[Contingent F in v]"
      hence "[\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr
        unfolding Contingent_def using Sucems cal orce
      hence "[(\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<not>\<^bold>\<box>(\<^bold>         FEx show ?case
        
      hence "[(\<^bold>\<diamond>\<^bold>\<not>(\<^bold>using is_path_f blast
        using KBasic2_2[
      thus "[(\<^bold>\<diamond>(\<^bold>\<exists> x.\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<diamond>(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        unfolding exists_def apply -
        apply (PLM_subst_method "\<lambda> x . \<lparr>F,x\<^sup>P\<rparr>" "\<lambda> x . \<^byer
        using oth_class_taut_4_b by auto
     <((concat \<circ> map subs) ^^ m) (concat (map subs l)) = ((concat \<circ> map subs) ^^ Suc m) l\<close>
      assume "[(\<^bold>\<diamond>(\<^bold>\<exists> x.\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<diamond>(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
      hence "[(\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        unfolding exists_def apply -
        apply (PLM_subst_goal_method
               "\<lambda> \<phi> . (\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<forall>x. \<phi> x))" "\<lambda> x . \<^bold>\<not>\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>")
        using oth_class_taut_4_b[equiv_sym] by auto
      hence "[(\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>& (\<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        using KBasic2_2[equiv_rl] "\<^bold>&I" "\<^bold>&E" by meson
      hence "[\<^bold>\<not>(\<^bold>\<box>(\<^bold>\<forall>x.\<lparr>F,x\<^sup>P\<rparr>) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<forall>x.\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) 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::\<Pi>\<^sub>1) \<^bold>\<equiv> Contingent (F\<^sup>-) 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]:
    "[\<^bold>\<diamond>(\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>))) \<^bold>\<equiv> \<^bold>\<diamond>(\<^bold>\<exists> x . ((\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>))in v]"
    proof -
      have "[\<^bold>\<diamond>(\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>))) in v]
             = [(\<^bold>\<exists> x . \<^bold>\<diamond>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>))) in v]"
        using "BF\<^bold>\<diamond>"[deduction] "CBF\<^bold>\<diamond>"[deduction] by fast
      also have "... = [\<^bold>\<exists> x . (\<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        apply (PLM_subst_method
               "\<lambda> x . \<^bold>\<diamond>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>))"
               "\<lambda> x . \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)")
        using S5Basic_12 by auto
      also have "... = [\<^bold>\<exists> x . \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> in v]" 
        apply (PLM_subst_method
               "\<lambda> x . \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)"
               "\<lambda> x . \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>")
        using oth_class_taut_3_b by auto
      also have "... = [\<^bold>\<exists> x . \<^bold>\<diamond>((\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>) in v]"
        apply (PLM_subst_method
               "\<lambda> x . \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>"
               "\<lambda> x . \<^bold>\<diamond>((\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>)")
        Basic_12ymyo
      also have "... = [\<^bold>\<diamond> (\<^bold>\<exists> x . ((\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        using "CBF\<^bold>\<diamond>"[deduction] "BF\<^bold>\<diamond>"[deduction] by fast
      finally show ?thesis using "\<^bold>\<equiv>I" CP by blast
    qed

  lemma lem_cont_e_2[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<exists> x . \<lparr>F,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)) \<^bold>\<equiv> \<^bold>\<diamond>(\<^bold>\<exists> x . \<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>)) in v]"
    apply (PLM_subst_method "\<lambda> x . \<lparr>F,x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>")
     using thm_relation_negation_2_1[equiv_sym] apply simp
    apply (PLM_subst_method "\<lambda> x . \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>" "\<lambda> x . \<lparr>F\<^sup>-,x\<^sup>P\<rparr>")
     using thm_relation_negation_1_1[equiv_sym] apply simp
    using lem_cont_e by simp

  lemma thm_cont_e_1[PLM]:
    "[\<^bold>\<diamond>(\<^bold>\<exists> x . ((\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>) \<^bold>& (\^bold>\<diamond>\<lparr>E,<sup><>v
    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] "\<^bold>&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]
    bythus "[\lparr>O!,\^sup>P\rparr>inv"

  lemma thm_cont_e_3[PLM]:
    "[Contingent (E!\<^sup>-) in v]"
    using thm_cont_e_2 thm_cont_prop_3[equiv_lr] by blast

  lemma thm_cont_e_4[PLM]:
    "[\<^bold>\<exists> (F::\<Pi>\<^sub>1) G . (F \<^bold>\<noteq> G \<^bold>& Contingent F \<^bold>&> Contingent G) in v]"
    apply (rule_tac \<alpha>="E!" in "\<^bold>\<exists>I", rule_tac \<alpha>="E!\<^sup>-" in "\<^bold>\<exists>I")
    using thm_cont_e_2 thm_cont_e_3 thm_relation_negation_5_1 "\<^bold>&I" by auto

  context
  begin
    qualified definition L where "L \<equiv> (\<^bold>\<lambda> x . \<lparr>E!, x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>E!, x\<^sup>P\<rparr>)"
    
    lemma thm_noncont_e_e_1[PLM]:
      "[Necessary L in v]"
      unfolding Necessary_defs L_def apply (rule RN, rule "\<^bold>\<forall>I")
      apply (rule lambda_predicates_2_1[axiom_instance, equiv_rl])
        apply show_proper
      using if_p_then_p .

    lemma thm_noncont_e_e_2[PLM]:
      "[Impossible (L\<^sup>-) in v]"
      unfolding Impossible_defs L_def apply (rule RN, rule "\<^bold>\<forall>I")
      apply(ule thm_relation_negation_2_1])
      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 "\<^bold>\<or>I"(1))

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

    lemma thm_noncont_e_e_5[PLM]:
      "[\<^bold>\<exists> (F::\<Pi>\<^sub>1) G . F \<^bold>\<noteq  <> NonContingent F \<^bold>& NonContingent G in v]"
      apply (rule_tac \<alpha>="L" in "\<^bold>\<exists>I", rule_tac \<alpha>="L\<^sup>-" in "\<^bold>\<exists>I")
      using "\<^bold>\<exists>I" thm_relation_negation_5_1 thm_noncont_e_e_3
            thm_noncont_e_e_4 "\<^bold>&I"
      by simp


  lemma four_distinct_1[PLM]:
    "[NonContingent (F::\<Pi>\<^sub>1) \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<exists> G . (Contingent G \<^bold>& G \<^bold>= F)) in v]"
    proof (rule CP)
      assume "[NonContingent F in v]"
      hence "[\<^bold>\<not>(Contingent F) in v]"
        unfolding NonContingent_def Contingent_def
        apply - by PLM_solver
      moreover {
         assume "[\<^bold>\<exists> G . Contingent G \<^bold>& G \<^bold>= F in v]"
         then obtain P where "[Contingent P \<^bold>& P \<^bold>= F in v]"
          by (rule "\<^bold>\<exists>E")
         hence "[Contingent F in v]"
             finally show ?thesis by simp
           \<open>Const\<close> may introduce
      }
       "[\<boldnot(\<^bold>\<exists>G. Contingent G \<^bold>& G \<^bold>= F) in
        using modus_tollens_1 CP by blast
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7

  lemma four_distinct_2[PLM]:
    "[Contingent (F::\<Pi>\<^sub>1) \<^bold>\<rightarrow> \<^bold>\<not>(\<^bold>\<exists> G . (NonContingent G \<^bold>& G \<^bold>= F)) in v]"
    proof (rule CP)
      assume "[Contingent F in v]"
      hence "[\<^bold>\<not>(NonContingent F) in v]"
        unfolding NonContingent_def Contingent_def
        apply - by PLM_solver
      moreover {
         assume "[\<^bold>\<exists> G . NonContingent G \<^bold>& G \<^bold>= F in v]"
         then obtain P where "[NonContingent P \<^bold>& P \<^bold>= F in v]"
          by (rule "\<^bold>\<exists>E")
         hence "[NonContingent F in v]"
           using "\<^bold>&E" l_identity[axiom_instance, deduction, deduction]
           by blast
      }
      ultimately show "[\<^bold>\<not>(\<^bold>\<exists>G. NonContingent G \<^bold>& G \<^bold>= F) in v]"
        using modus_tollens_1 CP by blast
    qed

    lemma four_distinct_3[PLM]:
      "[L \<^bold>\<noteq> (L\<^sup>-) \<^bold>& L \<^bold>\<noteq> E! \<^bold>& L \<^bold>\<noteq> (E!\<^sup>-) \<^bold>& (L\<^sup>-) \<^bold>\<noteq> E!
        \<^bold>& (L\<^sup>-) \<^bold>\<noteq> (E!\<^sup>-) \<^bold>& E! \<^bold>\<noteq> (E!\<^sup>-) in v]"
      proof (rule "\<^bold>&I")+
        show "[L \<^bold>\<noteq> (L\<^sup>-) in v]"
        by (rule thm_relation_negation_5_1)
     
