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AOT_PLM.thy
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AOT_PLM.thy
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(*<*)
theory AOT_PLM
imports AOT_Axioms
begin
(*>*)
section\<open>The Deductive System PLM\<close>
text\<open>\label{PLM: 9}\<close>
(* constrain sledgehammer to the abstraction layer *)
unbundle AOT_no_atp
subsection\<open>Primitive Rule of PLM: Modus Ponens\<close>
text\<open>\label{PLM: 9.1}\<close>
AOT_theorem "modus-ponens":
assumes \<open>\<phi>\<close> and \<open>\<phi> \<rightarrow> \<psi>\<close>
shows \<open>\<psi>\<close>
(* NOTE: semantics needed *)
using assms by (simp add: AOT_sem_imp)
lemmas MP = "modus-ponens"
subsection\<open>(Modally Strict) Proofs and Derivations\<close>
text\<open>\label{PLM: 9.2}\<close>
AOT_theorem "non-con-thm-thm":
assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>\<close>
shows \<open>\<^bold>\<turnstile> \<phi>\<close>
using assms by simp
AOT_theorem "vdash-properties:1[1]":
assumes \<open>\<phi> \<in> \<Lambda>\<close>
shows \<open>\<^bold>\<turnstile> \<phi>\<close>
(* NOTE: semantics needed *)
using assms unfolding AOT_model_act_axiom_def by blast
text\<open>Convenience attribute for instantiating modally-fragile axioms.\<close>
attribute_setup act_axiom_inst =
\<open>Scan.succeed (Thm.rule_attribute []
(K (fn thm => thm RS @{thm "vdash-properties:1[1]"})))\<close>
"Instantiate modally fragile axiom as modally fragile theorem."
lemmas "act_axiom_inst" = "vdash-properties:1[1]"
declare "act_axiom_inst"[AOT_inst AOT_model_act_axiom]
AOT_theorem "vdash-properties:1[2]":
assumes \<open>\<phi> \<in> \<Lambda>\<^sub>\<box>\<close>
shows \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>\<close>
(* NOTE: semantics needed *)
using assms unfolding AOT_model_axiom_def by blast
text\<open>Convenience attribute for instantiating modally-strict axioms.\<close>
attribute_setup axiom_inst =
\<open>Scan.succeed (Thm.rule_attribute []
(K (fn thm => thm RS @{thm "vdash-properties:1[2]"})))\<close>
"Instantiate axiom as theorem."
lemmas "axiom_inst" = "vdash-properties:1[2]"
declare "axiom_inst"[AOT_inst AOT_model_axiom]
text\<open>Convenience methods and theorem sets for applying "cqt:2".\<close>
method cqt_2_lambda_inst_prover =
(fast intro: AOT_instance_of_cqt_2_intro)
method "cqt:2[lambda]" =
(rule "cqt:2[lambda]"[axiom_inst]; cqt_2_lambda_inst_prover)
lemmas "cqt:2" =
"cqt:2[const_var]"[axiom_inst] "cqt:2[lambda]"[axiom_inst]
AOT_instance_of_cqt_2_intro
method "cqt:2" = (safe intro!: "cqt:2")
AOT_theorem "vdash-properties:3":
assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>\<close>
shows \<open>\<Gamma> \<^bold>\<turnstile> \<phi>\<close>
using assms by blast
AOT_theorem "vdash-properties:5":
assumes \<open>\<Gamma>\<^sub>1 \<^bold>\<turnstile> \<phi>\<close> and \<open>\<Gamma>\<^sub>2 \<^bold>\<turnstile> \<phi> \<rightarrow> \<psi>\<close>
shows \<open>\<Gamma>\<^sub>1, \<Gamma>\<^sub>2 \<^bold>\<turnstile> \<psi>\<close>
using MP assms by blast
AOT_theorem "vdash-properties:6":
assumes \<open>\<phi>\<close> and \<open>\<phi> \<rightarrow> \<psi>\<close>
shows \<open>\<psi>\<close>
using MP assms by blast
AOT_theorem "vdash-properties:8":
assumes \<open>\<Gamma> \<^bold>\<turnstile> \<phi>\<close> and \<open>\<phi> \<^bold>\<turnstile> \<psi>\<close>
shows \<open>\<Gamma> \<^bold>\<turnstile> \<psi>\<close>
using assms by argo
AOT_theorem "vdash-properties:9":
assumes \<open>\<phi>\<close>
shows \<open>\<psi> \<rightarrow> \<phi>\<close>
using MP "pl:1"[axiom_inst] assms by blast
AOT_theorem "vdash-properties:10":
assumes \<open>\<phi> \<rightarrow> \<psi>\<close> and \<open>\<phi>\<close>
shows \<open>\<psi>\<close>
using MP assms by blast
lemmas "\<rightarrow>E" = "vdash-properties:10"
declare "\<rightarrow>E"[AOT_elim AOT_imp]
subsection\<open>Two Fundamental Metarules: GEN and RN\<close>
text\<open>\label{PLM: 9.3}\<close>
AOT_theorem "rule-gen":
assumes \<open>for arbitrary \<alpha>: \<phi>{\<alpha>}\<close>
shows \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
(* NOTE: semantics needed *)
using assms by (metis AOT_var_of_term_inverse AOT_sem_denotes AOT_sem_forall)
lemmas GEN = "rule-gen"
AOT_theorem "RN[prem]":
assumes \<open>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<phi>\<close>
shows \<open>\<box>\<Gamma> \<^bold>\<turnstile>\<^sub>\<box> \<box>\<phi>\<close>
by (meson AOT_sem_box assms image_iff) (* NOTE: semantics needed *)
AOT_theorem RN:
assumes \<open>\<^bold>\<turnstile>\<^sub>\<box> \<phi>\<close>
shows \<open>\<box>\<phi>\<close>
using "RN[prem]" assms by blast
subsection\<open>The Inferential Role of Definitions\<close>
text\<open>\label{PLM: 9.4}\<close>
AOT_axiom "df-rules-formulas[1]":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close>
shows \<open>\<phi> \<rightarrow> \<psi>\<close>
(* NOTE: semantics needed *)
using assms
by (auto simp: assms AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
AOT_axiom "df-rules-formulas[2]":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close>
shows \<open>\<psi> \<rightarrow> \<phi>\<close>
(* NOTE: semantics needed *)
using assms
by (auto simp: AOT_model_axiomI AOT_model_equiv_def AOT_sem_imp)
(* NOTE: for convenience also state the above as regular theorems *)
AOT_theorem "df-rules-formulas[3]":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close>
shows \<open>\<phi> \<rightarrow> \<psi>\<close>
using "df-rules-formulas[1]"[axiom_inst, OF assms].
AOT_theorem "df-rules-formulas[4]":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close>
shows \<open>\<psi> \<rightarrow> \<phi>\<close>
using "df-rules-formulas[2]"[axiom_inst, OF assms].
AOT_axiom "df-rules-terms[1]":
assumes \<open>\<tau>{\<alpha>\<^sub>1...\<alpha>\<^sub>n} =\<^sub>d\<^sub>f \<sigma>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}\<close>
shows \<open>(\<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down> \<rightarrow> \<tau>{\<tau>\<^sub>1...\<tau>\<^sub>n} = \<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}) &
(\<not>\<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down> \<rightarrow> \<not>\<tau>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down>)\<close>
(* NOTE: semantics needed *)
using assms
by (simp add: AOT_model_axiomI AOT_sem_conj AOT_sem_imp AOT_sem_eq
AOT_sem_not AOT_sem_denotes AOT_model_id_def)
AOT_axiom "df-rules-terms[2]":
assumes \<open>\<tau> =\<^sub>d\<^sub>f \<sigma>\<close>
shows \<open>(\<sigma>\<down> \<rightarrow> \<tau> = \<sigma>) & (\<not>\<sigma>\<down> \<rightarrow> \<not>\<tau>\<down>)\<close>
by (metis "df-rules-terms[1]" case_unit_Unity assms)
(* NOTE: for convenience also state the above as regular theorems *)
AOT_theorem "df-rules-terms[3]":
assumes \<open>\<tau>{\<alpha>\<^sub>1...\<alpha>\<^sub>n} =\<^sub>d\<^sub>f \<sigma>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}\<close>
shows \<open>(\<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down> \<rightarrow> \<tau>{\<tau>\<^sub>1...\<tau>\<^sub>n} = \<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}) &
(\<not>\<sigma>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down> \<rightarrow> \<not>\<tau>{\<tau>\<^sub>1...\<tau>\<^sub>n}\<down>)\<close>
using "df-rules-terms[1]"[axiom_inst, OF assms].
