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AOT_misc.thy
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AOT_misc.thy
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theory AOT_misc
imports AOT_NaturalNumbers
begin
section\<open>Miscellaneous Theorems\<close>
AOT_theorem PossiblyNumbersEmptyPropertyImpliesZero:
\<open>\<diamond>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) \<rightarrow> x = 0\<close>
proof(rule "\<rightarrow>I")
AOT_have \<open>Rigid([\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
proof (safe intro!: "df-rigid-rel:1"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "cqt:2";
rule RN; safe intro!: GEN "\<rightarrow>I")
AOT_modally_strict {
fix x
AOT_assume \<open>[\<lambda>z O!z & z \<noteq>\<^sub>E z]x\<close>
AOT_hence \<open>O!x & x \<noteq>\<^sub>E x\<close> by (rule "\<beta>\<rightarrow>C")
moreover AOT_have \<open>x =\<^sub>E x\<close> using calculation[THEN "&E"(1)]
by (metis "ord=Eequiv:1" "vdash-properties:10")
ultimately AOT_have \<open>x =\<^sub>E x & \<not>x =\<^sub>E x\<close>
by (metis "con-dis-i-e:1" "con-dis-i-e:2:b" "intro-elim:3:a" "thm-neg=E")
AOT_thus \<open>\<box>[\<lambda>z O!z & z \<noteq>\<^sub>E z]x\<close> using "raa-cor:1" by blast
}
qed
AOT_hence \<open>\<box>\<forall>x (Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) \<rightarrow> \<box>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
by (safe intro!: "num-cont:2"[unvarify G, THEN "\<rightarrow>E"] "cqt:2")
AOT_hence \<open>\<forall>x \<box>(Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) \<rightarrow> \<box>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
using "BFs:2"[THEN "\<rightarrow>E"] by blast
AOT_hence \<open>\<box>(Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) \<rightarrow> \<box>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
using "\<forall>E"(2) by auto
moreover AOT_assume \<open>\<diamond>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
ultimately AOT_have \<open>\<^bold>\<A>Numbers(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
using "sc-eq-box-box:1"[THEN "\<equiv>E"(1), THEN "\<rightarrow>E", THEN "nec-imp-act"[THEN "\<rightarrow>E"]]
by blast
AOT_hence \<open>Numbers(x,[\<lambda>z \<^bold>\<A>[\<lambda>z O!z & z \<noteq>\<^sub>E z]z])\<close>
by (safe intro!: "eq-num:1"[unvarify G, THEN "\<equiv>E"(1)] "cqt:2")
AOT_hence \<open>x = #[\<lambda>z O!z & z \<noteq>\<^sub>E z]\<close>
by (safe intro!: "eq-num:2"[unvarify G, THEN "\<equiv>E"(1)] "cqt:2")
AOT_thus \<open>x = 0\<close>
using "cqt:2"(1) "rule-id-df:2:b[zero]" "rule=E" "zero:1" by blast
qed
AOT_define Numbers' :: \<open>\<tau> \<Rightarrow> \<tau> \<Rightarrow> \<phi>\<close> (\<open>Numbers'''(_,_')\<close>)
\<open>Numbers'(x, G) \<equiv>\<^sub>d\<^sub>f A!x & G\<down> & \<forall>F (x[F] \<equiv> F \<approx>\<^sub>E G)\<close>
AOT_theorem Numbers'equiv: \<open>Numbers'(x,G) \<equiv> A!x & \<forall>F (x[F] \<equiv> F \<approx>\<^sub>E G)\<close>
by (AOT_subst_def Numbers')
(auto intro!: "\<equiv>I" "\<rightarrow>I" "&I" "cqt:2" dest: "&E")
AOT_theorem Numbers'DistinctZeroes:
\<open>\<exists>x\<exists>y (\<diamond>Numbers'(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) & \<diamond>Numbers'(y,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) & x \<noteq> y)\<close>