        {
          assume "[L \<^bold>= E! in v]"
          hence "[NonContingent L \<^bold>& L \<^bold>= E! in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" by auto
          hence "[\<^bold>\<exists> G . NonContingent G \<^bold>& G \<^bold>= E! in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" "\<^bold>\<exists>I" by fast
        }
        thus "[L \<^bold>\<noteq> E! in v]"
          using four_distinct_2[deduction, OF thm_cont_e_2]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "[L \<^bold>= (E!\<^sup>-) in v]"
          hence "[NonContingent L \<^bold>& L \<^bold>= (E!\<^sup>-) in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" by auto
          hence "[\<^bold>\<exists> G . NonContingent G \<^bold>& G \<^bold>= (E!\<^sup>-) in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" "\<^bold>\<exists>I" by fast
        }
        thus "[L \<^bold>\<noteq> (E!\<^sup>-) in v]"
          using four_distinct_2[deduction, OF thm_cont_e_3]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "[(L\<^sup>-) \<^bold>= E! in v]"
          hence "[NonContingent (L\<^sup>-) \<^bold>& (L\<^sup>-) \<^bold>= E! in v]"
            using thm_noncont_e_e_4 "\<^bold>&I" by auto
          hence "[\<^bold>\<exists> G . NonContingent G \<^bold>& G \<^bold>= E! in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" "\<^bold>\<exists>I" by fast
        }
        thus "[(L\<^sup>-) \<^bold>\<noteq> E! in v]"
          using four_distinct_2[deduction, OF thm_cont_e_2]
                modus_tollens_1 CP
          by blast
      next
        {
          assume "[(L\<^sup>-) \<^bold>= (E!\<^sup>-) in v]"
          hence "[NonContingent (L\<^sup>-) \<^bold>& (L\<^sup>-) \<^bold>= (E!\<^sup>-) in v]"
            using thm_noncont_e_e_4 "\<^bold>&I" by auto
          hence "[\<^bold>\<exists> G . NonContingent G \<^bold>& G \<^bold>= (E!\<^sup>-) in v]"
            using thm_noncont_e_e_3 "\<^bold>&I" "\<^bold>\<exists>I" by fast
        }
        thus "[(L\<^sup>-) \<^bold>\<noteq> (E!\<^sup>-) in v]"
          using four_distinct_2[deduction, OF thm_cont_e_3]
                modus_tollens_1 CP
          by blast
      next
        show "[E! \<^bold>\<noteq> (E!\<^sup>-) in v]"
          by (rule thm_relation_negation_5_1)
      qed
  end

  lemma thm_cont_propos_1[PLM]:
    "[NonContingent (p::\<o>) \<^bold>\<equiv> NonContingent (p\<^sup>-) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[NonContingent p in v]"
      hence "[\<^bold>\<box>p \<^bold>\<or> \<^bold>\<box>\<^bold>\<not>p in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
      hence "[\<^bold>\<box>(\<^bold>\<not>(p\<^sup>-)) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>p) in v]"
        apply -
        apply (PLM_subst_method "p" "\<^bold>\<not>(p\<^sup>-)")
        using thm_relation_negation_4[equiv_sym] by auto
      hence "[\<^bold>\<box>(\<^bold>\<not>(p\<^sup>-)) \<^bold>\<or> \<^bold>\<box>(p\<^sup>-) in v]"
        apply -
        apply (PLM_subst_goal_method "\<lambda>\<phi> . \<^bold>\<box>(\<^bold>\<not>(p\<^sup>-)) \<^bold>\<or> \<^bold>\<box>(\<phi>)" "\<^bold>\<not>p")
        using thm_relation_negation_3[equiv_sym] by auto
      hence "[\<^bold>\<box>(p\<^sup>-) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>(p\<^sup>-)) in v]"
        by (rule oth_class_taut_3_e[equiv_lr])
      thus "[NonContingent (p\<^sup>-) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    next
      assume "[NonContingent (p\<^sup>-) in v]"
      hence "[\<^bold>\<box>(\<^bold>\<not>(p\<^sup>-)) \<^bold>\<or> \<^bold>\<box>(p\<^sup>-) in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs
        by (rule oth_class_taut_3_e[equiv_lr])
      hence "[\<^bold>\<box>(p) \<^bold>\<or> \<^bold>\<box>(p\<^sup>-) in v]"
ply
        apply (PLM_subst_goal_method  "\<lambda>\<phi> . \<^bold>\<box>\<phi> \<^bold>\<or> \<^bold>\<box>(p\<^sup>-)" "\<^bold>\<not>(p\<^sup>-)")
        using thm_relation_negation_4 by auto
      hence "[\<^bold>\<box>(p) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>p) in v]"
        apply -
        apply (PLM_subst_method "p\<^sup>-" "\<^bold>\<not>p")
        using thm_relation_negation_3 by auto
      thus "[NonContingent p in v]"
        unfolding NonContingent_def Necessary_defs Impossible_defs .
    qed

  lemma thm_cont_propos_2[PLM]:
    "[Contingent p \<^bold>\<equiv> \<^bold>\<diamond>p \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>p) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[Contingent p in v]"
      hence "[\<^bold>\<not>(\<^bold>\<box>p \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>p)) in v]"
        unfolding Contingent_def Necessary_defs Impossible_defs .
      hence "[(\<^bold>\<not>\<^bold>\<box>p) \<^bold>& (\<^bold>\<not>\<^bold>\<box>(\<^bold>\<not>p)) in v]"
        by (rule oth_class_taut_6_d[equiv_lr])
      hence "[(\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<not>p)) \<^bold>& (\<^bold>\<diamond>\<^bold>\<not>p) in v]"
        using KBasic2_2[equiv_lr] "\<^bold>&I" "\<^bold>&E" by meson
      thus "[(\<^bold>\<diamond>p) \<^bold>& (\<^bold>\<diamond>(\<^bold>\<not>p)) in v]"
        apply - apply PLM_solver
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<not>p"        using  cont_nec_fact2_1 cont_nec_fact2_4
        using oth_class_taut_4_b[equiv_sym] by auto
    next
      assume "[(\<^bold>\<diamond>p) \<^bold>& (\<^bold>\<diamond>\<^bold>\<not>(p)) in v]"
      hence "[(\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<not>p)) \<^bold>& (\<^bold>\<diamond>\<^bold>\<not>(p)) in v]"
        apply - apply PLM_solver
        apply (PLM_subst_method "p" "\<^bold>\<not>\<^bold>\<not>p")
        using oth_class_taut_4_b by auto
      hence "[(\<^bold>\<not>\<^bold>\<box>p) \<^bold>& (\<^bold>\<not>\<^bold>\<box>(\<^bold>\<not>p)) in v]"
        using KBasic2_2[equiv_rl] "\<^bold>&I" "\<^bold>&E" by meson
      hence "[\<^bold>\<not>(\<^bold>\<box>(p) \<^bold>\<or> \<^bold>\<box>(\<^bold>\<not>p)) in v]"
        by (rule oth_class_taut_6_d[equiv_rl])
      thus "[Contingent p in v]"
        unfolding Contingent_def Necessary_defs Impossible_defs .
    qed

  lemma thm_cont_propos_3[PLM]:
    "[Contingent (p::\<o>) \<^bold>\<equiv> Contingent (p\<^sup>-) 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 p\<^ub0 here
      "p\<^sub>0 \<equiv> \<^bold>\<forall>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>E!,x\<^sup>P\<rparr>"

    lemma thm_noncont_propos_1[PLM]:
      "[Necessary p\<^sub>0 in v]"
      unfolding Necessary_defs p\<^sub>0_def
      apply (rule RN, rule "\<^bold>\<forall>I")
      using if_p_then_p .

    lemma thm_noncont_propos_2[PLM]:
      "[Impossible (p\<^sub>0\<^sup>-) in v]"
     Impossible_defs
      apply (PLM_subst_method "\<^bold>\<not>p\<^sub>0" "p\<^sub>0\<^sup>-")
       using thm_relation_negation_3[equiv_sym] apply simp
      apply (PLM_subst_method "p\<^sub>0" "\<^bold>\<not>\<^bold>\<not>p\<^sub>0")
       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 (p\<^sub>0) in v]"
      unfolding NonContingent_def using thm_noncont_propos_1
by ^><or>I"(1))

    lemmathm_noncont_propos_4[PLM]:
      "[NonContingent (p\<^sub>0\<^sup>-) in v]"
      unfolding NonContingent_def using thm_noncont_propos_2
      by (rule "\<^bold>\<or>I"(2))

    lemma thm_noncont_propos_5[PLM]:
      "[\<^bold>\<exists> (p::\<o>) q . p \<^bold>\<noteq> q \<^bold>& NonContingent p \<^bold>& NonContingent q in v]"
      apply (rule_tac \<alpha>="p\<^sub>0" in "\<^bold>\<exists>I", rule_tac \<alpha>="p\<^sub>0\<^sup>-" in "\<^bold>\<exists>I")
      using "\<^bold>\<exists>I" thm_relation_negation_6 thm_noncont_propos_3
            thm_noncont_propos_4 "\<^bold>&I" by simp

    private definition q\<^sub>0 where
      "q\<^sub>0 \<equiv> \<^bold>\<exists> x . \<lparr>E!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>)"

    lemma basic_prop_1[PLM]:
      "[\<^bold>\<exists> p . \<^bold>\<diamond>p \<^bold>& \<^bold>\<diamond>(\<^bold>\<not>p) in v]"
      apply (rule_tac \<alpha>="q\<^sub>0" in "\<^bold>\<exists>I") unfolding q\<^sub>0_def
      using qml_4[axiom_instance] by simp

    lemma basic_prop_2[PLM]:
      "[Contingent q\<^sub>0 in v]"
      unfolding Contingent_def Necessary_defs Impossible_defs
      apply (rule oth_class_taut_6_d[equiv_rl])
      apply (PLM_subst_goal_method "\<lambda> \<phi> . (\<^bold>\<not>\<^bold>\<box>(\<phi>)) \<^bold>& \<^bold>\<not>\<^bold>\<box>\<^bold>\<not>q\<^sub>0" "\<^bold>\<not>\<^bold>\<not>q\<^sub>0")
       using oth_class_taut_4_b[equiv_sym] apply simp
      using qml_4[axiom_instance,conj_sym]
      unfolding q\<^sub>0_def diamond_def by simp

    lemma basic_prop_3[PLM]:
      "[Contingent (q\<^sub>0\<^sup>-) in v]"
      apply (rule thm_cont_propos_3[equiv_lr])
      using basic_prop_2 .