AOT_theorem "df-rules-terms[4]":
assumes \<open>\<tau> =\<^sub>d\<^sub>f \<sigma>\<close>
shows \<open>(\<sigma>\<down> \<rightarrow> \<tau> = \<sigma>) & (\<not>\<sigma>\<down> \<rightarrow> \<not>\<tau>\<down>)\<close>
using "df-rules-terms[2]"[axiom_inst, OF assms].
subsection\<open>The Theory of Negations and Conditionals\<close>
text\<open>\label{PLM: 9.5}\<close>
AOT_theorem "if-p-then-p": \<open>\<phi> \<rightarrow> \<phi>\<close>
by (meson "pl:1"[axiom_inst] "pl:2"[axiom_inst] MP)
AOT_theorem "deduction-theorem":
assumes \<open>\<phi> \<^bold>\<turnstile> \<psi>\<close>
shows \<open>\<phi> \<rightarrow> \<psi>\<close>
(* NOTE: semantics needed *)
using assms by (simp add: AOT_sem_imp)
lemmas CP = "deduction-theorem"
lemmas "\<rightarrow>I" = "deduction-theorem"
AOT_theorem "ded-thm-cor:1":
assumes \<open>\<Gamma>\<^sub>1 \<^bold>\<turnstile> \<phi> \<rightarrow> \<psi>\<close> and \<open>\<Gamma>\<^sub>2 \<^bold>\<turnstile> \<psi> \<rightarrow> \<chi>\<close>
shows \<open>\<Gamma>\<^sub>1, \<Gamma>\<^sub>2 \<^bold>\<turnstile> \<phi> \<rightarrow> \<chi>\<close>
using "\<rightarrow>E" "\<rightarrow>I" assms by blast
AOT_theorem "ded-thm-cor:2":
assumes \<open>\<Gamma>\<^sub>1 \<^bold>\<turnstile> \<phi> \<rightarrow> (\<psi> \<rightarrow> \<chi>)\<close> and \<open>\<Gamma>\<^sub>2 \<^bold>\<turnstile> \<psi>\<close>
shows \<open>\<Gamma>\<^sub>1, \<Gamma>\<^sub>2 \<^bold>\<turnstile> \<phi> \<rightarrow> \<chi>\<close>
using "\<rightarrow>E" "\<rightarrow>I" assms by blast
AOT_theorem "ded-thm-cor:3":
assumes \<open>\<phi> \<rightarrow> \<psi>\<close> and \<open>\<psi> \<rightarrow> \<chi>\<close>
shows \<open>\<phi> \<rightarrow> \<chi>\<close>
using "\<rightarrow>E" "\<rightarrow>I" assms by blast
declare "ded-thm-cor:3"[trans]
AOT_theorem "ded-thm-cor:4":
assumes \<open>\<phi> \<rightarrow> (\<psi> \<rightarrow> \<chi>)\<close> and \<open>\<psi>\<close>
shows \<open>\<phi> \<rightarrow> \<chi>\<close>
using "\<rightarrow>E" "\<rightarrow>I" assms by blast
lemmas "Hypothetical Syllogism" = "ded-thm-cor:3"
AOT_theorem "useful-tautologies:1": \<open>\<not>\<not>\<phi> \<rightarrow> \<phi>\<close>
by (metis "pl:3"[axiom_inst] "\<rightarrow>I" "Hypothetical Syllogism")
AOT_theorem "useful-tautologies:2": \<open>\<phi> \<rightarrow> \<not>\<not>\<phi>\<close>
by (metis "pl:3"[axiom_inst] "\<rightarrow>I" "ded-thm-cor:4")
AOT_theorem "useful-tautologies:3": \<open>\<not>\<phi> \<rightarrow> (\<phi> \<rightarrow> \<psi>)\<close>
by (meson "ded-thm-cor:4" "pl:3"[axiom_inst] "\<rightarrow>I")
AOT_theorem "useful-tautologies:4": \<open>(\<not>\<psi> \<rightarrow> \<not>\<phi>) \<rightarrow> (\<phi> \<rightarrow> \<psi>)\<close>
by (meson "pl:3"[axiom_inst] "Hypothetical Syllogism" "\<rightarrow>I")
AOT_theorem "useful-tautologies:5": \<open>(\<phi> \<rightarrow> \<psi>) \<rightarrow> (\<not>\<psi> \<rightarrow> \<not>\<phi>)\<close>
by (metis "useful-tautologies:4" "Hypothetical Syllogism" "\<rightarrow>I")
AOT_theorem "useful-tautologies:6": \<open>(\<phi> \<rightarrow> \<not>\<psi>) \<rightarrow> (\<psi> \<rightarrow> \<not>\<phi>)\<close>
by (metis "\<rightarrow>I" MP "useful-tautologies:4")
AOT_theorem "useful-tautologies:7": \<open>(\<not>\<phi> \<rightarrow> \<psi>) \<rightarrow> (\<not>\<psi> \<rightarrow> \<phi>)\<close>
by (metis "\<rightarrow>I" MP "useful-tautologies:3" "useful-tautologies:5")
AOT_theorem "useful-tautologies:8": \<open>\<phi> \<rightarrow> (\<not>\<psi> \<rightarrow> \<not>(\<phi> \<rightarrow> \<psi>))\<close>
by (metis "\<rightarrow>I" MP "useful-tautologies:5")
AOT_theorem "useful-tautologies:9": \<open>(\<phi> \<rightarrow> \<psi>) \<rightarrow> ((\<not>\<phi> \<rightarrow> \<psi>) \<rightarrow> \<psi>)\<close>
by (metis "\<rightarrow>I" MP "useful-tautologies:6")
AOT_theorem "useful-tautologies:10": \<open>(\<phi> \<rightarrow> \<not>\<psi>) \<rightarrow> ((\<phi> \<rightarrow> \<psi>) \<rightarrow> \<not>\<phi>)\<close>
by (metis "\<rightarrow>I" MP "pl:3"[axiom_inst])
AOT_theorem "dn-i-e:1":
assumes \<open>\<phi>\<close>
shows \<open>\<not>\<not>\<phi>\<close>
using MP "useful-tautologies:2" assms by blast
lemmas "\<not>\<not>I" = "dn-i-e:1"
AOT_theorem "dn-i-e:2":
assumes \<open>\<not>\<not>\<phi>\<close>
shows \<open>\<phi>\<close>
using MP "useful-tautologies:1" assms by blast
lemmas "\<not>\<not>E" = "dn-i-e:2"
AOT_theorem "modus-tollens:1":
assumes \<open>\<phi> \<rightarrow> \<psi>\<close> and \<open>\<not>\<psi>\<close>
shows \<open>\<not>\<phi>\<close>
using MP "useful-tautologies:5" assms by blast
AOT_theorem "modus-tollens:2":
assumes \<open>\<phi> \<rightarrow> \<not>\<psi>\<close> and \<open>\<psi>\<close>
shows \<open>\<not>\<phi>\<close>
using "\<not>\<not>I" "modus-tollens:1" assms by blast
lemmas MT = "modus-tollens:1" "modus-tollens:2"
AOT_theorem "contraposition:1[1]":
assumes \<open>\<phi> \<rightarrow> \<psi>\<close>
shows \<open>\<not>\<psi> \<rightarrow> \<not>\<phi>\<close>
using "\<rightarrow>I" MT(1) assms by blast
AOT_theorem "contraposition:1[2]":
assumes \<open>\<not>\<psi> \<rightarrow> \<not>\<phi>\<close>
shows \<open>\<phi> \<rightarrow> \<psi>\<close>
using "\<rightarrow>I" "\<not>\<not>E" MT(2) assms by blast
AOT_theorem "contraposition:2":
assumes \<open>\<phi> \<rightarrow> \<not>\<psi>\<close>
shows \<open>\<psi> \<rightarrow> \<not>\<phi>\<close>
using "\<rightarrow>I" MT(2) assms by blast
AOT_theorem "reductio-aa:1":
assumes \<open>\<not>\<phi> \<^bold>\<turnstile> \<not>\<psi>\<close> and \<open>\<not>\<phi> \<^bold>\<turnstile> \<psi>\<close>
shows \<open>\<phi>\<close>
using "\<rightarrow>I" "\<not>\<not>E" MT(2) assms by blast
AOT_theorem "reductio-aa:2":
assumes \<open>\<phi> \<^bold>\<turnstile> \<not>\<psi>\<close> and \<open>\<phi> \<^bold>\<turnstile> \<psi>\<close>
shows \<open>\<not>\<phi>\<close>
using "reductio-aa:1" assms by blast
lemmas "RAA" = "reductio-aa:1" "reductio-aa:2"
AOT_theorem "exc-mid": \<open>\<phi> \<or> \<not>\<phi>\<close>
using "df-rules-formulas[4]" "if-p-then-p" MP
"conventions:2" by blast
AOT_theorem "non-contradiction": \<open>\<not>(\<phi> & \<not>\<phi>)\<close>
using "df-rules-formulas[3]" MT(2) "useful-tautologies:2"
"conventions:1" by blast
AOT_theorem "con-dis-taut:1": \<open>(\<phi> & \<psi>) \<rightarrow> \<phi>\<close>
by (meson "\<rightarrow>I" "df-rules-formulas[3]" MP RAA(1) "conventions:1")
AOT_theorem "con-dis-taut:2": \<open>(\<phi> & \<psi>) \<rightarrow> \<psi>\<close>
by (metis "\<rightarrow>I" "df-rules-formulas[3]" MT(2) RAA(2)
"\<not>\<not>E" "conventions:1")
lemmas "Conjunction Simplification" = "con-dis-taut:1" "con-dis-taut:2"