proof -
AOT_obtain w\<^sub>1 where \<open>\<exists>w w\<^sub>1 \<noteq> w\<close>
using "two-worlds-exist:4" "PossibleWorld.\<exists>E"[rotated] by fast
then AOT_obtain w\<^sub>2 where distinct_worlds: \<open>w\<^sub>1 \<noteq> w\<^sub>2\<close>
using "PossibleWorld.\<exists>E"[rotated] by blast
AOT_obtain x where x_prop:
\<open>A!x & \<forall>F (x[F] \<equiv> w\<^sub>1 \<Turnstile> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
using "A-objects"[axiom_inst] "\<exists>E"[rotated] by fast
moreover AOT_obtain y where y_prop:
\<open>A!y & \<forall>F (y[F] \<equiv> w\<^sub>2 \<Turnstile> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
using "A-objects"[axiom_inst] "\<exists>E"[rotated] by fast
moreover {
fix x w
AOT_assume x_prop: \<open>A!x & \<forall>F (x[F] \<equiv> w \<Turnstile> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
AOT_have \<open>\<forall>F w \<Turnstile> (x[F] \<equiv> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
proof(safe intro!: GEN "conj-dist-w:4"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2",THEN "\<equiv>E"(2)] "\<equiv>I" "\<rightarrow>I")
fix F
AOT_assume \<open>w \<Turnstile> x[F]\<close>
AOT_hence \<open>\<diamond>x[F]\<close>
using "fund:1"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(2),
OF "PossibleWorld.\<exists>I"] by blast
AOT_hence \<open>x[F]\<close>
by (metis "en-eq:3[1]" "intro-elim:3:a")
AOT_thus \<open>w \<Turnstile> (F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
using x_prop[THEN "&E"(2), THEN "\<forall>E"(2), THEN "\<equiv>E"(1)] by blast
next
fix F
AOT_assume \<open>w \<Turnstile> (F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
AOT_hence \<open>x[F]\<close>
using x_prop[THEN "&E"(2), THEN "\<forall>E"(2), THEN "\<equiv>E"(2)] by blast
AOT_hence \<open>\<box>x[F]\<close>
using "pre-en-eq:1[1]"[THEN "\<rightarrow>E"] by blast
AOT_thus \<open>w \<Turnstile> x[F]\<close>
using "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)]
"PossibleWorld.\<forall>E" by fast
qed
AOT_hence \<open>w \<Turnstile> \<forall>F (x[F] \<equiv> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
using "conj-dist-w:5"[THEN "\<equiv>E"(2)] by fast
moreover {
AOT_have \<open>\<box>[\<lambda>z O!z & z \<noteq>\<^sub>E z]\<down>\<close>
by (safe intro!: RN "cqt:2")
AOT_hence \<open>w \<Turnstile> [\<lambda>z O!z & z \<noteq>\<^sub>E z]\<down>\<close>
using "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1),
THEN "PossibleWorld.\<forall>E"] by blast
}
moreover {
AOT_have \<open>\<box>A!x\<close>
using x_prop[THEN "&E"(1)] by (metis "oa-facts:2" "\<rightarrow>E")
AOT_hence \<open>w \<Turnstile> A!x\<close>
using "fund:2"[unvarify p, OF "log-prop-prop:2",
THEN "\<equiv>E"(1), THEN "PossibleWorld.\<forall>E"] by blast
}
ultimately AOT_have \<open>w \<Turnstile> (A!x & [\<lambda>z O!z & z \<noteq>\<^sub>E z]\<down> &
\<forall>F (x[F] \<equiv> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
using "conj-dist-w:1"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(2), OF "&I"] by auto
AOT_hence \<open>\<exists>w w \<Turnstile> (A!x & [\<lambda>z O!z & z \<noteq>\<^sub>E z]\<down> &