    lemma basic_prop_4[PLM]:
      "[\<^bold>\<exists> (p::\<o>) q . p \<^bold>\<noteq> q \<^bold>& Contingent p \<^bold>& Contingent q in v]"
      apply (rule_tac \<alpha>="q\<^sub>0" in "\<^bold>\<exists>I", rule_tac \<alpha>="q\<^sub>0\<^sup>-" in "\<^bold>\<exists>I")
      using thm_relation_negation_6 basic_prop_2 basic_prop_3 "\<^bold>&I" by simp

    lemma four_distinct_props_1[PLM]:
      "[NonContingent (p::\<Pi>\<^sub>0) \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<exists> q . Contingent q \<^bold>& q \<^bold>= p)) in v]"
      proof (rule CP)
        assume "[NonContingent p in v]"
        hence "[\<^bold>\<not>(Contingent p) in v]"
          unfolding NonContingent_def Contingent_def
          apply - by PLM_solver
        moreover {
           assume "[\<^bold>\<exists> q . Contingent q \  by (rule refines_bind_no_throw_strong)
           then obtain r where "[Contingent r \<^bold>& r \<^bold>= p in v]"
            by (rule "\<^bold>\<exists>E")
           hence "[Contingent p in v]"
             using "\<^bold>&E" l_identity[axiom_instance, deduction, deduction]
             by blast
        }
        ultimately show "[\<^bold>\<not>(\<^bold>\<exists>q. Contingent q \<^bold>& q \<^bold>= p) in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma four_distinct_props_2[PLM]:
      "[Contingent (p::\<o>    "[((\<lpha>:(>>\beta)in ]"
      proof (rule CP)
        assume "[Contingent p in v]"
        hence "[\<^bold>\<not>(NonContingent p) in v]"
unfoldingjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
          apply - by PLM_solver
        moreover
           assume "(assume_on_exit <> Pcleanup)s java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
           then obtain r where "[NonContingent r \<^bold>& r \<^bold>= p in v]"
            by (rule "\<^bold>\<exists>E")
           hence "[NonContingent p in v]"
             using "\<^bold>&E" l_identity[axiom_instance, deduction, deduction]
             by blast
        }
        ultimately show "[\<^bold>\<not>(\<^bold>\<exists>q. NonContingent q \<^bold>& q \<^bold>= p) in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma four_distinct_props_4[PLM]:
      "[p\<^sub>0 \<^bold>\<noteq> (p\<^sub>0\<^sup>-) \<^bold>& p\<
        \<^bold>& (p\<^sub>0\<^sup>-) \<^bold>\<noteq> (q\<^sub>text\<open>labelTAO_PLM_Objects}\<close>
      proof (rule "\<^bold>&I")+
        show "[p\<^sub>0 \<^bold>\<noteq> (p\<^sub>0\<^sup>-) in v]"
          by (rule thm_relation_negation_6)
        next
          {
            assume "[p\<^sub>0 \<^bold>= q\<^sub>0 in v]"
            hence "[\<^bold>\<exists> q . NonContingentrow> 'a" and
              using "\<^bold>&I" thm_noncont_propos_3 "\<^bold
              by simp
          }
          thus "[p\<^sub>0 \<^bold>\<noteq> q\<^sub>0 in v]"
            using four_distinct_props_2[deduction, OF basic_prop_2]
                  modus_tollens_1 CP
            by blast
        next
          definitionassume_stack_alloc: nat\Rightarrow>(' \Rightarrow> 'a:xmem_type)listset <> (e:default ' ,')spec_monad java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
            assume "[p\<^sub>0 \<^bold>= (q\<^sub>0\<^sup>-) in v]"
 ^\exists   NonContingentq\^>  <bold=(\^>\^>)inv]java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
              using thm_noncont_propos_3 "\<^bold>&I" "\<^bold>\<exists>I"[where \<alpha>=p\<^sub>0] by simp
}
          thus "[p\<^sub>0 \<^bold>\<noteq> (q\<^sub>0\<^sup>-) in v]"
            using four_distinct_props_2[deduction, OF basic_prop_3]
                  modus_tollens_1 CP
          by blast
        next
          {
            assume "[(p\<^sub>0\<^sup>-) \<^bold>= q\<^sub>0 in v]"
            :
              using thm_noncont_propos_4 "\<^bold>&I" "\<^bold>\<exists>I"[where \<alpha>="p\<^sub>0\<^sup>-"] by auto
          }
          thus "[(p\<^sub>0\<^sup>-) \<^bold>\<noteq> q\<^sub>0 in v]"
            using four_distinct_props_2[deduction, OF basic_prop_2]
                  modus_tollens_1 CP
            by blast
        next
          {
            assume "[(p\<^sub>0\<^sup>-) \<^bold>= (q\<^sub>0\<^sup>-) in v]"
            hence "[\<^bold>\<exists> q . NonContingent q \<^bold>& q \<^bold>= (q\<^sub>0\<^sup>-) in v]"
              using thm_noncont_propos_4 "\<^bold>&I" "\<^bold>\<exists>I"[where \<alpha>="p\<^sub>0\<^sup>-"] by auto
          }
          thus "[(p\<^sub>0\<^sup>-) \<^bold>\<noteq> (q\<^sub>0\<^sup>-) in v]"
            using four_distinct_props_2[deduction, OF basic_prop_3]
                  modus_tollens_1 CP
            by blast
        next
          show "[q\<^sub>0 \<^bold>\<noteq> (q\<^sub>0\<^sup>-) in v]"
            by (rule thm_relation_negation_6: 'a 'c list
        qed

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

lemma[PLM]
      "[ContingentlyTrue q\<^sub>0 \<^bold>\<or> ContingentlyFalse q\<^sub>0 in v]"
      proof -
        have "[q\<^sub>0 \<^bold>\<or> \<^bold>\<not>q\<^sub>0 in v]"
          by PLM_solver
        moreover {
          assume "[q\<^sub>0 in v]"
          hence "[q\<^      in if not has_comb thenraiseBind else () end
            unfolding q\<^sub>0_def
            using qml_4[axiom_instance,conj2] "\<^bold>&I"
 
end
           Termadd_tfreesTconstrT [|>  |> 
          assume "[\<^        __),') . NONE]]((.empty,[) ) lthy'
hence[<><>\^>) <bold> <bold><>q<sub0in v]"
unfoldingsub0_def
fun Inr_const LT java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
        }
 thesis
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
          using "\<else (.Inr_const(generate_sum_prodT [snd )] generate_sum_prodT( sndcs)]
      qed

    lemma cont_tf_thm_2[PLM]:
      "[ContingentlyFalse q\<^sub>0 \           (((ctr_typarams .name),NoSynjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
      using cont_tf_thm_1 cont_true_cont_3[where p="q\<^sub>0"]
            cont_true_cont_4[where p="q\<^sub>0"]
      apply - by PLM_solver

    
      "[\<^bold>\<exists> p . ContingentlyTrue p in v]"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        show "[ContingentlyTrue q\<^sub>0 \<^
 
      next
        assume "[ContingentlyTrue q\<^sub>0 in v]"
        thus ?thesis
          using "\<^bold>\<exists>I" by metis
      next
        assume "[ContingentlyFalse q\<^sub>0 in v]"
        hence "[ContingentlyTrue (q\<^sub>0\<^sup>-) in v]"
          using cont_true_cont_4[equiv_lr] by simp
         ?thesis
          using "\<^bold>\<exists>I" by metis
      qed

    lemma cont_tf_thm_4[PLM]:
      "[\<^bold>\<exists> p . ContingentlyFalse p in v]"
      proof (rule "\<^bold>\<or>E"(1); (rule CP)?)
        show "[ContingentlyTrue q\<^sub>0 \<^bold>\<or> ContingentlyFalse q\<^sub>0 in v]"
          using cont_tf_thm_1 .
      next
        assume "[ContingentlyTrue q\<^sub>0 in v]"
        hence "[ContingentlyFalse (q\<^sub>0\<^sup>-) in v]"
          using cont_true_cont_3[equiv_lr] by simp
        thus ?thesis
          using "\<^bold>\<exists>I" by metis
      next
        assume "[ContingentlyFalse q\<^sub>0 in v]"
        thus ?thesis
          using "\<^bold>\<exists>I" by metis
      qed

    lemma cont_tf_thm_5[PLM]:
      "[ContingentlyTrue p \<^bold>& Necessary q \<^bold>\<rightarrow> p \<^bold>\<noteq> q in v]"
      proof (rule CP)
        assume "[ContingentlyTrue p \<^bold>& Necessary q in v]"
        hence 1: "[\<^bold>\<diamond>(\<^bold>\<not>p) \<^bold>& \<^bold>\<box> q in v]"
          unfolding ContingentlyTrue_def Necessary_defs
          using "\<^bold>&E" "\<^bold>&I" by blast
        hence "[\<^bold>\<not>\<^bold>\<box>p in v]"
          apply - apply (drule "\<^bold>&E"(1))
          unfolding diamond_def
          apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<not>p" "p")
          using oth_class_taut_4_b[equiv_sym] by auto
        moreover {
          assume "[p \<^bold>= q in v]"
          hence "[\<^bold>\<box>p in v]"
            using l_identity[where \<alpha>="q" and \<beta>="p" and \<phi>="\<lambda> x . \<^bold>\<box> x",
                             axiom_instance, deduction, deduction]
                  1[conj2] id_eq_prop_prop_8_b[deduction]
            by blast
        }
        ultimately show "[p \<^bold>\<noteq> q in v]"
          using modus_tollens_1 CP by blast
      qed

    lemma cont_tf_thm_6[PLM]:
      "[(ContingentlyFalse p \<^bold>& Impossible q) \<^bold>\<rightarrow> p \<^bold>\<noteq> q in v]"
      proof (rule CP)
        assume "[ContingentlyFalse p \<^bold>& Impossible q in v]"
        hence 1: "[\<^bold>\<diamond>p \<^bold>& \<^bold>\<box>(\<^bold>\<not>q) in v]"
          unfolding ContingentlyFalse_def Impossible_defs
          using "\<^bold>&E" "\<^bold>&I" by blast
        hence "[\<^bold>\<not>\<^bold>\<diamond>q in v]"
          unfolding diamond_def apply - by PLM_solver
        moreover {
          assume "[p \<^bold>= q in v]"
          hence "[\<^bold>\<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 \<^bold>\<noteq> q in v]"
          using modus_tollens_1 CP by blast
      qed
  

  lemmate.\exists>b.a}
    "[O! \<^bold>\<noteq> A! in v]"
    proof -
      {
        assume "[O! \<^bold>= A! in v]"
        hence "[(\<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>) in v]"
          unfolding Ordinary_def Abstract_def .
        moreover have "[\<lparr>(\<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>), x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
          apply (rule beta_C_meta_1)
          by show_proper
        ultimately have "[\<lparr>(\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>), x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
          using l_identity[axiom_instance, deduction, deduction] by fast
        moreover have "[\<lparr>(\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>), x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
          apply (rule beta_C_meta_1)
          by show_proper
        ultimately have "[\<^bold>\<diamond    using o_objects_exist_1 "BF\<^bold\diamond>"[deduction]byblast
          apply - by PLM_solver
      }
  thus?
        using oth_class_taut_1_b modus_tollens_1 CP
        by blast
    qed