AOT_theorem "con-dis-taut:3": \<open>\<phi> \<rightarrow> (\<phi> \<or> \<psi>)\<close>
by (meson "contraposition:1[2]" "df-rules-formulas[4]"
MP "\<rightarrow>I" "conventions:2")
AOT_theorem "con-dis-taut:4": \<open>\<psi> \<rightarrow> (\<phi> \<or> \<psi>)\<close>
using "Hypothetical Syllogism" "df-rules-formulas[4]"
"pl:1"[axiom_inst] "conventions:2" by blast
lemmas "Disjunction Addition" = "con-dis-taut:3" "con-dis-taut:4"
AOT_theorem "con-dis-taut:5": \<open>\<phi> \<rightarrow> (\<psi> \<rightarrow> (\<phi> & \<psi>))\<close>
by (metis "contraposition:2" "Hypothetical Syllogism" "\<rightarrow>I"
"df-rules-formulas[4]" "conventions:1")
lemmas Adjunction = "con-dis-taut:5"
AOT_theorem "con-dis-taut:6": \<open>(\<phi> & \<phi>) \<equiv> \<phi>\<close>
by (metis Adjunction "\<rightarrow>I" "df-rules-formulas[4]" MP
"Conjunction Simplification"(1) "conventions:3")
lemmas "Idempotence of &" = "con-dis-taut:6"
AOT_theorem "con-dis-taut:7": \<open>(\<phi> \<or> \<phi>) \<equiv> \<phi>\<close>
proof -
{
AOT_assume \<open>\<phi> \<or> \<phi>\<close>
AOT_hence \<open>\<not>\<phi> \<rightarrow> \<phi>\<close>
using "conventions:2"[THEN "df-rules-formulas[3]"] MP by blast
AOT_hence \<open>\<phi>\<close> using "if-p-then-p" RAA(1) MP by blast
}
moreover {
AOT_assume \<open>\<phi>\<close>
AOT_hence \<open>\<phi> \<or> \<phi>\<close> using "Disjunction Addition"(1) MP by blast
}
ultimately AOT_show \<open>(\<phi> \<or> \<phi>) \<equiv> \<phi>\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"] MP
by (metis Adjunction "\<rightarrow>I")
qed
lemmas "Idempotence of \<or>" = "con-dis-taut:7"
AOT_theorem "con-dis-i-e:1":
assumes \<open>\<phi>\<close> and \<open>\<psi>\<close>
shows \<open>\<phi> & \<psi>\<close>
using Adjunction MP assms by blast
lemmas "&I" = "con-dis-i-e:1"
declare "&I"[AOT_intro]
AOT_theorem "con-dis-i-e:2:a":
assumes \<open>\<phi> & \<psi>\<close>
shows \<open>\<phi>\<close>
using "Conjunction Simplification"(1) MP assms by blast
AOT_theorem "con-dis-i-e:2:b":
assumes \<open>\<phi> & \<psi>\<close>
shows \<open>\<psi>\<close>
using "Conjunction Simplification"(2) MP assms by blast
lemmas "&E" = "con-dis-i-e:2:a" "con-dis-i-e:2:b"
declare "&E"(1)[AOT_elim AOT_conj]
declare "&E"(2)[AOT_elim AOT_conj]
AOT_theorem "con-dis-i-e:3:a":
assumes \<open>\<phi>\<close>
shows \<open>\<phi> \<or> \<psi>\<close>
using "Disjunction Addition"(1) MP assms by blast
AOT_theorem "con-dis-i-e:3:b":
assumes \<open>\<psi>\<close>
shows \<open>\<phi> \<or> \<psi>\<close>
using "Disjunction Addition"(2) MP assms by blast
AOT_theorem "con-dis-i-e:3:c":
assumes \<open>\<phi> \<or> \<psi>\<close> and \<open>\<phi> \<rightarrow> \<chi>\<close> and \<open>\<psi> \<rightarrow> \<Theta>\<close>
shows \<open>\<chi> \<or> \<Theta>\<close>
by (metis "con-dis-i-e:3:a" "Disjunction Addition"(2)
"df-rules-formulas[3]" MT(1) RAA(1)
"conventions:2" assms)
lemmas "\<or>I" = "con-dis-i-e:3:a" "con-dis-i-e:3:b" "con-dis-i-e:3:c"
AOT_theorem "con-dis-i-e:4:a":
assumes \<open>\<phi> \<or> \<psi>\<close> and \<open>\<phi> \<rightarrow> \<chi>\<close> and \<open>\<psi> \<rightarrow> \<chi>\<close>
shows \<open>\<chi>\<close>
by (metis MP RAA(2) "df-rules-formulas[3]" "conventions:2" assms)
AOT_theorem "con-dis-i-e:4:b":
assumes \<open>\<phi> \<or> \<psi>\<close> and \<open>\<not>\<phi>\<close>
shows \<open>\<psi>\<close>
using "con-dis-i-e:4:a" RAA(1) "\<rightarrow>I" assms by blast
AOT_theorem "con-dis-i-e:4:c":
assumes \<open>\<phi> \<or> \<psi>\<close> and \<open>\<not>\<psi>\<close>
shows \<open>\<phi>\<close>
using "con-dis-i-e:4:a" RAA(1) "\<rightarrow>I" assms by blast
lemmas "\<or>E" = "con-dis-i-e:4:a" "con-dis-i-e:4:b" "con-dis-i-e:4:c"
AOT_theorem "raa-cor:1":
assumes \<open>\<not>\<phi> \<^bold>\<turnstile> \<psi> & \<not>\<psi>\<close>
shows \<open>\<phi>\<close>
using "&E" "\<or>E"(3) "\<or>I"(2) RAA(2) assms by blast
AOT_theorem "raa-cor:2":
assumes \<open>\<phi> \<^bold>\<turnstile> \<psi> & \<not>\<psi>\<close>
shows \<open>\<not>\<phi>\<close>
using "raa-cor:1" assms by blast
AOT_theorem "raa-cor:3":
assumes \<open>\<phi>\<close> and \<open>\<not>\<psi> \<^bold>\<turnstile> \<not>\<phi>\<close>
shows \<open>\<psi>\<close>
using RAA assms by blast
AOT_theorem "raa-cor:4":
assumes \<open>\<not>\<phi>\<close> and \<open>\<not>\<psi> \<^bold>\<turnstile> \<phi>\<close>
shows \<open>\<psi>\<close>
using RAA assms by blast
AOT_theorem "raa-cor:5":
assumes \<open>\<phi>\<close> and \<open>\<psi> \<^bold>\<turnstile> \<not>\<phi>\<close>
shows \<open>\<not>\<psi>\<close>
using RAA assms by blast
AOT_theorem "raa-cor:6":
assumes \<open>\<not>\<phi>\<close> and \<open>\<psi> \<^bold>\<turnstile> \<phi>\<close>
shows \<open>\<not>\<psi>\<close>
using RAA assms by blast
AOT_theorem "oth-class-taut:1:a": \<open>(\<phi> \<rightarrow> \<psi>) \<equiv> \<not>(\<phi> & \<not>\<psi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&E" "&I" "raa-cor:3" "\<rightarrow>I" MP)
AOT_theorem "oth-class-taut:1:b": \<open>\<not>(\<phi> \<rightarrow> \<psi>) \<equiv> (\<phi> & \<not>\<psi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&E" "&I" "raa-cor:3" "\<rightarrow>I" MP)
AOT_theorem "oth-class-taut:1:c": \<open>(\<phi> \<rightarrow> \<psi>) \<equiv> (\<not>\<phi> \<or> \<psi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&I" "\<or>I"(1, 2) "\<or>E"(3) "\<rightarrow>I" MP "raa-cor:1")
AOT_theorem "oth-class-taut:2:a": \<open>(\<phi> & \<psi>) \<equiv> (\<psi> & \<phi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(meson "&I" "&E" "\<rightarrow>I")
lemmas "Commutativity of &" = "oth-class-taut:2:a"
AOT_theorem "oth-class-taut:2:b": \<open>(\<phi> & (\<psi> & \<chi>)) \<equiv> ((\<phi> & \<psi>) & \<chi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&I" "&E" "\<rightarrow>I")
lemmas "Associativity of &" = "oth-class-taut:2:b"
AOT_theorem "oth-class-taut:2:c": \<open>(\<phi> \<or> \<psi>) \<equiv> (\<psi> \<or> \<phi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&I" "\<or>I"(1, 2) "\<or>E"(1) "\<rightarrow>I")
lemmas "Commutativity of \<or>" = "oth-class-taut:2:c"
AOT_theorem "oth-class-taut:2:d": \<open>(\<phi> \<or> (\<psi> \<or> \<chi>)) \<equiv> ((\<phi> \<or> \<psi>) \<or> \<chi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"])
(metis "&I" "\<or>I"(1, 2) "\<or>E"(1) "\<rightarrow>I")
lemmas "Associativity of \<or>" = "oth-class-taut:2:d"
AOT_theorem "oth-class-taut:2:e": \<open>(\<phi> \<equiv> \<psi>) \<equiv> (\<psi> \<equiv> \<phi>)\<close>
by (rule "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"]; rule "&I";