\<forall>F (x[F] \<equiv> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
using "PossibleWorld.\<exists>I" by auto
AOT_hence \<open>\<diamond>(A!x & [\<lambda>z O!z & z \<noteq>\<^sub>E z]\<down> & \<forall>F (x[F] \<equiv> F \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]))\<close>
using "fund:1"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(2)] by blast
AOT_hence \<open>\<diamond>Numbers'(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
by (AOT_subst_def Numbers')
}
ultimately AOT_have \<open>\<diamond>Numbers'(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
and \<open>\<diamond>Numbers'(y,[\<lambda>z O!z & z \<noteq>\<^sub>E z])\<close>
by auto
moreover AOT_have \<open>x \<noteq> y\<close>
proof (rule "ab-obey:2"[THEN "\<rightarrow>E"])
AOT_have \<open>\<box>\<not>\<exists>u [\<lambda>z O!z & z \<noteq>\<^sub>E z]u\<close>
proof (safe intro!: RN "raa-cor:2")
AOT_modally_strict {
AOT_assume \<open>\<exists>u [\<lambda>z O!z & z \<noteq>\<^sub>E z]u\<close>
then AOT_obtain u where \<open>[\<lambda>z O!z & z \<noteq>\<^sub>E z]u\<close>
using "Ordinary.\<exists>E"[rotated] by blast
AOT_hence \<open>O!u & u \<noteq>\<^sub>E u\<close>
by (rule "\<beta>\<rightarrow>C")
AOT_hence \<open>\<not>(u =\<^sub>E u)\<close>
by (metis "con-dis-taut:2" "intro-elim:3:d" "modus-tollens:1"
"raa-cor:3" "thm-neg=E")
AOT_hence \<open>u =\<^sub>E u & \<not>u =\<^sub>E u\<close>
by (metis "modus-tollens:1" "ord=Eequiv:1" "raa-cor:3" Ordinary.\<psi>)
AOT_thus \<open>p & \<not>p\<close> for p
by (metis "raa-cor:1")
}
qed
AOT_hence nec_not_ex: \<open>\<forall>w w \<Turnstile> \<not>\<exists>u [\<lambda>z O!z & z \<noteq>\<^sub>E z]u\<close>
using "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)] by blast
AOT_have \<open>\<box>([\<lambda>y p]x \<equiv> p)\<close> for x p
by (safe intro!: RN "beta-C-meta"[THEN "\<rightarrow>E"] "cqt:2")
AOT_hence \<open>\<forall>w w \<Turnstile> ([\<lambda>y p]x \<equiv> p)\<close> for x p
using "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)] by blast
AOT_hence world_prop_beta: \<open>\<forall>w (w \<Turnstile> [\<lambda>y p]x \<equiv> w \<Turnstile> p)\<close> for x p
using "conj-dist-w:4"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)]
"PossibleWorld.\<forall>E" "PossibleWorld.\<forall>I" by meson
AOT_have \<open>\<exists>p (w\<^sub>1 \<Turnstile> p & \<not>w\<^sub>2 \<Turnstile> p)\<close>
proof(rule "raa-cor:1")
AOT_assume 0: \<open>\<not>\<exists>p (w\<^sub>1 \<Turnstile> p & \<not>w\<^sub>2 \<Turnstile> p)\<close>
AOT_have 1: \<open>w\<^sub>1 \<Turnstile> p \<rightarrow> w\<^sub>2 \<Turnstile> p\<close> for p
proof(safe intro!: GEN "\<rightarrow>I")
AOT_assume \<open>w\<^sub>1 \<Turnstile> p\<close>
AOT_thus \<open>w\<^sub>2 \<Turnstile> p\<close>
using 0 "con-dis-i-e:1" "\<exists>I"(2) "raa-cor:4" by fast
qed
moreover AOT_have \<open>w\<^sub>2 \<Turnstile> p \<rightarrow> w\<^sub>1 \<Turnstile> p\<close> for p
proof(safe intro!: GEN "\<rightarrow>I")
AOT_assume \<open>w\<^sub>2 \<Turnstile> p\<close>
AOT_hence \<open>\<not>w\<^sub>2 \<Turnstile> \<not>p\<close>
using "coherent:2" "intro-elim:3:a" by blast
AOT_hence \<open>\<not>w\<^sub>1 \<Turnstile> \<not>p\<close>
using 1["\<forall>I" p, THEN "\<forall>E"(1), OF "log-prop-prop:2"]