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

  lemma oa_contingent_3[PLM]:
    "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr> 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 "\<^bold>&I")
    subgoal
      thus ?thesis
      apply (PLM_subst_method "\<lambda> x . \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>" "\<lambda> x . \<lparr>\<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>")
    usingnocoderaxiom_instancecontraposition_2
        apply show_proper
      using "BF\<^bold>\<diamond>"[deduction, OF thm_cont_prop_2[equiv_lr, OF thm_cont_e_2, conj1]]
      by (rule "T\<^bold>\<diamond>"[deduction])
    subgoal
      apply (PLM_subst_method "\<lambda> x . \<lparr>A!,x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr>")
       using oa_contingent_3 apply simp
      using cqt_further_5[deduction,conj1, OF A_objects[axiom_instance]]
      by (rule "T\<^bold>\<diamond>"[deduction])
    done

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

  lemma oa_contingent_6[PLM]:
    "[(O!\<^sup>-) blast
    proof -
      {
assume[O\^>) \bold= A!<sup-) in v]"
        [\<>\<lambda>x. \^>\<not><lparr>O!x^supP\<rparr>) \<^bold>= (\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>) in v]"
          unfolding propnot_defs .
        moreover have "[\<lparr>(\<^bold>\<lambda>x. \<^bold>\<not>\<java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
          apply (rule beta_C_meta_1)
          by show_proper
><bold>\<lambda>x. <bold\<>\<lparr>A!x\^supP<>,x\<^up>\<rparr> \<^bold>\<equiv>  \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr> in v]"
          dentity_tancedeductioneduction
by fast
        hence "[(\<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>) \<^bold>\<equiv> \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr> in v]"
          apply -
          apply (PLM_subst_method "\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>" "(\<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>)")
           apply (safe intro!: beta_C_meta_1)
          by aveq_in_\>q\<n\P> 'v" usingdommths_eq_defauto
        hence "[\<lparr>O!,x\<^sup>P\<             3 X \in>T" using T by blast
usingoa_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]:
    "[\<lparr>O!\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>A!\<^sup>-,x\<^sup>P\<rparr> in v]"
    proof -
      have "[(\<^bold>\<not>\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>
        apply (PLM_subst_method "(\<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>)" "\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>A!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>")
         apply (safe intro!: beta_C_meta_1[equiv_sym])
          apply show_proper
        using oth_class_taut_4_b[equiv_sym] by auto
      moreover have "[\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr> in v]"
        apply (rule beta_C_meta_1)
        by show_proper
      ultimately show ?thesis
        unfolding propnot_defs
        using oa_contingent_3
        apply - by PLM_solver
    qed

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

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

  lemma oa_facts_1[PLM]:
    "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>O!,x\<^sup>P\<rparr> in v]"
    proof (rule CP)
      assume "[\<lparr>O!,x\<^sup>P\<rparr> in v]"
      hence "[\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
        unfolding Ordinary_def apply -
        apply (rule beta_C_meta_1[equiv_lr])
        by show_proper
      hence "[\<^bold>\<box>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
        using qml_3axiom_instanceionbyto
"bold\<lparr>O!,x\<^sup>P\<rparr> in v]"
        unfolding Ordinary_def
apply
        apply (PLM_subst_method "\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>" "\<lparr>\<^bold>\<lambda>x. \<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>")
         apply (safe intro!: beta_C_meta_1[equiv_sym])
        per
    qed

  lemma oa_facts_2LM

    proof (rule CP)
       \!,x\<^sup>P\<rparr> in v]"
      hence "[\<^bold>\<not>\<^thenhave_anguage'o"
        stract_def apply -
        apply (rule beta_C_meta_1[equiv_lr])
        by show_proper
<<box>\<^bold>\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr> in v]"
        using KBasic2_4equivalence_relation_on_states_ranetset_iff
      hence "[\<^bold>\<box>\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]
        apply -
_bold\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>" "\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^uprparr")
      Basic2_4 yto
      thus "[\<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
        unfolding Abstract_def
        apply -
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>" "\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>")
         apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

  lemma oa_facts_3[PLM]:
    "[\<^bold>\<diamond>\<lparr>O!,x\<^sup>P\<rparr> \<^
    using oa_facts_1 by (rule derived_S5_rules_2_b)

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

  lemma oa_facts_5[PLM]:
    "[\<^bold>\<diamond>\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<box>\<lparr>O!,x\<^sup>P\<rparr> in v]"
    using oa_facts_1[deduction, OF oa_facts_3[deduction]]
      "T\<^bold>\<diamond>"[deduction, OF qml_2[axiom_instance, deduction]]
      "\<^bold>\<equiv>I" CP by blast

  lemma oa_facts_6[PLM]:
    "[\<^bold>\<diamond>\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
    using oa_facts_2[deduction, OF oa_facts_4[deduction]]
      "T\<^bold>\<diamond>"[deduction, OF qml_2[axiom_instance, deduction]]
      "\<^bold>\<equiv>I" CP by blast

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

  lemma oa_facts_8[PLM]:
    "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<A>\<lparr>A!,x\<^sup>P\<rparr> quiv> \phi F) in vjava.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
    apply (rule "\<^bold>\<equiv>I"; rule CP)
     apply (rule nec_imp_act[deduction, OF oa_facts_2[deduction]]; assumption)
    proof -
      assume "[\<^bold>\<A>\<lparr>A!,x\<^sup>P\<rparr> in v]"
      hence "[\<^bold>\<A>(\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>) in v]"
        unfolding Abstract_def apply -
        apply (PLM_subst_method "\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>" "\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>")
        apply (safe intro!: beta_C_meta_1)
        by show_proper
      hence "[\<^bold>\<A>(\<^bold>\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>) in v]"
        apply -
        apply (PLM_subst_method "(\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>)" "(\<^bold>\<box>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>)")
        using KBasic2_4[equiv_sym] by auto
      hence "[\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr> in v]"
        using qml_act_2[axiom_instance, equiv_rl] KBasic2_4[equiv_lr] by auto
      thus "[\<lparr>A!,x\<^sup>P\<rparr> in v]"
        unfolding Abstract_def apply -
        apply (PLM_subst_method "\<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>" "\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>,x\<^sup>P\<rparr>")
        apply (safe intro!: beta_C_meta_1[equiv_sym])
        by show_proper
    qed

  lemma cont_nec_fact1_1[PLM]:
    "[WeaklyContingent F \<^bold>\<equiv> WeaklyContingent (F\<^sup>-) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[WeaklyContingent F in v]"
      hence wc_def: "[Contingent F \<^bold>& (\<^bold>\<forall> x . (\<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F,x\<^sup>P\<rparr>)) in v]"
        unfolding WeaklyContingent_def .
      have "[Contingent (F\<^sup>-) in v]"
        using wc_def[conj1] by (rule thm_cont_prop_3[equiv_lr])
      moreover {
        {
          fix x
          assume "[\<^bold>\<diamond>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> in v]"
          hence "[\<^bold>\<not>\<^bold>\<box>\<lparr>F,x\<^sup>P\<rparr> in v]"
            unfolding diamond_def apply -
            apply (PLM_subst_method "\<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>" "\<lparr>F,x\<^sup>P\<rparr>")
             thm_relation_negation_2_1 byauto
          moreover {
            assume "[\<^bold>\<not>\<^bold>\<box>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> in v]"
            hence "[\<^bold>\<not>\<^bold>\<box>\<lparr>\<^bold>\<lambda>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>,x\<^sup>P\<rparr> in v]"
              unfolding propnot_defs .
            hence "[\<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> in v]"
              unfolding diamond_def
applyapply(LM_subst_method \<parr><^bold><lambda>x \^bold\<not>\lparr>,x^>P<>,x<supP<>""\^bold><>\<parr>,x\^sup>\<parr>)
              apply (safe intro!: beta_C_meta_1)
              by show_proper
            hence "[\<^bold>\<box>\<lparr>F,x\<^sup>P\<rparr> in v]"
              using wc_def[conj2] cqt_1[axiom_instance, deduction]
                    modus_ponens by fast
          }
          ultimately have "[\<^bold>\<box>\<lparr>F\<^sup>-, x\<^sup>P\<rparr> in v]"
            using "\<^bold>\<not>\<^bold>\<not>E" modus_tollens_1 CP by blast
        }
        hence "[\<^bold>\<forall> x . \<^bold>\<diamond>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F\<^sup>-, x\<^sup>P\<rparr> in v]"
          using "\<^bold>\<forall>I" CP by fast
      }
      ultimately show "[WeaklyContingent (F\<^sup>-) in v]"
        unfolding WeaklyContingent_def by (rule "\<^bold>&I")
    next
      assume "[WeaklyContingent (F\<^sup>-) in v]"
      hence wc_def: "[Contingent (F\<^sup>-) \<^bold>& (\<^bold>\<forall> x . (\<^bold>\<diamond>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>)) 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 "[\<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> in v]"
          hence "[\<^bold>\<not>\<^bold>\<box>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> in v]"
            unfolding diamond_def apply -
            apply (PLM_subst_method "\<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>" "\<lparr>F\<^sup>-,x\<^sup>P\<rparr>")
            using thm_relation_negation_1_1[equiv_sym] by auto
          moreover {
            assume "[\<^bold>\<not>\<^bold>\<box>\<lparr>F,x\<^sup>P\<rparr> in v]"
            hence "[\<^bold>\<diamond>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> in v]"
              unfolding diamond_def
              apply - apply (PLM_subst_method "\<lparr>F,x\<^sup>P\<rparr>" "\<^bold>\<not>\<lparr>F\<^sup>-,x\<^sup>P\<rparr>")
              using thm_relation_negation_2_1[equiv_sym] by auto
            hence "[\<^bold>\<box>\<lparr>F\<^sup>-,x\<^sup>P\<rparr> in v]"
              using wc_def[conj2] cqt_1[axiom_instance, deduction]
                    modus_ponens by fast
          }
          ultimately have "[\<^bold>\<box>\<lparr>F, x\<^sup>P\<rparr> in v]"
            using "\<^bold>\<not>\<^bold>\<not>E" modus_tollens_1 CP by blast
        }
        hence "[\<^bold>\<forall> x . \<^bold>\<diamond>\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>\<lparr>F, x\<^sup>P\<rparr> in v]"
          using "\<^bold>\<forall>I" CP by fast
      }
      ultimately show "[WeaklyContingent (F) in v]"
        unfolding WeaklyContingent_def by (rule "\<^bold>&I")
    qed