metis "&I" "df-rules-formulas[4]" "conventions:3" "&E"
"Hypothetical Syllogism" "\<rightarrow>I" "df-rules-formulas[3]")
lemmas "Commutativity of \<equiv>" = "oth-class-taut:2:e"
AOT_theorem "oth-class-taut:2:f": \<open>(\<phi> \<equiv> (\<psi> \<equiv> \<chi>)) \<equiv> ((\<phi> \<equiv> \<psi>) \<equiv> \<chi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I"
by metis
lemmas "Associativity of \<equiv>" = "oth-class-taut:2:f"
AOT_theorem "oth-class-taut:3:a": \<open>\<phi> \<equiv> \<phi>\<close>
using "&I" "vdash-properties:6" "if-p-then-p"
"df-rules-formulas[4]" "conventions:3" by blast
AOT_theorem "oth-class-taut:3:b": \<open>\<phi> \<equiv> \<not>\<not>\<phi>\<close>
using "&I" "useful-tautologies:1" "useful-tautologies:2" "\<rightarrow>E"
"df-rules-formulas[4]" "conventions:3" by blast
AOT_theorem "oth-class-taut:3:c": \<open>\<not>(\<phi> \<equiv> \<not>\<phi>)\<close>
by (metis "&E" "\<rightarrow>E" RAA "df-rules-formulas[3]" "conventions:3")
AOT_theorem "oth-class-taut:4:a": \<open>(\<phi> \<rightarrow> \<psi>) \<rightarrow> ((\<psi> \<rightarrow> \<chi>) \<rightarrow> (\<phi> \<rightarrow> \<chi>))\<close>
by (metis "\<rightarrow>E" "\<rightarrow>I")
AOT_theorem "oth-class-taut:4:b": \<open>(\<phi> \<equiv> \<psi>) \<equiv> (\<not>\<phi> \<equiv> \<not>\<psi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" RAA by metis
AOT_theorem "oth-class-taut:4:c": \<open>(\<phi> \<equiv> \<psi>) \<rightarrow> ((\<phi> \<rightarrow> \<chi>) \<equiv> (\<psi> \<rightarrow> \<chi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" by metis
AOT_theorem "oth-class-taut:4:d": \<open>(\<phi> \<equiv> \<psi>) \<rightarrow> ((\<chi> \<rightarrow> \<phi>) \<equiv> (\<chi> \<rightarrow> \<psi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" by metis
AOT_theorem "oth-class-taut:4:e": \<open>(\<phi> \<equiv> \<psi>) \<rightarrow> ((\<phi> & \<chi>) \<equiv> (\<psi> & \<chi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" by metis
AOT_theorem "oth-class-taut:4:f": \<open>(\<phi> \<equiv> \<psi>) \<rightarrow> ((\<chi> & \<phi>) \<equiv> (\<chi> & \<psi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" by metis
AOT_theorem "oth-class-taut:4:g": \<open>(\<phi> \<equiv> \<psi>) \<equiv> ((\<phi> & \<psi>) \<or> (\<not>\<phi> & \<not>\<psi>))\<close>
proof(safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"]
"&I" "\<rightarrow>I"
dest!: "conventions:3"[THEN "df-rules-formulas[3]", THEN "\<rightarrow>E"])
AOT_show \<open>\<phi> & \<psi> \<or> (\<not>\<phi> & \<not>\<psi>)\<close> if \<open>(\<phi> \<rightarrow> \<psi>) & (\<psi> \<rightarrow> \<phi>)\<close>
using "&E" "\<or>I" "\<rightarrow>E" "&I" "raa-cor:1" "\<rightarrow>I" "\<or>E" that by metis
next
AOT_show \<open>\<psi>\<close> if \<open>\<phi> & \<psi> \<or> (\<not>\<phi> & \<not>\<psi>)\<close> and \<open>\<phi>\<close>
using that "\<or>E" "&E" "raa-cor:3" by blast
next
AOT_show \<open>\<phi>\<close> if \<open>\<phi> & \<psi> \<or> (\<not>\<phi> & \<not>\<psi>)\<close> and \<open>\<psi>\<close>
using that "\<or>E" "&E" "raa-cor:3" by blast
qed
AOT_theorem "oth-class-taut:4:h": \<open>\<not>(\<phi> \<equiv> \<psi>) \<equiv> ((\<phi> & \<not>\<psi>) \<or> (\<not>\<phi> & \<psi>))\<close>
proof (safe intro!: "conventions:3"[THEN "df-rules-formulas[4]", THEN "\<rightarrow>E"]
"&I" "\<rightarrow>I")
AOT_show \<open>\<phi> & \<not>\<psi> \<or> (\<not>\<phi> & \<psi>)\<close> if \<open>\<not>(\<phi> \<equiv> \<psi>)\<close>
by (metis that "&I" "\<or>I"(1, 2) "\<rightarrow>I" MT(1) "df-rules-formulas[4]"
"raa-cor:3" "conventions:3")
next
AOT_show \<open>\<not>(\<phi> \<equiv> \<psi>)\<close> if \<open>\<phi> & \<not>\<psi> \<or> (\<not>\<phi> & \<psi>)\<close>
by (metis that "&E" "\<or>E"(2) "\<rightarrow>E" "df-rules-formulas[3]"
"raa-cor:3" "conventions:3")
qed
AOT_theorem "oth-class-taut:5:a": \<open>(\<phi> & \<psi>) \<equiv> \<not>(\<not>\<phi> \<or> \<not>\<psi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
AOT_theorem "oth-class-taut:5:b": \<open>(\<phi> \<or> \<psi>) \<equiv> \<not>(\<not>\<phi> & \<not>\<psi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
AOT_theorem "oth-class-taut:5:c": \<open>\<not>(\<phi> & \<psi>) \<equiv> (\<not>\<phi> \<or> \<not>\<psi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
AOT_theorem "oth-class-taut:5:d": \<open>\<not>(\<phi> \<or> \<psi>) \<equiv> (\<not>\<phi> & \<not>\<psi>)\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
lemmas DeMorgan = "oth-class-taut:5:c" "oth-class-taut:5:d"
AOT_theorem "oth-class-taut:6:a":
\<open>(\<phi> & (\<psi> \<or> \<chi>)) \<equiv> ((\<phi> & \<psi>) \<or> (\<phi> & \<chi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
AOT_theorem "oth-class-taut:6:b":
\<open>(\<phi> \<or> (\<psi> & \<chi>)) \<equiv> ((\<phi> \<or> \<psi>) & (\<phi> \<or> \<chi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E" RAA by metis
AOT_theorem "oth-class-taut:7:a": \<open>((\<phi> & \<psi>) \<rightarrow> \<chi>) \<rightarrow> (\<phi> \<rightarrow> (\<psi> \<rightarrow> \<chi>))\<close>
by (metis "&I" "\<rightarrow>E" "\<rightarrow>I")
lemmas Exportation = "oth-class-taut:7:a"
AOT_theorem "oth-class-taut:7:b": \<open>(\<phi> \<rightarrow> (\<psi> \<rightarrow>\<chi>)) \<rightarrow> ((\<phi> & \<psi>) \<rightarrow> \<chi>)\<close>
by (metis "&E" "\<rightarrow>E" "\<rightarrow>I")
lemmas Importation = "oth-class-taut:7:b"
AOT_theorem "oth-class-taut:8:a":
\<open>(\<phi> \<rightarrow> (\<psi> \<rightarrow> \<chi>)) \<equiv> (\<psi> \<rightarrow> (\<phi> \<rightarrow> \<chi>))\<close>
using "conventions:3"[THEN "df-rules-formulas[4]"] "\<rightarrow>I" "\<rightarrow>E" "&E" "&I"
by metis
lemmas Permutation = "oth-class-taut:8:a"
AOT_theorem "oth-class-taut:8:b":
\<open>(\<phi> \<rightarrow> \<psi>) \<rightarrow> ((\<phi> \<rightarrow> \<chi>) \<rightarrow> (\<phi> \<rightarrow> (\<psi> & \<chi>)))\<close>
by (metis "&I" "\<rightarrow>E" "\<rightarrow>I")
lemmas Composition = "oth-class-taut:8:b"
AOT_theorem "oth-class-taut:8:c":
\<open>(\<phi> \<rightarrow> \<chi>) \<rightarrow> ((\<psi> \<rightarrow> \<chi>) \<rightarrow> ((\<phi> \<or> \<psi>) \<rightarrow> \<chi>))\<close>
by (metis "\<or>E"(2) "\<rightarrow>E" "\<rightarrow>I" RAA(1))
AOT_theorem "oth-class-taut:8:d":
\<open>((\<phi> \<rightarrow> \<psi>) & (\<chi> \<rightarrow> \<Theta>)) \<rightarrow> ((\<phi> & \<chi>) \<rightarrow> (\<psi> & \<Theta>))\<close>
by (metis "&E" "&I" "\<rightarrow>E" "\<rightarrow>I")
lemmas "Double Composition" = "oth-class-taut:8:d"
AOT_theorem "oth-class-taut:8:e":