by (metis "modus-tollens:1")
AOT_thus \<open>w\<^sub>1 \<Turnstile> p\<close>
using "coherent:1" "intro-elim:3:b" "reductio-aa:1" by blast
qed
ultimately AOT_have \<open>w\<^sub>1 \<Turnstile> p \<equiv> w\<^sub>2 \<Turnstile> p\<close> for p
by (metis "intro-elim:2")
AOT_hence \<open>w\<^sub>1 = w\<^sub>2\<close>
using "sit-identity"[unconstrain s, THEN "\<rightarrow>E",
OF PossibleWorld.\<psi>[THEN "world:1"[THEN "\<equiv>\<^sub>d\<^sub>fE"], THEN "&E"(1)],
unconstrain s', THEN "\<rightarrow>E",
OF PossibleWorld.\<psi>[THEN "world:1"[THEN "\<equiv>\<^sub>d\<^sub>fE"], THEN "&E"(1)],
THEN "\<equiv>E"(2)] GEN by fast
AOT_thus \<open>w\<^sub>1 = w\<^sub>2 & \<not>w\<^sub>1 = w\<^sub>2\<close>
using "=-infix" "\<equiv>\<^sub>d\<^sub>fE" "con-dis-i-e:1" distinct_worlds by blast
qed
then AOT_obtain p where 0: \<open>w\<^sub>1 \<Turnstile> p & \<not>w\<^sub>2 \<Turnstile> p\<close>
using "\<exists>E"[rotated] by blast
AOT_have \<open>y[\<lambda>y p]\<close>
proof (safe intro!: y_prop[THEN "&E"(2), THEN "\<forall>E"(1), THEN "\<equiv>E"(2)] "cqt:2")
AOT_show \<open>w\<^sub>2 \<Turnstile> [\<lambda>y p] \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]\<close>
proof (safe intro!: "cqt:2" "empty-approx:1"[unvarify F H, THEN RN,
THEN "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)],
THEN "PossibleWorld.\<forall>E",
THEN "conj-dist-w:2"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(1)],
THEN "\<rightarrow>E"]
"conj-dist-w:1"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(2)] "&I")
AOT_have \<open>\<not>w\<^sub>2 \<Turnstile> \<exists>u [\<lambda>y p]u\<close>
proof (rule "raa-cor:2")
AOT_assume \<open>w\<^sub>2 \<Turnstile> \<exists>u [\<lambda>y p]u\<close>
AOT_hence \<open>\<exists>x w\<^sub>2 \<Turnstile> (O!x & [\<lambda>y p]x)\<close>
by (metis "conj-dist-w:6" "intro-elim:3:a")
then AOT_obtain x where \<open>w\<^sub>2 \<Turnstile> (O!x & [\<lambda>y p]x)\<close>
using "\<exists>E"[rotated] by blast
AOT_hence \<open>w\<^sub>2 \<Turnstile> [\<lambda>y p]x\<close>
using "conj-dist-w:1"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(1), THEN "&E"(2)] by blast
AOT_hence \<open>w\<^sub>2 \<Turnstile> p\<close>
using world_prop_beta[THEN "PossibleWorld.\<forall>E", THEN "\<equiv>E"(1)] by blast
AOT_thus \<open>w\<^sub>2 \<Turnstile> p & \<not>w\<^sub>2 \<Turnstile> p\<close>
using 0[THEN "&E"(2)] "&I" by blast
qed
AOT_thus \<open>w\<^sub>2 \<Turnstile> \<not>\<exists>u [\<lambda>y p]u\<close>
by (safe intro!: "coherent:1"[unvarify p, OF "log-prop-prop:2",
THEN "\<equiv>E"(2)])
next
AOT_show \<open>w\<^sub>2 \<Turnstile> \<not>\<exists>v [\<lambda>z O!z & z \<noteq>\<^sub>E z]v\<close>
using nec_not_ex[THEN "PossibleWorld.\<forall>E"] by blast
qed
qed
moreover AOT_have \<open>\<not>x[\<lambda>y p]\<close>
proof(rule "raa-cor:2")
AOT_assume \<open>x[\<lambda>y p]\<close>
AOT_hence "w\<^sub>1 \<Turnstile> [\<lambda>y p] \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]"
using x_prop[THEN "&E"(2), THEN "\<forall>E"(1), THEN "\<equiv>E"(1)]
"prop-prop2:2" by blast
AOT_hence "\<not>w\<^sub>1 \<Turnstile> \<not>[\<lambda>y p] \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z]"
using "coherent:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)] by blast