  lemma cont_nec_fact1_2[PLM]:
    "[(WeaklyContingent F \<^bold>& \<^bold>\<not>(WeaklyContingent G)) \<^bold>\<rightarrow> (F \<^bold>\<noteq> G) in v]"
    using l_identity[axiom_instance,deduction,deduction] "\<^bold>&E" "\<^bold>&I"
          modus_tollens_1 CP by metis

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

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

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

  lemma cont_nec_fact2_4[PLM]:
    "[\<^bold>\<not>(WeaklyContingent (PLM.L)) in v]"
    proof -
      {
        assume "[WeaklyContingent PLM.L in v]"
        hence "[Contingent PLM.L in v]"
          unfolding WeaklyContingent_def using "\<^bold>&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

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

  lemma cont_nec_fact2_6[PLM]:
    "[A! \<^bold>\<noteq> E! \<^bold>& A! \<^bold>\<noteq> (E!\<^sup>-) \<^bold>& A! \<^bold>\<noteq> PLM.L \<^bold>& A! \<^bold>\<noteq> (PLM.L\<^sup>-) in v]"
    proof ((rule "\<^bold>&I")+)
      show "[A! \<^bold>\<noteq> E! in v]"
        using cont_nec_fact2_2 cont_nec_fact2_3
              cont_nec_fact1_2[deduction] "\<^bold>&I" by simp
    next
       [<^bold>\<ot( (E!\<sup-) inv"
        using cont_nec_fact1_1[THEN oth_class_taut_5_d[equiv_lr], equiv_lr]
              cont_nec_fact2_3 by auto
      thus "[A! \<^bold>\<noteq> (E!\<^sup>-) in v]"
        using cont_nec_fact2_2 cont_nec_fact1_2[deduction] "\<^bold>&I" by simp
            hence"\<braceb\<supP,(<bold<>   <>,z\<>P,a\^>P<rparr)\<rbrace>in v]"
      show "[A! \<^bold>\<noteq> PLM.L in v]"
        using cont_nec_fact2_2 cont_nec_fact2_4
              cont_nec_fact1_2[deduction] "\<^bold>&I" by simp
    next
      have "[\<^bold>\<not>(WeaklyContingent (PLM.L\<^sup>-)) 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! \<^bold>\<noteq> (PLM.L\<^sup>-) in v]"
        using cont_nec_fact2_2 cont_nec_fact1_2[deduction] "\<^bold>&I" by simp
    qed

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

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

  lemma thm_neg_eqE[PLM]:
    "[((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> (\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) in v]"
    proof -
      have "[(x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P) in v] = [\<lparr>(\<^bold>\<lambda>\<^sup>2 (\<lambda> x y . (x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)))\<^sup>-, x\<^sup>P, y\<^sup>P\<rparr> in v]"
        unfolding not_identical\<^sub>E_def by simp
      also have "... = [\<^bold>\<not>\<lparr>(\<^bold>\<lambda>\<^sup>2 (\<lambda> x y . (x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))), x\<^sup>P, y\<^sup>P\<rparr> in v]"
        unfolding propnot_defs
        apply (safe intro!: beta_C_meta_2[equiv_lr] beta_C_meta_2[equiv_rl])
        by show_proper+
      also have "... = [\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) in v]"
        apply (PLM_subst_method
               "\<lparr>(\<^bold>\<lambda>\<^sup>2 (\<lambda> x y . (x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))), x\<^sup>P, y\<^sup>P\<rparr>"
               "(x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)")
         apply (safe intro!: beta_C_meta_2)
        unfolding identity_defs by show_proper
      finally show ?thesis
        using "\<^bold>\<equiv>I" CP by presburger
    qed

  lemma id_nec4_1[PLM]:
    "[((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> \<^bold>\<box>((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P        using reductio_aa_1CP if_p_then_p byblast
    proof -
      have "[(\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) in v]"
        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 -
        apply (PLM_subst_method "(\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)))" "(x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)")
        using thm_neg_eqE[equiv_sym] by auto
    qed

  lemma id_nec4_2[PLM]:
    "[\<^bold>\<diamond>((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> ((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)) in v]"
    using "\<^bold>\<equiv>I" id_nec4_1[equiv_lr] derived_S5_rules_2_b CP "T\<^bold>\<diamond>" by simp

  lemma id_act_1[PLM]:
    "[((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> (\<^bold>\<A>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) in v]"
    proof (rule "\<^bold>\<equiv>I"; rule CP)
      assume "[(x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P) in v]"
      hence "[\<^bold>\<box>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) in v]"
        using id_nec3_1[equiv_lr] by auto
      thus "[\<^bold>\<A>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) in v]"
        using nec_imp_act[deduction] by fast
    next
      assume "[\<^bold>\<A>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) in v]"
      hence "[\<^bold>\<A>(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> \<^bold>&pan> \<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>)) in v]"
        apply -
        apply (PLM_subst_method
               "(x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)"
               "(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>))")
        using eq_E_simple_1 by auto
      hence "[\<^bold>\<A>\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<A>\<lparr>O!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<A>(\<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>)) in v]"
        using Act_Basic_2[equiv_lr] "\<^bold>&I" "\<^bold>&E" by meson
      thus "[(x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P) in v]"
        apply - apply (rule eq_E_simple_1[equiv_rl])
        using oa_facts_7[equiv_rl] qml_act_2[axiom_instance, equiv_rl]
              "\<^bold>&I" "\<^bold>&E" by meson
    qed

  lemma id_act_2[PLM]:
    "[((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P)) \<^bold>\<equiv> (\<^bold>\<A>((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>P))) in v]"
    apply (PLM_subst_method "(\<^bold>\<not>((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)))" "((x\<^sup>P) \<^bold>\<noteq>\<^sub>E (y\<^sup>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: "[\<^bold>\<A>(\<alpha> \<^bold>= \<beta>) in v] \<Longrightarrow> [(\<alpha> \<^bold>= \<beta>) in v]"

instantiation \<nu> :: id_act
begin
  instance proof
    interpret PLM .
    fix x::\<nu> and y::\<nu> and v::i
    assume "[\<^bold>\<A>(x \<^bold>= y) in v]"<^upP\rparr in v]
    hence "[\<^bold>\<A>(((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P)) \<^bold>\<or> (\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr>
            \<^bold>& \<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>))) in v]"
      unfolding identity_defs by auto
    hence "[\<^bold>\<A>(((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) \<^bold>\<or> \<^bold>\<A>((\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr>
            \<^bold>& \<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>))) in v]"
      using Act_Basic_10[equiv_lr] by auto
    moreover {
       assume "[\<^bold>\<A>(((x\<^sup>P) \<^bold>=\<^sub>E (y\<^sup>P))) in v]"
       hence "[(x\<^sup>P) \<^bold>= (y\<^sup>P) in v]"
        using id_act_1[equiv_rl] eq_E_simple_2[deduction] by auto
    }
    moreover {
       assume "[\<^bold>\<A>(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>&span> \<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>)) in v]"
       hence "[\<^bold>\<A>\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<A>\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<A>(\<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>)) in v]"
         using Act_Basic_2[equiv_lr] "\<^bold>&I" "\<^bold>&E" by meson
       hence "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>)) in v]"
         using oa_facts_8[equiv_rl] qml_act_2[axiom_instance,equiv_rl]
           "\<^bold>&I" "\<^bold>&E" by meson
       hence "[(x\<^sup>P) \<^bold>= (y\<^sup>P) in v]"
        unfolding identity_defs using "\<^bold>\<or>I" by auto
    }
    ultimately have "[(x\<^sup>P) \<^bold>= (y\<^sup>P) in v]"
      using intro_elim_4_a CP by meson
    thus "[x \<^bold>= y in v]"
      unfolding identity_defs by auto
  qed
end

instantiation \<Pi>\<^sub>1 :: id_act
begin
  instance proof
    interpret PLM .
    fix F::\<Pi>\<^sub>1 and G::\<Pi>\<^sub>1 and v::i
    show "[\<^bold>\<A>(F \<^bold>= G) in v] \<Longrightarrow> [(F \<^bold>= G) in v]" 
      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 "[\<^bold>\<A>(p \<^bold>= q) in v] \<Longrightarrow> [p \<^bold>= q in v]"
      unfolding identity\<^sub>\<o>_def using id_act_prop by blast
  qed
end