\<open>((\<phi> & \<psi>) \<equiv> (\<phi> & \<chi>)) \<equiv> (\<phi> \<rightarrow> (\<psi> \<equiv> \<chi>))\<close>
by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I")
AOT_theorem "oth-class-taut:8:f":
\<open>((\<phi> & \<psi>) \<equiv> (\<chi> & \<psi>)) \<equiv> (\<psi> \<rightarrow> (\<phi> \<equiv> \<chi>))\<close>
by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I")
AOT_theorem "oth-class-taut:8:g":
\<open>(\<psi> \<equiv> \<chi>) \<rightarrow> ((\<phi> \<or> \<psi>) \<equiv> (\<phi> \<or> \<chi>))\<close>
by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E"(1))
AOT_theorem "oth-class-taut:8:h":
\<open>(\<psi> \<equiv> \<chi>) \<rightarrow> ((\<psi> \<or> \<phi>) \<equiv> (\<chi> \<or> \<phi>))\<close>
by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I" "\<or>I" "\<or>E"(1))
AOT_theorem "oth-class-taut:8:i":
\<open>(\<phi> \<equiv> (\<psi> & \<chi>)) \<rightarrow> (\<psi> \<rightarrow> (\<phi> \<equiv> \<chi>))\<close>
by (metis "conventions:3"[THEN "df-rules-formulas[4]"]
"conventions:3"[THEN "df-rules-formulas[3]"]
"\<rightarrow>I" "\<rightarrow>E" "&E" "&I")
AOT_theorem "intro-elim:1":
assumes \<open>\<phi> \<or> \<psi>\<close> and \<open>\<phi> \<equiv> \<chi>\<close> and \<open>\<psi> \<equiv> \<Theta>\<close>
shows \<open>\<chi> \<or> \<Theta>\<close>
by (metis assms "\<or>I"(1, 2) "\<or>E"(1) "\<rightarrow>I" "\<rightarrow>E" "&E"(1)
"conventions:3"[THEN "df-rules-formulas[3]"])
AOT_theorem "intro-elim:2":
assumes \<open>\<phi> \<rightarrow> \<psi>\<close> and \<open>\<psi> \<rightarrow> \<phi>\<close>
shows \<open>\<phi> \<equiv> \<psi>\<close>
by (meson "&I" "conventions:3" "df-rules-formulas[4]" MP assms)
lemmas "\<equiv>I" = "intro-elim:2"
AOT_theorem "intro-elim:3:a":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<phi>\<close>
shows \<open>\<psi>\<close>
by (metis "\<or>I"(1) "\<rightarrow>I" "\<or>E"(1) "intro-elim:1" assms)
AOT_theorem "intro-elim:3:b":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<psi>\<close>
shows \<open>\<phi>\<close>
using "intro-elim:3:a" "Commutativity of \<equiv>" assms by blast
AOT_theorem "intro-elim:3:c":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<not>\<phi>\<close>
shows \<open>\<not>\<psi>\<close>
using "intro-elim:3:b" "raa-cor:3" assms by blast
AOT_theorem "intro-elim:3:d":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<not>\<psi>\<close>
shows \<open>\<not>\<phi>\<close>
using "intro-elim:3:a" "raa-cor:3" assms by blast
AOT_theorem "intro-elim:3:e":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<psi> \<equiv> \<chi>\<close>
shows \<open>\<phi> \<equiv> \<chi>\<close>
by (metis "\<equiv>I" "\<rightarrow>I" "intro-elim:3:a" "intro-elim:3:b" assms)
declare "intro-elim:3:e"[trans]
AOT_theorem "intro-elim:3:f":
assumes \<open>\<phi> \<equiv> \<psi>\<close> and \<open>\<phi> \<equiv> \<chi>\<close>
shows \<open>\<chi> \<equiv> \<psi>\<close>
by (metis "\<equiv>I" "\<rightarrow>I" "intro-elim:3:a" "intro-elim:3:b" assms)
lemmas "\<equiv>E" = "intro-elim:3:a" "intro-elim:3:b" "intro-elim:3:c"
"intro-elim:3:d" "intro-elim:3:e" "intro-elim:3:f"
declare "\<equiv>E"(1)[AOT_elim AOT_equiv]
declare "\<equiv>E"(2)[AOT_elim AOT_equiv]
declare "Commutativity of \<equiv>"[THEN "\<equiv>E"(1), sym]
AOT_theorem "rule-eq-df:1":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close>
shows \<open>\<phi> \<equiv> \<psi>\<close>
by (simp add: "\<equiv>I" "df-rules-formulas[3]" "df-rules-formulas[4]" assms)
lemmas "\<equiv>Df" = "rule-eq-df:1"
AOT_theorem "rule-eq-df:2":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close> and \<open>\<phi>\<close>
shows \<open>\<psi>\<close>
using "\<equiv>Df" "\<equiv>E"(1) assms by blast
lemmas "\<equiv>\<^sub>d\<^sub>fE" = "rule-eq-df:2"
declare "\<equiv>\<^sub>d\<^sub>fE"[AOT_elim AOT_model_equiv_def]
AOT_theorem "rule-eq-df:3":
assumes \<open>\<phi> \<equiv>\<^sub>d\<^sub>f \<psi>\<close> and \<open>\<psi>\<close>
shows \<open>\<phi>\<close>
using "\<equiv>Df" "\<equiv>E"(2) assms by blast
lemmas "\<equiv>\<^sub>d\<^sub>fI" = "rule-eq-df:3"
declare "\<equiv>\<^sub>d\<^sub>fI"[AOT_elim AOT_model_equiv_def]
AOT_theorem "df-simplify:1":
assumes \<open>\<phi> \<equiv> (\<psi> & \<chi>)\<close> and \<open>\<psi>\<close>
shows \<open>\<phi> \<equiv> \<chi>\<close>
by (metis "&E"(2) "&I" "\<equiv>E"(1, 2) "\<equiv>I" "\<rightarrow>I" assms)
(* Note: this is a slight variation from PLM *)
AOT_theorem "df-simplify:2":
assumes \<open>\<phi> \<equiv> (\<psi> & \<chi>)\<close> and \<open>\<chi>\<close>
shows \<open>\<phi> \<equiv> \<psi>\<close>
by (metis "&E"(1) "&I" "\<equiv>E"(1, 2) "\<equiv>I" "\<rightarrow>I" assms)
lemmas "\<equiv>S" = "df-simplify:1" "df-simplify:2"
subsection\<open>The Theory of Quantification\<close>
text\<open>\label{PLM: 9.6}\<close>
AOT_theorem "rule-ui:1":
assumes \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close> and \<open>\<tau>\<down>\<close>
shows \<open>\<phi>{\<tau>}\<close>
using "\<rightarrow>E" "cqt:1"[axiom_inst] assms by blast
AOT_theorem "rule-ui:2[const_var]":
assumes \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
shows \<open>\<phi>{\<beta>}\<close>
by (simp add: "rule-ui:1" "cqt:2[const_var]"[axiom_inst] assms)
AOT_theorem "rule-ui:2[lambda]":
assumes \<open>\<forall>F \<phi>{F}\<close> and \<open>INSTANCE_OF_CQT_2(\<psi>)\<close>
shows \<open>\<phi>{[\<lambda>\<nu>\<^sub>1...\<nu>\<^sub>n \<psi>{\<nu>\<^sub>1...\<nu>\<^sub>n}]}\<close>
by (simp add: "rule-ui:1" "cqt:2[lambda]"[axiom_inst] assms)
AOT_theorem "rule-ui:3":
assumes \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
shows \<open>\<phi>{\<alpha>}\<close>
by (simp add: "rule-ui:2[const_var]" assms)
lemmas "\<forall>E" = "rule-ui:1" "rule-ui:2[const_var]"
"rule-ui:2[lambda]" "rule-ui:3"
declare "\<forall>E"(1)[AOT_elim AOT_forall]
AOT_theorem "cqt-orig:1[const_var]": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<phi>{\<beta>}\<close>
by (simp add: "\<forall>E"(2) "\<rightarrow>I")
AOT_theorem "cqt-orig:1[lambda]":
assumes \<open>INSTANCE_OF_CQT_2(\<psi>)\<close>
shows \<open>\<forall>F \<phi>{F} \<rightarrow> \<phi>{[\<lambda>\<nu>\<^sub>1...\<nu>\<^sub>n \<psi>{\<nu>\<^sub>1...\<nu>\<^sub>n}]}\<close>
by (simp add: "\<forall>E"(3) "\<rightarrow>I" assms)
AOT_theorem "cqt-orig:2": \<open>\<forall>\<alpha> (\<phi> \<rightarrow> \<psi>{\<alpha>}) \<rightarrow> (\<phi> \<rightarrow> \<forall>\<alpha> \<psi>{\<alpha>})\<close>
by (metis "\<rightarrow>I" GEN "vdash-properties:6" "\<forall>E"(4))
AOT_theorem "cqt-orig:3": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>}\<close>
using "cqt-orig:1[const_var]".