moreover AOT_have "w\<^sub>1 \<Turnstile> \<not>([\<lambda>y p] \<approx>\<^sub>E [\<lambda>z O!z & z \<noteq>\<^sub>E z])"
proof (safe intro!: "cqt:2" "empty-approx:2"[unvarify F H, THEN RN,
THEN "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)],
THEN "PossibleWorld.\<forall>E",
THEN "conj-dist-w:2"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(1)], THEN "\<rightarrow>E"]
"conj-dist-w:1"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(2)] "&I")
fix u
AOT_have \<open>w\<^sub>1 \<Turnstile> O!u\<close>
using Ordinary.\<psi>[THEN RN,
THEN "fund:2"[unvarify p, OF "log-prop-prop:2", THEN "\<equiv>E"(1)],
THEN "PossibleWorld.\<forall>E"] by simp
moreover AOT_have \<open>w\<^sub>1 \<Turnstile> [\<lambda>y p]u\<close>
by (safe intro!: world_prop_beta[THEN "PossibleWorld.\<forall>E", THEN "\<equiv>E"(2)]
0[THEN "&E"(1)])
ultimately AOT_have \<open>w\<^sub>1 \<Turnstile> (O!u & [\<lambda>y p]u)\<close>
using "conj-dist-w:1"[unvarify p q, OF "log-prop-prop:2",
OF "log-prop-prop:2", THEN "\<equiv>E"(2),
OF "&I"] by blast
AOT_hence \<open>\<exists>x w\<^sub>1 \<Turnstile> (O!x & [\<lambda>y p]x)\<close>
by (rule "\<exists>I")
AOT_thus \<open>w\<^sub>1 \<Turnstile> \<exists>u [\<lambda>y p]u\<close>
by (metis "conj-dist-w:6" "intro-elim:3:b")
next
AOT_show \<open>w\<^sub>1 \<Turnstile> \<not>\<exists>v [\<lambda>z O!z & z \<noteq>\<^sub>E z]v\<close>
using "PossibleWorld.\<forall>E" nec_not_ex by fastforce
qed
ultimately AOT_show \<open>p & \<not>p\<close> for p
using "raa-cor:3" by blast
qed
ultimately AOT_have \<open>y[\<lambda>y p] & \<not>x[\<lambda>y p]\<close>
using "&I" by blast
AOT_hence \<open>\<exists>F (y[F] & \<not>x[F])\<close>
by (metis "existential:1" "prop-prop2:2")
AOT_thus \<open>\<exists>F (x[F] & \<not>y[F]) \<or> \<exists>F (y[F] & \<not>x[F])\<close>
by (rule "\<or>I")
qed
ultimately AOT_have \<open>\<diamond>Numbers'(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) &
\<diamond>Numbers'(y,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) & x \<noteq> y\<close>
using "&I" by blast
AOT_thus \<open>\<exists>x\<exists>y (\<diamond>Numbers'(x,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) &
\<diamond>Numbers'(y,[\<lambda>z O!z & z \<noteq>\<^sub>E z]) & x \<noteq> y)\<close>
using "\<exists>I"(2)[where \<beta>=x] "\<exists>I"(2)[where \<beta>=y] by auto
qed
AOT_theorem restricted_identity:
\<open>x =\<^sub>\<R> y \<equiv> (InDomainOf(x,\<R>) & InDomainOf(y,\<R>) & x = y)\<close>
by (auto intro!: "\<equiv>I" "\<rightarrow>I" "&I"
dest: "id-R-thm:2"[THEN "\<rightarrow>E"] "&E"
"id-R-thm:3"[THEN "\<rightarrow>E"]
"id-R-thm:4"[THEN "\<rightarrow>E", OF "\<or>I"(1), THEN "\<equiv>E"(2)])
AOT_theorem induction': \<open>\<forall>F ([F]0 & \<forall>n([F]n \<rightarrow> [F]n\<^bold>') \<rightarrow> \<forall>n [F]n)\<close>
proof(rule GEN; rule "\<rightarrow>I")
fix F n
AOT_assume A: \<open>[F]0 & \<forall>n([F]n \<rightarrow> [F]n\<^bold>')\<close>
AOT_have \<open>\<forall>n\<forall>m([\<bbbP>]nm \<rightarrow> ([F]n \<rightarrow> [F]m))\<close>
proof(safe intro!: "Number.GEN" "\<rightarrow>I")
fix n m
AOT_assume \<open>[\<bbbP>]nm\<close>
moreover AOT_have \<open>[\<bbbP>]n n\<^bold>'\<close>
using "suc-thm".