instantiation \<Pi>\<^sub>2 :: id_act
begin
  instance proof
    interpret PLM .
    fix F::\<Pi>\<^sub>2 and G::\<Pi>\<^sub>2 and v::i
    assume a: "[\<^bold>\<A>(F \<^bold>= G) in v]"
    {
      fix x
      have "[\<^bold>\<A>((\<^bold>\<lambda>y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>y. \<lparr>G,x\<^sup>P,y\<^sup>P\<rparr>)
\&^>lambda.\Fy\supx<suprparr)<bold> \^bold\lambda> <>G,\<^>Px\^upP<rparr>  ]"
        using a logic_actual_nec_3[axiom_instance, equiv_lr] cqt_basic_4[equiv_lr] "\<^bold>\<forall>E"
        unfolding identity\<^sub>2_def by fast
      hence "[((\<^bold>\<lambda>y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>y. \<lparr>G,x\<^sup>P,y\<^sup>P\<rparr>))
              \<^bold>& ((\<^bold>\<lambda>y. \<lparr>F,y\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>y. \<lparr>G,y\<^sup>P,x\<^sup>P\<rparr>)) in v]"
        using "\<^bold>&I" "\<^bold>&E" id_act_prop Act_Basic_2[equiv_lr] by metis
    }
    thus "[F \<^bold>= G in v]" unfolding identity_defs by (rule "\<^bold>\<forall>I")
  qed
end

instantiation \<Pi>\<^sub>3 :: id_act
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  instance proof
    interpret PLM .
    fix F::\<Pi>\<^sub>3 and G::\<Pi>\<^sub>3 and v::i
    assume a: "[\<^bold>\<A>(F \<^bold>= G) in v]"
    let ?p = "\<lambda> x y . (\<^bold>\<lambda>z. \<lparr>F,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,z\<^sup>P,x\<^sup>P,y\<^sup>P\<rparr>)
                    \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,z\<^sup>P,y\<^sup>P\<rparr>)
                    \<^bold>& (\<^bold>\<lambda>z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>G,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>)"
    {
      fix x
      {
        fix y
        have "[\<^bold>\<A>(?p x y) in v]"
          using a logic_actual_nec_3[axiom_instance, equiv_lr]
                cqt_basic_4[equiv_lr] "\<^bold>\<forall>E"[where 'a=\<nu>]
          unfolding identity\<^sub>3_def by blast
        hence "[?p x y in v]"
          using "\<^bold>&I" "\<^bold>&E" id_act_prop Act_Basic_2[equiv_lr] by metis
      }
      hence "[\<^bold>\<forall> y . ?p x y in v]"
        by (rule "\<^bold>\<forall>I")
    }
    thus "[F \<^bold>= G in v]"
      unfolding identity\<^sub>3_def by (rule "\<^bold>\<forall>I")
  qed
end

context PLM
begin
  lemma id_act_3[PLM]:
    "[((\<alpha>::('a::id_act)) \<^bold>= \<beta>) \<^bold>\<equiv> \<^bold>\<A>(\<alpha> \<^bold>= \<beta>) in v]"
    using "\<^bold>\<equiv>I" CP id_nec[equiv_lr, THEN nec_imp_act[deduction]]
          id_act_prop by metis

  lemma id_act_4[PLM]:
    "[((\<alpha>::('a::id_act)) \<^bold>\<noteq> \<beta>) \<^bold>\<equiv> \<^bold>\<A>(\<alpha> \<^bold>\<noteq> \<beta>) 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\<^sup>P) \<^bold>= (\<^bold>\<iota>x . x \<^bold>= y) in v]"
    using descriptions[axiom_instance,equiv_rl]
          id_act_3[equiv_sym] "\<^bold>\<forall>I" by fast

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

  lemma eta_conversion_lemma_0[PLM]:
    "[(\<^bold>\<lambda>\<^sup>0 p) \<^bold>= p in v]"
    using lambda_predicates_3_0[axiom_instance] .

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

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

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

subsection\<open>The Theory of Objects\<close>
text\<open>\label{TAO_PLM_Objects}\<close>

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

  lemma partition_2[PLM]:
    "[\<^bold>\<not>(\<^bold>\<exists> x . \<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,x\<^sup>P\<rparr>) in v]"
    proof -
      {
        assume "[\<^bold>\<exists> x . \<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,x\<^sup>P\<rparr> in v]"
        then obtain b where "[\<lparr>O!,b\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> in v]"
          by (rule "\<^bold>\<exists>E")
        hence ?thesis
          using "\<^bold>&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]:
    "[\<lparr>O!,x\<rparr> \<^bold>\<rightarrow> (x \<^bold>=\<^sub>E x) in v]"
    proof (rule CP)
      assume "[\<lparr>O!,x\<rparr> in v]"
      moreover have "[\<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,x\<rparr>) in v]"
        by PLM_solver
      ultimately show "[(x) \<^bold>=\<^sub>E (x) in v]"
        using "\<^bold>&I" eq_E_simple_1[equiv_rl] by blast
    qed

  lemma ord_eq_Eequiv_2[PLM]:
    "[(x \<^bold>=\<^sub>E y) \<^bold>\<rightarrow> (y \<^bold>=\<^sub>E x) in v]"
    proof (rule CP)
      assume "[x \<^bold>=\<^sub>E y in v]"
      hence 1: "[\<lparr>O!,x\<rparr> \<^bold>& \<lparr>O!,y\<rparr> \<^bold>& \<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) in v]"
        using eq_E_simple_1[equiv_lr] by simp
      have "[\<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,y\<rparr> \<^bold>\<equiv> \<lparr>F,x\<rparr>) in v]"
        apply (PLM_subst_method
               "\<lambda> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>"
               "\<lambda> F . \<lparr>F,y\<rparr> \<^bold>\<equiv> \<lparr>F,x\<rparr>")
        using oth_class_taut_3_g 1[conj2] by auto
      thus "[y \<^bold>=\<^sub>E x "\<bold>\not>\bold\<exists>p.(\^> (\^>\>   ))java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
        using eq_E_simple_1[equiv_rl] 1[conj1]
              "\<^bold>&E" "\<^bold>&I" by meson
    qed

  lemma ord_eq_Eequiv_3[PLM]:
    "[((x \<^bold>=\<^sub>E y) \<^bold>& (y \<^bold>=\<^sub>E z)) \<^bold>\<rightarrow> (x \<^bold>=\<^sub>E z) in v]"
    proof (rule CP)
      assume a: "[(x \<^bold>=\<^sub>E y) \<^bold>& (y \<^bold>=\<^sub>E z) in v]"
      have "[\<^bold>\<box>((\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) \<^bold>& (\<^bold>\<forall> F . \<lparr>F,y\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>)) in v]"
        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]] "\<^bold>&I" by blast
      moreover {
        {
          fix w
          have "[((\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) \<^bold>& (\<^bold>\<forall> F . \<lparr>F,y\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>))
                  \<^bold>\<rightarrow> (\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>) in w]"
            by PLM_solver
        }
        hence "[\<^bold>\<box>(((\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,y\<rparr>) \<^bold>& (\<^bold>\<forall> F . \<lparr>F,y\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>))
                \<^bold>\<rightarrow> (\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>)) in v]"
          by (rule RN)
      }
      ultimately have "[\<^bold>\<box>(\<^bold>\<forall> F . \<lparr>F,x\<rparr> \<^bold>\<equiv> \<lparr>F,z\<rparr>) in v]"
        using qml_1[axiom_instance,deduction,deduction] by blast
      thus "[x \<^bold>=\<^sub>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] "\<^bold>&I"
        by presburger
    qed

  lemma ord_eq_E_eq[PLM]:
    "[(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<or> \<lparr>O!,y\<^sup>P\<rparr>) \<^bold>\<rightarrow> ((x\<^sup>P \<^bold>= y\<^sup>P) \<^bold>\<equiv> (x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P)) in v]"
    proof (rule CP)
      assume "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<or> \<lparr>O!,y\<^sup>P\<rparr> in v]"
      moreover {
        assume "[\<lparr>O!,x\<^sup>P\<rparr> in v]"
        hence "[(x\<^sup>P \<^bold>= y\<^sup>P) \<^bold>\<equiv> (x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
          using "\<^bold>\<equiv>I" CP l_identity[axiom_instance, deduction, deduction]
                ord_eq_Eequiv_1[deduction] eq_E_simple_2[deduction] by metis
      }
      moreover {
        assume "[\<lparr>O!,y\<^sup>P\<rparr> in v]"
        hence "[(x\<^sup>P \<^bold>= y\<^sup>P) \<^bold>\<equiv> (x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
          using "\<^bold>\<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_\<nu>_def by metis
      }
      ultimately show "[(x\<^sup>P \<^bold>= y\<^sup>P) \<^bold>\<equiv> (x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
        using intro_elim_4_a CP by blast
    qed

  lemma ord_eq_E[PLM]:
    "[(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr>) \<^bold>\<rightarrow> ((\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) \<^bold>\<rightarrow> x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
    proof (rule CP; rule CP)
      assume ord_xy: "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> in v]"
      assume "[\<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr> in v]"
      hence "[\<lparr>\<^bold>\<lambda> z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P, x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda> z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P, y\<^sup>P\<rparr> in v]"
        by (rule "\<^bold>\<forall>E")
      moreover have "[\<lparr>\<^bold>\<lambda> z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P, x\<^sup>P\<rparr> in v]"
        apply (rule beta_C_meta_1[equiv_rl])
        unfolding identity\<^sub>E_infix_def
         apply show_proper
        using ord_eq_Eequiv_1[deduction] ord_xy[conj1]
        unfolding identity\<^sub>E_infix_def by simp
      ultimately have "[\<lparr>\<^bold>\<lambda> z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P, y\<^sup>P\<rparr> in v]"
        using "\<^bold>\<equiv>E" by blast
      hence "[y\<^sup>P \<^bold>=\<^sub>E x\<^sup>P in v]"
        unfolding identity\<^sub>E_infix_def
        apply (safe intro!:
            beta_C_meta_1[where \<phi> = "\<lambda> z . \<lparr>basic_identity\<^sub>E,z,x\<^sup>P\<rparr>", equiv_lr])
        by show_proper
      thus "[x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P in v]"
        by (rule ord_eq_Eequiv_2[deduction])
    qed