AOT_theorem universal:
assumes \<open>for arbitrary \<beta>: \<phi>{\<beta>}\<close>
shows \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
using GEN assms .
lemmas "\<forall>I" = universal
(* Generalized mechanism for \<forall>I followed by \<forall>E *)
ML\<open>
fun get_instantiated_allI' ctxt match thm = let
val trm = Thm.concl_of thm
val trm =
case trm of (@{const Trueprop} $ (@{const AOT_model_valid_in} $ _ $ x)) => x
| _ => raise Term.TERM ("Expected simple theorem.", [trm])
fun extractVars (Const (\<^const_name>\<open>AOT_term_of_var\<close>, _) $ Var v) =
(if match v then [Var v] else [])
| extractVars (t1 $ t2) = extractVars t1 @ extractVars t2
| extractVars (Abs (_, _, t)) = extractVars t
| extractVars _ = []
val vars = extractVars trm
val vars = fold Term.add_vars vars []
val var = hd vars
val trmty =
case (snd var) of (Type (\<^type_name>\<open>AOT_var\<close>, [t])) => (t)
| _ => raise Term.TYPE ("Expected variable type.", [snd var], [Var var])
val trm = Abs (Term.string_of_vname (fst var), trmty, Term.abstract_over (
Const (\<^const_name>\<open>AOT_term_of_var\<close>, Type ("fun", [snd var, trmty]))
$ Var var, trm))
val trm = Thm.cterm_of (Context.proof_of ctxt) trm
val ty = hd (Term.add_tvars (Thm.prop_of @{thm "\<forall>I"}) [])
val typ = Thm.ctyp_of (Context.proof_of ctxt) trmty
val allthm = Drule.instantiate_normalize (TVars.make [(ty, typ)], Vars.empty) @{thm "\<forall>I"}
val phi = hd (Term.add_vars (Thm.prop_of allthm) [])
val allthm = Drule.instantiate_normalize (TVars.empty, Vars.make [(phi,trm)]) allthm
val allthm = Thm.put_name_hint ("unvarify_"^fst (fst var)) allthm
in
allthm
end
fun get_instantiated_allI ctxt varname thm = get_instantiated_allI' ctxt (fn v => fst (fst v) = fst varname) thm
\<close>
local_setup\<open>AOT_add_varify_rule (\<^const_name>\<open>AOT_var.AOT_term_of_var\<close>,
fn ctxt => (
fn (Var arg) => (fn thm => SOME (get_instantiated_allI' ctxt (fn var => var = arg) thm))
| x => K NONE
))\<close>
attribute_setup "\<forall>I" =
\<open>Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
(fn ctxt => fn thm => fold (fn arg => fn thm =>
thm RS get_instantiated_allI ctxt arg thm) args thm))\<close>
"Quantify over a variable in a theorem using GEN."
attribute_setup "unvarify" =
\<open>Scan.lift (Scan.repeat1 Args.var) >> (fn args => Thm.rule_attribute []
(fn ctxt => fn thm =>
let
fun get_inst_allI arg thm = thm RS get_instantiated_allI ctxt arg thm
val thm = fold get_inst_allI args thm
val thm = fold (K (fn thm => thm RS @{thm "\<forall>E"(1)})) args thm
in
Object_Logic.rulify (Context.proof_of ctxt) thm
end))\<close>
"Generalize a statement about variables to a statement about denoting terms."
(* Note: rereplace-lem does not apply to the embedding *)
AOT_theorem "cqt-basic:1": \<open>\<forall>\<alpha>\<forall>\<beta> \<phi>{\<alpha>,\<beta>} \<equiv> \<forall>\<beta>\<forall>\<alpha> \<phi>{\<alpha>,\<beta>}\<close>
by (metis "\<equiv>I" "\<forall>E"(2) "\<forall>I" "\<rightarrow>I")
AOT_theorem "cqt-basic:2":
\<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) \<equiv> (\<forall>\<alpha>(\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}) & \<forall>\<alpha>(\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>}))\<close>
proof (rule "\<equiv>I"; rule "\<rightarrow>I")
AOT_assume \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
AOT_hence \<open>\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}\<close> for \<alpha> using "\<forall>E"(2) by blast
AOT_hence \<open>\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}\<close> and \<open>\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>}\<close> for \<alpha>
using "\<equiv>E"(1,2) "\<rightarrow>I" by blast+
AOT_thus \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}) & \<forall>\<alpha>(\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>})\<close>
by (auto intro: "&I" "\<forall>I")
next
AOT_assume \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}) & \<forall>\<alpha>(\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>})\<close>
AOT_hence \<open>\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}\<close> and \<open>\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>}\<close> for \<alpha>
using "\<forall>E"(2) "&E" by blast+
AOT_hence \<open>\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}\<close> for \<alpha>
using "\<equiv>I" by blast
AOT_thus \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close> by (auto intro: "\<forall>I")
qed
AOT_theorem "cqt-basic:3": \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) \<rightarrow> (\<forall>\<alpha> \<phi>{\<alpha>} \<equiv> \<forall>\<alpha> \<psi>{\<alpha>})\<close>
proof(rule "\<rightarrow>I")
AOT_assume \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
AOT_hence 1: \<open>\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}\<close> for \<alpha> using "\<forall>E"(2) by blast
{
AOT_assume \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
AOT_hence \<open>\<forall>\<alpha> \<psi>{\<alpha>}\<close> using 1 "\<forall>I" "\<forall>E"(4) "\<equiv>E" by metis
}
moreover {
AOT_assume \<open>\<forall>\<alpha> \<psi>{\<alpha>}\<close>
AOT_hence \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close> using 1 "\<forall>I" "\<forall>E"(4) "\<equiv>E" by metis
}
ultimately AOT_show \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<equiv> \<forall>\<alpha> \<psi>{\<alpha>}\<close>
using "\<equiv>I" "\<rightarrow>I" by auto
qed
AOT_theorem "cqt-basic:4": \<open>\<forall>\<alpha>(\<phi>{\<alpha>} & \<psi>{\<alpha>}) \<equiv> (\<forall>\<alpha> \<phi>{\<alpha>} & \<forall>\<alpha> \<psi>{\<alpha>})\<close>
proof(safe intro!: "\<equiv>I" "\<rightarrow>I")
AOT_assume 0: \<open>\<forall>\<alpha>(\<phi>{\<alpha>} & \<psi>{\<alpha>})\<close>
AOT_have \<open>\<phi>{\<alpha>}\<close> and \<open>\<psi>{\<alpha>}\<close> for \<alpha> using "\<forall>E"(2) 0 "&E" by blast+