ultimately AOT_have m_eq_suc_n: \<open>m = n\<^bold>'\<close>
using "pred-func:1"[unvarify z, OF "def-suc[den2]", THEN "\<rightarrow>E", OF "&I"]
by blast
AOT_assume \<open>[F]n\<close>
AOT_hence \<open>[F]n\<^bold>'\<close>
using A[THEN "&E"(2), THEN "Number.\<forall>E", THEN "\<rightarrow>E"] by blast
AOT_thus \<open>[F]m\<close>
using m_eq_suc_n[symmetric] "rule=E" by fast
qed
AOT_thus \<open>\<forall>n[F]n\<close>
using induction[THEN "\<forall>E"(2), THEN "\<rightarrow>E", OF "&I", OF A[THEN "&E"(1)]]
by simp
qed
AOT_define ExtensionOf :: \<open>\<tau> \<Rightarrow> \<Pi> \<Rightarrow> \<phi>\<close> (\<open>ExtensionOf'(_,_')\<close>)
"exten-property:1": \<open>ExtensionOf(x,[G]) \<equiv>\<^sub>d\<^sub>f A!x & G\<down> & \<forall>F(x[F] \<equiv> \<forall>z([F]z \<equiv> [G]z))\<close>
AOT_define OrdinaryExtensionOf :: \<open>\<tau> \<Rightarrow> \<Pi> \<Rightarrow> \<phi>\<close> (\<open>OrdinaryExtensionOf'(_,_')\<close>)
\<open>OrdinaryExtensionOf(x,[G]) \<equiv>\<^sub>d\<^sub>f A!x & G\<down> & \<forall>F(x[F] \<equiv> \<forall>z(O!z \<rightarrow> ([F]z \<equiv> [G]z)))\<close>
AOT_theorem BeingOrdinaryExtensionOfDenotes:
\<open>[\<lambda>x OrdinaryExtensionOf(x,[G])]\<down>\<close>
proof(rule "safe-ext"[axiom_inst, THEN "\<rightarrow>E", OF "&I"])
AOT_show \<open>[\<lambda>x A!x & G\<down> & [\<lambda>x \<forall>F(x[F] \<equiv> \<forall>z(O!z \<rightarrow> ([F]z \<equiv> [G]z)))]x]\<down>\<close>
by "cqt:2"
next
AOT_show \<open>\<box>\<forall>x (A!x & G\<down> & [\<lambda>x \<forall>F (x[F] \<equiv> \<forall>z (O!z \<rightarrow> ([F]z \<equiv> [G]z)))]x \<equiv>
OrdinaryExtensionOf(x,[G]))\<close>
proof(safe intro!: RN GEN)
AOT_modally_strict {
fix x
AOT_modally_strict {
AOT_have \<open>[\<lambda>x \<forall>F (x[F] \<equiv> \<forall>z (O!z \<rightarrow> ([F]z \<equiv> [G]z)))]\<down>\<close>
proof (safe intro!: "Comprehension_3"[THEN "\<rightarrow>E"] RN GEN
"\<rightarrow>I" "\<equiv>I" Ordinary.GEN)
AOT_modally_strict {
fix F H u
AOT_assume \<open>\<box>H \<equiv>\<^sub>E F\<close>
AOT_hence \<open>\<forall>u([H]u \<equiv> [F]u)\<close>
using eqE[THEN "\<equiv>\<^sub>d\<^sub>fE", THEN "&E"(2)] "qml:2"[axiom_inst, THEN "\<rightarrow>E"]
by blast
AOT_hence 0: \<open>[H]u \<equiv> [F]u\<close> using "Ordinary.\<forall>E" by fast
{
AOT_assume \<open>\<forall>u([F]u \<equiv> [G]u)\<close>
AOT_hence 1: \<open>[F]u \<equiv> [G]u\<close> using "Ordinary.\<forall>E" by fast
AOT_show \<open>[G]u\<close> if \<open>[H]u\<close> using 0 1 "\<equiv>E"(1) that by blast
AOT_show \<open>[H]u\<close> if \<open>[G]u\<close> using 0 1 "\<equiv>E"(2) that by blast
}
{
AOT_assume \<open>\<forall>u([H]u \<equiv> [G]u)\<close>
AOT_hence 1: \<open>[H]u \<equiv> [G]u\<close> using "Ordinary.\<forall>E" by fast
AOT_show \<open>[G]u\<close> if \<open>[F]u\<close> using 0 1 "\<equiv>E"(1,2) that by blast
AOT_show \<open>[F]u\<close> if \<open>[G]u\<close> using 0 1 "\<equiv>E"(1,2) that by blast
}
}
qed
}
AOT_thus \<open>(A!x & G\<down> & [\<lambda>x \<forall>F (x[F] \<equiv> \<forall>z (O!z \<rightarrow> ([F]z \<equiv> [G]z)))]x) \<equiv>