  lemma ord_eq_E2[PLM]:
    "[(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr>) \<^bold>\<rightarrow>
      ((x\<^sup>P \<^bold>\<noteq> y\<^sup>P) \<^bold>\<equiv> (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P) \<^bold>\<noteq> (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P)) in v]"
    proof (rule CP; rule "\<^bold>\<equiv>I"; rule CP)
      assume ord_xy: "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> in v]"
      assume "[x\<^sup>P \<^bold>\<noteq> y\<^sup>P in v]"
      hence "[\<^bold>\<not>(x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
        using eq_E_simple_2 modus_tollens_1 by fast
      moreover {
        assume "[(\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P) \<^bold>= (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
        moreover have "[\<lparr>\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P, x\<^sup>P\<rparr> in v]"
          apply (rule beta_C_meta_1[equiv_rl])
          unfolding identity\<^sub>E_infix_def
           apply show_proper
          using ord_eq_Eequiv_1[deduction] ord_xy[conj1]
          unfolding identity\<^sub>E_infix_def by presburger
        ultimately have "[\<lparr>\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P, x\<^sup>P\<rparr> in v]"
          using l_identity[axiom_instance, deduction, deduction] by fast
        hence "[x\<^sup>P \<^bold>=\<^sub>E y\<^sup>P in v]"
          unfolding identity\<^sub>E_infix_def
          apply (safe intro!:
              beta_C_meta_1[where \<phi> = "\<lambda> z . \<lparr>basic_identity\<^sub>E,z,y\<^sup>P\<rparr>", equiv_lr])
          by show_proper
      }
      ultimately show "[(\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P) \<^bold>\<noteq> (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
        using modus_tollens_1 CP by blast
    next
      assume ord_xy: "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>& \<lparr>O!,y\<^sup>P\<rparr> in v]"
      assume "[(\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P) \<^bold>\<noteq> (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
      moreover {
        assume "[x\<^sup>P \<^bold>= y\<^sup>P in v]"
        hence "[(\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E x\<^sup>P) \<^bold>= (\<^bold>\<lambda>z . z\<^sup>P \<^bold>=\<^sub>E y\<^sup>P) in v]"
          using id_eq_1 l_identity[axiom_instance, deduction, deduction]
          by fast
      }
      ultimately show "[x\<^sup>P \<^bold>\<noteq> y\<^sup>P in v]"
        using modus_tollens_1 CP by blast
    qed

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

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

  lemma ordnecfail[PLM]:
    "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P, F\<rbrace>)) in v]"
    proof (rule CP)
      assume "[\<lparr>O!,x\<^sup>P\<rparr> in v]"
      hence "[\<^bold>\<box>\<lparr>O!,x\<^sup>P\<rparr> in v]"
        using oa_facts_1[deduction] by simp
      moreover hence "[\<^bold>\<box>(\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<rightarrow> (\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P, F\<rbrace>))) in v]"
        using nocoder[axiom_necessitation, axiom_instance] by simp
      ultimately show "[\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<exists> F . \<lbrace>x\<^sup>P, F\<rbrace>)) in v]"
        using qml_1[axiom_instance, deduction, deduction] by fast
    qed

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

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

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

  lemma a_objects_exist_1[PLM]:
    "[\<^bold>\<box>(\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr>) in v]"
    proof -
      {
        fix v
        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 A_objects[axiom_instance] by simp
        hence "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> in v]"
          using cqt_further_5[deduction,conj1] by fast
      }
      thus ?thesis by (rule RN)
    qed

  lemma a_objects_exist_2[PLM]:
    "[\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>O!,x\<^sup>P\<rparr>)) in v]"
    apply (PLM_subst_method "(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr>)" "\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>O!,x\<^sup>P\<rparr>)")
     using cqt_further_2[equiv_sym] apply fast
    apply (PLM_subst_method "\<lambda> x . \<lparr>A!,x\<^sup>P\<rparr>" "\<lambda> x . \<^bold>\<not>\<lparr>O!,x\<^sup>P\<rparr>")
     using oa_contingent_3 a_objects_exist_1 by auto

  lemma a_objects_exist_3[PLM]:
    "[\<^bold>\<box>(\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>E!,x\<^sup>P\<rparr>)) in v]"
    proof -
      {
        fix v
        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 A_objects[axiom_instance] by simp
        hence "[\<^bold>\<exists> x . \<lparr>A!,x\<^sup>P\<rparr> in v]"
          using cqt_further_5[deduction,conj1] by fast
        then obtain a where
          "[\<lparr>A!,a\<^sup>P\<rparr> in v]"
          by (rule "\<^bold>\<exists>E")
        hence "[\<^bold>\<not>(\<^bold>\<diamond>\<lparr>E!,a\<^sup>P\<rparr>) in v]"
          unfolding Abstract_def
          apply (safe intro!: beta_C_meta_1[equiv_lr])
          by show_proper
        hence "[(\<^bold>\<not>\<lparr>E!,a\<^sup>P\<rparr>) in v]"
          using KBasic2_4[equiv_rl] qml_2[axiom_instance,deduction]
          by simp
        hence "[\<^bold>\<not>(\<^bold>\<forall> x . \<lparr>E!,x\<^sup>P\<rparr>) in v]"
          using "\<^bold>\<exists>I" cqt_further_2[equiv_rl]
          by fast  
      }
      thus ?thesis
        by (rule RN)
    qed

  lemma encoders_are_abstract[PLM]:
    "[(\<^bold>\<exists> F . \<lbrace>x\<^sup>P, F\<rbrace>) \<^bold>\<rightarrow> \<lparr>A!,x\<^sup>P\<rparr> in v]"
    using nocoder[axiom_instance] 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]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) 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> \<phi> F) in v]"
        using A_objects[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> \<phi> F) in v]" by (rule "\<^bold>\<exists>E")
      moreover have "[\<^bold>\<forall> y . \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) \<^bold>\<rightarrow> (y \<^bold>= a) in v]"
        proof (rule "\<^bold>\<forall>I"; rule CP)
          fix b
          assume b_prop: "[\<lparr>A!,b\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>b\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
          {
            fix P
            have "[\<lbrace>b\<^sup>P,P\<rbrace> \<^bold>\<equiv> \<lbrace>a\<^sup>P, P\<rbrace> in v]"
              using a_prop[conj2] b_prop[conj2] "\<^bold>\<equiv>I" "\<^bold>\<equiv>E"(1) "\<^bold>\<equiv>E"(2)
                    CP vdash_properties_10 "\<^bold>\<forall>E" by metis
          }
          hence "[\<^bold>\<forall> F . \<lbrace>b\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>a\<^sup>P, F\<rbrace> in v]"
            using "\<^bold>\<forall>I" by fast
          thus "[b \<^bold>= a in v]"
            unfolding identity_\<nu>_def
            using ab_obey_1[deduction, deduction]
                  a_prop[conj1] b_prop[conj1] "\<^bold>&I" by blast
        qed
      ultimately show ?thesis
        unfolding exists_unique_def
        using "\<^bold>&I" "\<^bold>\<exists>I" by fast
    qed

  lemma obj_oth_1[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<lparr>F, y\<^sup>P\<rparr>) in v]"
    using A_objects_unique .

  lemma obj_oth_2[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<lparr>F, y\<^sup>P\<rparr> \<^bold>& \<lparr>F, z\<^sup>P\<rparr>)) in v]"
    using A_objects_unique .

  lemma obj_oth_3[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<lparr>F, y\<^sup>P\<rparr> \<^bold>\<or> \<lparr>F, z\<^sup>P\<rparr>)) in v]"
    using A_objects_unique .

  lemma obj_oth_4[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<^bold>\<box>\<lparr>F, y\<^sup>P\<rparr>)) in v]"
    using A_objects_unique .

  lemma obj_oth_5[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (F \<^bold>= G)) in v]"
    using A_objects_unique .

  lemma obj_oth_6[PLM]:
    "[\<^bold>\<exists>! x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>(\<^bold>\<forall> y . \<lparr>G, y\<^sup>P\<rparr> \<^bold>\<rightarrow> \<lparr>F, y\<^sup>P\<rparr>)) in v]"
    using A_objects_unique .

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

  lemma A_Exists_2[PLM]:
    "[(\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<phi> x)) \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<exists>!x . \<phi> x) in v]"
    using actual_desc_1 A_Exists_1[equiv_sym]
          intro_elim_6_e by blast

  lemma A_descriptions[PLM]:
    "[\<^bold>\<exists> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
    using A_objects_unique[THEN RN, THEN nec_imp_act[deduction]]
          A_Exists_2[equiv_rl] by auto

  lemma thm_can_terms2[PLM]:
    "[(y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)))
      \<^bold>\<rightarrow> (\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in dw]"
    using y_in_2 by auto

  lemma can_ab2[PLM]:
    "[(y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F))) \<^bold>\<rightarrow> \<lparr>A!,y\<^sup>P\<rparr> in v]"
    proof (rule CP)
      assume "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
      hence "[\<^bold>\<A>\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& \<^bold>\<A>(\<^bold>\<forall> F . \<lbrace>y\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
        using nec_hintikka_scheme[equiv_lr, conj1]
              Act_Basic_2[equiv_lr] by blast
      thus "[\<lparr>A!,y\<^sup>P\<rparr> in v]"
        using oa_facts_8[equiv_rl] "\<^bold>&E" by blast
    qed

  lemma desc_encode[PLM]:
    "[\<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>\<equiv> \<phi> G in dw]"
    proof -
      obtain a where
        "[a\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in dw]"
        using A_descriptions by (rule "\<^bold>\<exists>E")
      moreover hence "[\<lbrace>a\<^sup>P, G\<rbrace> \<^bold>\<equiv> \<phi> G in dw]"
        using hintikka[equiv_lr, conj1] "\<^bold>&E" "\<^bold>\<forall>E" by fast
      ultimately show ?thesis
        using l_identity[axiom_instance, deduction, deduction] by fast
    qed