AOT_thus \<open>\<forall>\<alpha> \<phi>{\<alpha>} & \<forall>\<alpha> \<psi>{\<alpha>}\<close>
by (auto intro: "\<forall>I" "&I")
next
AOT_assume 1: \<open>\<forall>\<alpha> \<phi>{\<alpha>} & \<forall>\<alpha> \<psi>{\<alpha>}\<close>
AOT_hence \<open>\<phi>{\<alpha>}\<close> and \<open>\<psi>{\<alpha>}\<close> for \<alpha>
using "\<forall>E" "&E" apply blast
using "1" "con-dis-i-e:2:b" "rule-ui:3" by blast
AOT_thus \<open>\<forall>\<alpha> (\<phi>{\<alpha>} & \<psi>{\<alpha>})\<close>
using "&I" "\<forall>I" by meson
qed
AOT_theorem "cqt-basic:5": \<open>(\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n(\<phi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n})) \<rightarrow> \<phi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}\<close>
using "cqt-orig:3" by blast
AOT_theorem "cqt-basic:6": \<open>\<forall>\<alpha>\<forall>\<alpha> \<phi>{\<alpha>} \<equiv> \<forall>\<alpha> \<phi>{\<alpha>}\<close>
by (meson "\<equiv>I" "\<rightarrow>I" GEN "cqt-orig:1[const_var]")
AOT_theorem "cqt-basic:7": \<open>(\<phi> \<rightarrow> \<forall>\<alpha> \<psi>{\<alpha>}) \<equiv> \<forall>\<alpha>(\<phi> \<rightarrow> \<psi>{\<alpha>})\<close>
by (metis "\<rightarrow>I" "vdash-properties:6" "rule-ui:3" "\<equiv>I" GEN)
AOT_theorem "cqt-basic:8": \<open>(\<forall>\<alpha> \<phi>{\<alpha>} \<or> \<forall>\<alpha> \<psi>{\<alpha>}) \<rightarrow> \<forall>\<alpha> (\<phi>{\<alpha>} \<or> \<psi>{\<alpha>})\<close>
by (simp add: "\<or>I"(3) "\<rightarrow>I" GEN "cqt-orig:1[const_var]")
AOT_theorem "cqt-basic:9":
\<open>(\<forall>\<alpha> (\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}) & \<forall>\<alpha> (\<psi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})) \<rightarrow> \<forall>\<alpha>(\<phi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})\<close>
proof -
{
AOT_assume \<open>\<forall>\<alpha> (\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>})\<close>
moreover AOT_assume \<open>\<forall>\<alpha> (\<psi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})\<close>
ultimately AOT_have \<open>\<phi>{\<alpha>} \<rightarrow> \<psi>{\<alpha>}\<close> and \<open>\<psi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>}\<close> for \<alpha>
using "\<forall>E" by blast+
AOT_hence \<open>\<phi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>}\<close> for \<alpha> by (metis "\<rightarrow>E" "\<rightarrow>I")
AOT_hence \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<rightarrow> \<chi>{\<alpha>})\<close> using "\<forall>I" by fast
}
thus ?thesis using "&I" "\<rightarrow>I" "&E" by meson
qed
AOT_theorem "cqt-basic:10":
\<open>(\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) & \<forall>\<alpha>(\<psi>{\<alpha>} \<equiv> \<chi>{\<alpha>})) \<rightarrow> \<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<chi>{\<alpha>})\<close>
proof(rule "\<rightarrow>I"; rule "\<forall>I")
fix \<beta>
AOT_assume \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) & \<forall>\<alpha>(\<psi>{\<alpha>} \<equiv> \<chi>{\<alpha>})\<close>
AOT_hence \<open>\<phi>{\<beta>} \<equiv> \<psi>{\<beta>}\<close> and \<open>\<psi>{\<beta>} \<equiv> \<chi>{\<beta>}\<close> using "&E" "\<forall>E" by blast+
AOT_thus \<open>\<phi>{\<beta>} \<equiv> \<chi>{\<beta>}\<close> using "\<equiv>I" "\<equiv>E" by blast
qed
AOT_theorem "cqt-basic:11": \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}) \<equiv> \<forall>\<alpha> (\<psi>{\<alpha>} \<equiv> \<phi>{\<alpha>})\<close>
proof (rule "\<equiv>I"; rule "\<rightarrow>I")
AOT_assume 0: \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
{
fix \<alpha>
AOT_have \<open>\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}\<close> using 0 "\<forall>E" by blast
AOT_hence \<open>\<psi>{\<alpha>} \<equiv> \<phi>{\<alpha>}\<close> using "\<equiv>I" "\<equiv>E" "\<rightarrow>I" "\<rightarrow>E" by metis
}
AOT_thus \<open>\<forall>\<alpha>(\<psi>{\<alpha>} \<equiv> \<phi>{\<alpha>})\<close> using "\<forall>I" by fast
next
AOT_assume 0: \<open>\<forall>\<alpha>(\<psi>{\<alpha>} \<equiv> \<phi>{\<alpha>})\<close>
{
fix \<alpha>
AOT_have \<open>\<psi>{\<alpha>} \<equiv> \<phi>{\<alpha>}\<close> using 0 "\<forall>E" by blast
AOT_hence \<open>\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>}\<close> using "\<equiv>I" "\<equiv>E" "\<rightarrow>I" "\<rightarrow>E" by metis
}
AOT_thus \<open>\<forall>\<alpha>(\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close> using "\<forall>I" by fast
qed
AOT_theorem "cqt-basic:12": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> (\<psi>{\<alpha>} \<rightarrow> \<phi>{\<alpha>})\<close>
by (simp add: "\<forall>E"(2) "\<rightarrow>I" GEN)
AOT_theorem "cqt-basic:13": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<equiv> \<forall>\<beta> \<phi>{\<beta>}\<close>
using "\<equiv>I" "\<rightarrow>I" by blast
AOT_theorem "cqt-basic:14":
\<open>(\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n (\<phi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n} \<rightarrow> \<psi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n})) \<rightarrow>
((\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n \<phi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}) \<rightarrow> (\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n \<psi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}))\<close>
using "cqt:3"[axiom_inst] by auto
AOT_theorem "cqt-basic:15":
\<open>(\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n (\<phi> \<rightarrow> \<psi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n})) \<rightarrow> (\<phi> \<rightarrow> (\<forall>\<alpha>\<^sub>1...\<forall>\<alpha>\<^sub>n \<psi>{\<alpha>\<^sub>1...\<alpha>\<^sub>n}))\<close>
using "cqt-orig:2" by auto
AOT_theorem "universal-cor":
assumes \<open>for arbitrary \<beta>: \<phi>{\<beta>}\<close>
shows \<open>\<forall>\<alpha> \<phi>{\<alpha>}\<close>
using GEN assms .