  lemma desc_nec_encode[PLM]:
    "[\<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>\<equiv> \<^bold>\<A>(\<phi> G) in v]"
    proof -
      obtain a where
        "[a\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
        using A_descriptions by (rule "\<^bold>\<exists>E")
      moreover {
        hence "[\<^bold>\<A>(\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
          using nec_hintikka_scheme[equiv_lr, conj1] by fast
        hence "[\<^bold>\<A>(\<^bold>\<forall> F . \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
          using Act_Basic_2[equiv_lr,conj2] by blast
        hence "[\<^bold>\<forall> F . \<^bold>\<A>( \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
          using logic_actual_nec_3[axiom_instance, equiv_lr] by blast
        hence "[\<^bold>\<A>(\<lbrace>a\<^sup>P, G\<rbrace> \<^bold>\<equiv> \<phi> G) in v]"
          using "\<^bold>\<forall>E" by fast
        hence "[\<^bold>\<A>\<lbrace>a\<^sup>P, G\<rbrace> \<^bold>\<equiv> \<^bold>\<A>(\<phi> G) in v]"
          using Act_Basic_5[equiv_lr] by fast
        hence "[\<lbrace>a\<^sup>P, G\<rbrace> \<^bold>\<equiv> \<^bold>\<A>(\<phi> G) in v]"
          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 = "\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists> q . q \<^bold>& F \<^bold>= (\<^bold>\<lambda> y . q)))"
      have "[\<^bold>\<box>(\<^bold>\<exists> p . ContingentlyTrue p) in v]"
        using cont_tf_thm_3 RN by auto
      hence "[\<^bold>\<A>(\<^bold>\<exists> p . ContingentlyTrue p) in v]"
        using nec_imp_act[deduction] by simp
      hence "[\<^bold>\<exists> p . \<^bold>\<A>(ContingentlyTrue p) in v]"
        using Act_Basic_11[equiv_lr] by auto
      then obtain p\<^sub>1 where
        "[\<^bold>\<A>(ContingentlyTrue p\<^sub>1) in v]"
        by (rule "\<^bold>\<exists>E")
      hence "[\<^bold>\<A>p\<^sub>1 in v]"
        unfolding ContingentlyTrue_def
        using Act_Basic_2[equiv_lr] "\<^bold>&E" by fast
      hence "[\<^bold>\<A>p\<^sub>1 \<^bold>& \<^bold>\<A>((\<^bold>\<lambda> y . p\<^sub>1) \<^bold>= (\<^bold>\<lambda> y . p\<^sub>1)) in v]"
        using "\<^bold>&I" id_eq_1[THEN RN, THEN nec_imp_act[deduction]] by fast
      hence "[\<^bold>\<A>(p\<^sub>1 \<^bold>& (\<^bold>\<lambda> y . p\<^sub>1) \<^bold>= (\<^bold>\<lambda> y . p\<^sub>1)) in v]"
        using Act_Basic_2[equiv_rl] by fast
      hence "[\<^bold>\<exists> q . \<^bold>\<A>( q \<^bold>& (\<^bold>\<lambda> y . p\<^sub>1) \<^bold>= (\<^bold>\<lambda> y . q)) in v]"
        using "\<^bold>\<exists>I" by fast
      hence "[\<^bold>\<A>(\<^bold>\<exists> q . q \<^bold>& (\<^bold>\<lambda> y . p\<^sub>1) \<^bold>= (\<^bold>\<lambda> y . q)) in v]"
        using Act_Basic_11[equiv_rl] by fast
      moreover have "[\<lbrace>?x, \<^bold>\<lambda> y . p\<^sub>1\<rbrace> \<^bold>\<equiv> \<^bold>\<A>(\<^bold>\<exists> q . q \<^bold>& (\<^bold>\<lambda> y . p\<^sub>1) \<^bold>= (\<^bold>\<lambda> y . q)) in v]"
        using desc_nec_encode by fast
      ultimately have "[\<lbrace>?x, \<^bold>\<lambda> y . p\<^sub>1\<rbrace> in v]"
        using "\<^bold>\<equiv>E" by blast
  end

  lemma Box_desc_encode_1[PLM]:
    "[\<^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]"
    proof (rule CP)
      assume "[\<^bold>\<box>(\<phi> G) in v]"
      hence "[\<^bold>\<A>(\<phi> G) in v]"
        using nec_imp_act[deduction] by auto
      thus "[\<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 desc_nec_encode[equiv_rl] by simp
    qed

  lemma Box_desc_encode_2[PLM]:
    "[\<^bold>\<box>(\<phi> G) \<^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> \<^bold>\<equiv> \<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]"
          using a Box_desc_encode_1[deduction] by auto
        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
      }
      ultimately show "[\<^bold>\<box>(\<lbrace>(\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>&n> (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)), G\<rbrace>
                        \<^bold>\<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]"
    shows "[(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) \<^bold>\<rightarrow> \<^bold>\<box>(\<lparr>A!,x\<^sup>P\<rparr>
            \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
    proof (rule CP)
      assume a: "[(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
      have "[\<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
        using oa_facts_2[deduction] a[conj1] by auto
      moreover have "[\<^bold>\<box>(\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
        proof (rule BF[deduction]; rule "\<^bold>\<forall>I")
          fix F
          have \<theta>: "[\<^bold>\<box>(\<phi> F \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> F)) in v]"
            using assms[THEN CBF[deduction]] by (rule "\<^bold>\<forall>E")
          moreover have "[\<^bold>\<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
          moreover have "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>(\<phi> F) in v]"
            proof (rule "\<^bold>\<equiv>I"; rule CP)
              assume "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace> in v]"
              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>\<forall>E"[where 'a=\<Pi>\<^sub>1] "\<^bold>\<equiv>E" by blast
              thus "[\<^bold>\<box>(\<phi> F) in v]"
                using \<theta>[THEN qml_2[axiom_instance, deduction], deduction] by simp
            next
              assume "[\<^bold>\<box>(\<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[conj2] "\<^bold>\<forall>E"[where 'a=\<Pi>\<^sub>1] "\<^bold>\<equiv>E" by blast
              thus "[\<^bold>\<box>\<lbrace>x\<^sup>P, F\<rbrace> in v]"
                using encoding[axiom_instance, deduction] by 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
      ultimately show "[\<^bold>\<box>(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
       using "\<^bold>&I" KBasic_3[equiv_rl] by blast
    qed

  lemma box_phi_a_2[PLM]:
    assumes "[\<^bold>\<box>(\<^bold>\<forall> F . \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> F)) in v]"
    shows "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F. \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F))
            \<^bold>\<rightarrow> (\<lparr>A!,y\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>y\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
    proof -
      let ?\<psi> = "\<lambda> x . \<lparr>A!,x\<^sup>P\<rparr> \<^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]"
        using box_phi_a_1[OF assms] "\<^bold>\<forall>I" by fast
      hence "[(\<^bold>\<exists>! x . ?\<psi> x) \<^bold>\<rightarrow> (\<^bold>\<forall> y . y\<^sup>P \<^bold>= (\<^bold>\<iota>x . ?\<psi> x) \<^bold>\<rightarrow> ?\<psi> y) in v]"
        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 A_objects_unique modus_ponens by blast
      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>\<^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>\<equiv> \<phi> G in v]"
    proof -
      obtain a where
        "[a\<^sup>P \<^bold>= (\<^bold>\<iota>x . \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
        using A_descriptions by (rule "\<^bold>\<exists>E")
      moreover {
        hence "[(\<^bold>\<forall> F . \<lbrace>a\<^sup>P, F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
          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")
      }
      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>\<forall> F . \<lbrace>x\<^sup>P, F\<rbrace> \<^bold>\<equiv> (F \<^bold>\<noteq> F)) in v]"
        using A_objects[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>\<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")
            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 A_objects[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)
      assume "[Null (x\<^sup>P) in v]"
      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\<^sup>P) in v]"
        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 "\<^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_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 . \<lparr>A!,y\<^sup>P\<rparr>
          \<^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>)
                \<^bold>\<rightarrow> \<lbrace>a\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,a\<^sup>P\<rparr>)\<rbrace> in v]"
          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>= (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,b\<^sup>P\<rparr>)
          \<^bold>& \<^bold>\<not>\<lbrace>b\<^sup>P, (\<^bold>\<lambda> z . \<lparr>R,z\<^sup>P,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,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 . \<lparr>R,y\<^sup>P,z\<^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,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
          using \<xi>[conj1, conj2] by auto
        hence "[(\<^bold>\<lambda>\<^sup>0 \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>\<^sup>0 \<lparr>R,b\<^sup>P\<rparr>) in v]"
          using lambda_p_q_p_eq_q[equiv_rl] by simp
      }
      ultimately have "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& \<lparr>A!,b\<^sup>P\<rparr> \<^bold>&pan> a \<^bold>\<noteq> b
                \<^bold>& ((\<^bold>\<lambda>\<^sup>0 \<lparr>R,a\<^sup>P\<rparr>) \<^bold>=(\<^bold>\<lambda>\<^sup>0 \<lparr>R,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>\<^sup>0 \<lparr>R,a\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>\<^sup>0 \<lparr>R,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>\<^sup>0 \<lparr>R,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>\<^sup>0 \<lparr>R,y\<^sup>P\<rparr>) in v]"
        using "\<^bold>\<exists>I" by fast
    qed

  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>2 (\<lambda> x y . \<^bold>\<forall> F . \<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>)"
      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
        using oth_class_taut_4_a[THEN "\<^bold>\<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 . \<^bold>\<exists> F . F \<^bold>= (\<^bold>\<lambda> x . p) in v]"
    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>& \<^bold>\<not>\<lparr>F,y\<^sup>P\<rparr>) in v]"
          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> y . p)) in w]"
        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

  lemma prop_in_f_2:
    "[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 (rule 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]]
              by blast
    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 prop_prop_nec_1:
    "[\<^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 p where "[\<^bold>\<diamond>(F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
        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>) in v]"
        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>= (\<^bold>\<lambda> x . p)) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p)) in v]"
    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\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
      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>
      \<^bold>\<rightarrow> (\<^bold>\<exists> p . F \<^bold>= (\<^bold>\<lambda> x . p))) in v]"
    using derived_S5_rules_1_b enc_prop_nec_1 by blast
end
end