AOT_theorem "existential:1":
assumes \<open>\<phi>{\<tau>}\<close> and \<open>\<tau>\<down>\<close>
shows \<open>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
proof(rule "raa-cor:1")
AOT_assume \<open>\<not>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
AOT_hence \<open>\<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>
using "\<equiv>\<^sub>d\<^sub>fI" "conventions:4" RAA "&I" by blast
AOT_hence \<open>\<not>\<phi>{\<tau>}\<close> using assms(2) "\<forall>E"(1) "\<rightarrow>E" by blast
AOT_thus \<open>\<phi>{\<tau>} & \<not>\<phi>{\<tau>}\<close> using assms(1) "&I" by blast
qed
AOT_theorem "existential:2[const_var]":
assumes \<open>\<phi>{\<beta>}\<close>
shows \<open>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
using "existential:1" "cqt:2[const_var]"[axiom_inst] assms by blast
AOT_theorem "existential:2[lambda]":
assumes \<open>\<phi>{[\<lambda>\<nu>\<^sub>1...\<nu>\<^sub>n \<psi>{\<nu>\<^sub>1...\<nu>\<^sub>n}]}\<close> and \<open>INSTANCE_OF_CQT_2(\<psi>)\<close>
shows \<open>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
using "existential:1" "cqt:2[lambda]"[axiom_inst] assms by blast
lemmas "\<exists>I" = "existential:1" "existential:2[const_var]"
"existential:2[lambda]"
AOT_theorem "instantiation":
assumes \<open>for arbitrary \<beta>: \<phi>{\<beta>} \<^bold>\<turnstile> \<psi>\<close> and \<open>\<exists>\<alpha> \<phi>{\<alpha>}\<close>
shows \<open>\<psi>\<close>
by (metis (no_types, lifting) "\<equiv>\<^sub>d\<^sub>fE" GEN "raa-cor:3" "conventions:4" assms)
lemmas "\<exists>E" = "instantiation"
lemmas "\<exists>E'" = "\<exists>E"[rotated]
declare "\<exists>E'"[AOT_elim AOT_exists]
AOT_theorem "cqt-further:1": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<exists>\<alpha> \<phi>{\<alpha>}\<close>
using "\<forall>E"(4) "\<exists>I"(2) "\<rightarrow>I" by metis
AOT_theorem "cqt-further:2": \<open>\<not>\<forall>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<exists>\<alpha> \<not>\<phi>{\<alpha>}\<close>
using "\<forall>I" "\<exists>I"(2) "\<rightarrow>I" RAA by metis
AOT_theorem "cqt-further:3": \<open>\<forall>\<alpha> \<phi>{\<alpha>} \<equiv> \<not>\<exists>\<alpha> \<not>\<phi>{\<alpha>}\<close>
using "\<forall>E"(4) "\<exists>E" "\<rightarrow>I" RAA
by (metis "cqt-further:2" "\<equiv>I" "modus-tollens:1")
AOT_theorem "cqt-further:4": \<open>\<not>\<exists>\<alpha> \<phi>{\<alpha>} \<rightarrow> \<forall>\<alpha> \<not>\<phi>{\<alpha>}\<close>
using "\<forall>I" "\<exists>I"(2)"\<rightarrow>I" RAA by metis
AOT_theorem "cqt-further:5": \<open>\<exists>\<alpha> (\<phi>{\<alpha>} & \<psi>{\<alpha>}) \<rightarrow> (\<exists>\<alpha> \<phi>{\<alpha>} & \<exists>\<alpha> \<psi>{\<alpha>})\<close>
by (metis (no_types, lifting) "&E" "&I" "\<exists>E" "\<exists>I"(2) "\<rightarrow>I")
AOT_theorem "cqt-further:6": \<open>\<exists>\<alpha> (\<phi>{\<alpha>} \<or> \<psi>{\<alpha>}) \<rightarrow> (\<exists>\<alpha> \<phi>{\<alpha>} \<or> \<exists>\<alpha> \<psi>{\<alpha>})\<close>
by (metis (mono_tags, lifting) "\<exists>E" "\<exists>I"(2) "\<or>E"(3) "\<or>I"(1, 2) "\<rightarrow>I" RAA(2))
(* NOTE: vacuous in the embedding *)
AOT_theorem "cqt-further:7": \<open>\<exists>\<alpha> \<phi>{\<alpha>} \<equiv> \<exists>\<beta> \<phi>{\<beta>}\<close>
by (simp add: "oth-class-taut:3:a")
AOT_theorem "cqt-further:8":
\<open>(\<forall>\<alpha> \<phi>{\<alpha>} & \<forall>\<alpha> \<psi>{\<alpha>}) \<rightarrow> \<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
by (metis (mono_tags, lifting) "&E" "\<equiv>I" "\<forall>E"(2) "\<rightarrow>I" GEN)
AOT_theorem "cqt-further:9":
\<open>(\<not>\<exists>\<alpha> \<phi>{\<alpha>} & \<not>\<exists>\<alpha> \<psi>{\<alpha>}) \<rightarrow> \<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
by (metis (mono_tags, lifting) "&E" "\<equiv>I" "\<exists>I"(2) "\<rightarrow>I" GEN "raa-cor:4")
AOT_theorem "cqt-further:10":
\<open>(\<exists>\<alpha> \<phi>{\<alpha>} & \<not>\<exists>\<alpha> \<psi>{\<alpha>}) \<rightarrow> \<not>\<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
proof(rule "\<rightarrow>I"; rule "raa-cor:2")
AOT_assume 0: \<open>\<exists>\<alpha> \<phi>{\<alpha>} & \<not>\<exists>\<alpha> \<psi>{\<alpha>}\<close>
then AOT_obtain \<alpha> where \<open>\<phi>{\<alpha>}\<close> using "\<exists>E" "&E"(1) by metis
moreover AOT_assume \<open>\<forall>\<alpha> (\<phi>{\<alpha>} \<equiv> \<psi>{\<alpha>})\<close>
ultimately AOT_have \<open>\<psi>{\<alpha>}\<close> using "\<forall>E"(4) "\<equiv>E"(1) by blast
AOT_hence \<open>\<exists>\<alpha> \<psi>{\<alpha>}\<close> using "\<exists>I" by blast
AOT_thus \<open>\<exists>\<alpha> \<psi>{\<alpha>} & \<not>\<exists>\<alpha> \<psi>{\<alpha>}\<close> using 0 "&E"(2) "&I" by blast
qed
AOT_theorem "cqt-further:11": \<open>\<exists>\<alpha>\<exists>\<beta> \<phi>{\<alpha>,\<beta>} \<equiv> \<exists>\<beta>\<exists>\<alpha> \<phi>{\<alpha>,\<beta>}\<close>
using "\<equiv>I" "\<rightarrow>I" "\<exists>I"(2) "\<exists>E" by metis
subsection\<open>Logical Existence, Identity, and Truth\<close>
text\<open>\label{PLM: 9.7}\<close>
AOT_theorem "log-prop-prop:1": \<open>[\<lambda> \<phi>]\<down>\<close>
using "cqt:2[lambda0]"[axiom_inst] by auto
AOT_theorem "log-prop-prop:2": \<open>\<phi>\<down>\<close>
by (rule "\<equiv>\<^sub>d\<^sub>fI"[OF "existence:3"]) "cqt:2[lambda]"
declare "log-prop-prop:2"[AOT_intro]
AOT_theorem "exist-nec": \<open>\<tau>\<down> \<rightarrow> \<box>\<tau>\<down>\<close>
proof -
AOT_have \<open>\<forall>\<beta> \<box>\<beta>\<down>\<close>
by (simp add: GEN RN "cqt:2[const_var]"[axiom_inst])
AOT_thus \<open>\<tau>\<down> \<rightarrow> \<box>\<tau>\<down>\<close>
using "cqt:1"[axiom_inst] "\<rightarrow>E" by blast
qed
(* TODO: replace this mechanism by a "proof by types" command *)
class AOT_Term_id = AOT_Term +
assumes "t=t-proper:1"[AOT]: \<open>[v \<Turnstile> \<tau> = \<tau>' \<rightarrow> \<tau>\<down>]\<close>
and "t=t-proper:2"[AOT]: \<open>[v \<Turnstile> \<tau> = \<tau>' \<rightarrow> \<tau>'\<down>]\<close>
instance \<kappa> :: AOT_Term_id
proof
AOT_modally_strict {
AOT_show \<open>\<kappa> = \<kappa>' \<rightarrow> \<kappa>\<down>\<close> for \<kappa> \<kappa>'
proof(rule "\<rightarrow>I")
AOT_assume \<open>\<kappa> = \<kappa>'\<close>
AOT_hence \<open>O!\<kappa> \<or> A!\<kappa>\<close>
by (rule "\<or>I"(3)[OF "\<equiv>\<^sub>d\<^sub>fE"[OF "identity:1"]])
(meson "\<rightarrow>I" "\<or>I"(1) "&E"(1))+
AOT_thus \<open>\<kappa>\<down>\<close>
by (rule "\<or>E"(1))
(metis "cqt:5:a"[axiom_inst] "\<rightarrow>I" "\<rightarrow>E" "&E"(2))+
qed
}
next
AOT_modally_strict {
AOT_show \<open>\<kappa> = \<kappa>' \<rightarrow> \<kappa>'\<down>\<close> for \<kappa> \<kappa>'
proof(rule "\<rightarrow>I")
AOT_assume \<open>\<kappa> = \<kappa>'\<close>
AOT_hence \<open>O!\<kappa>' \<or> A!\<kappa>'\<close>
by (rule "\<or>I"(3)[OF "\<equiv>\<^sub>d\<^sub>fE"[OF "identity:1"]])
(meson "\<rightarrow>I" "\<or>I" "&E")+
AOT_thus \<open>\<kappa>'\<down>\<close>
by (rule "\<or>E"(1))
(metis "cqt:5:a"[axiom_inst] "\<rightarrow>I" "\<rightarrow>E" "&E"(2))+
qed
}
qed
instance rel :: (AOT_\<kappa>s) AOT_Term_id
proof
AOT_modally_strict {
AOT_show \<open>\<Pi> = \<Pi>' \<rightarrow> \<Pi>\<down>\<close> for \<Pi> \<Pi>' :: \<open><'a>\<close>
proof(rule "\<rightarrow>I")
AOT_assume \<open>\<Pi> = \<Pi>'\<close>