OrdinaryExtensionOf(x,[G])\<close>
apply (AOT_subst_def OrdinaryExtensionOf)
apply (AOT_subst \<open>[\<lambda>x \<forall>F (x[F] \<equiv> \<forall>z (O!z \<rightarrow> ([F]z \<equiv> [G]z)))]x\<close>
\<open>\<forall>F (x[F] \<equiv> \<forall>z (O!z \<rightarrow> ([F]z \<equiv> [G]z)))\<close>)
by (auto intro!: "beta-C-meta"[THEN "\<rightarrow>E"] simp: "oth-class-taut:3:a")
}
qed
qed
text\<open>Fragments of PLM's theory of Concepts.\<close>
AOT_define FimpG :: \<open>\<Pi> \<Rightarrow> \<Pi> \<Rightarrow> \<phi>\<close> (infixl \<open>\<Rightarrow>\<close> 50)
"F-imp-G": \<open>[G] \<Rightarrow> [F] \<equiv>\<^sub>d\<^sub>f F\<down> & G\<down> & \<box>\<forall>x ([G]x \<rightarrow> [F]x)\<close>
AOT_define concept :: \<open>\<Pi>\<close> (\<open>C!\<close>)
concepts: \<open>C! =\<^sub>d\<^sub>f A!\<close>
AOT_register_rigid_restricted_type
Concept: \<open>C!\<kappa>\<close>
proof
AOT_modally_strict {
AOT_have \<open>\<exists>x A!x\<close>
using "o-objects-exist:2" "qml:2"[axiom_inst] "\<rightarrow>E" by blast
AOT_thus \<open>\<exists>x C!x\<close>
using "rule-id-df:1[zero]"[OF concepts, OF "oa-exist:2"] "rule=E" id_sym
by fast
}
next
AOT_modally_strict {
AOT_show \<open>C!\<kappa> \<rightarrow> \<kappa>\<down>\<close> for \<kappa>
using "cqt:5:a"[axiom_inst, THEN "\<rightarrow>E", THEN "&E"(2)] "\<rightarrow>I"
by blast
}
next
AOT_modally_strict {
AOT_have \<open>\<forall>x(A!x \<rightarrow> \<box>A!x)\<close>
by (simp add: "oa-facts:2" GEN)
AOT_thus \<open>\<forall>x(C!x \<rightarrow> \<box>C!x)\<close>
using "rule-id-df:1[zero]"[OF concepts, OF "oa-exist:2"] "rule=E" id_sym
by fast
}
qed
AOT_register_variable_names
Concept: c d e
AOT_theorem "concept-comp:1": \<open>\<exists>x(C!x & \<forall>F(x[F] \<equiv> \<phi>{F}))\<close>
using concepts[THEN "rule-id-df:1[zero]", OF "oa-exist:2", symmetric]
"A-objects"[axiom_inst]
"rule=E" by fast
AOT_theorem "concept-comp:2": \<open>\<exists>!x(C!x & \<forall>F(x[F] \<equiv> \<phi>{F}))\<close>
using concepts[THEN "rule-id-df:1[zero]", OF "oa-exist:2", symmetric]
"A-objects!"
"rule=E" by fast
AOT_theorem "concept-comp:3": \<open>\<^bold>\<iota>x(C!x & \<forall>F(x[F] \<equiv> \<phi>{F}))\<down>\<close>
using "concept-comp:2" "A-Exists:2"[THEN "\<equiv>E"(2)] "RA[2]" by blast
AOT_theorem "concept-comp:4":
\<open>\<^bold>\<iota>x(C!x & \<forall>F(x[F] \<equiv> \<phi>{F})) = \<^bold>\<iota>x(A!x & \<forall>F(x[F] \<equiv> \<phi>{F}))\<close>
using "=I"(1)[OF "concept-comp:3"]
"rule=E"[rotated]
concepts[THEN "rule-id-df:1[zero]", OF "oa-exist:2"]
by fast
AOT_define conceptInclusion :: \<open>\<tau> \<Rightarrow> \<tau> \<Rightarrow> \<phi>\<close> (infixl \<open>\<preceq>\<close> 100)
"con:1": \<open>c \<preceq> d \<equiv>\<^sub>d\<^sub>f \<forall>F(c[F] \<rightarrow> d[F])\<close>
AOT_define conceptOf :: \<open>\<tau> \<Rightarrow> \<tau> \<Rightarrow> \<phi>\<close> (\<open>ConceptOf'(_,_')\<close>)
"concept-of-G": \<open>ConceptOf(c,G) \<equiv>\<^sub>d\<^sub>f G\<down> & \<forall>F (c[F] \<equiv> [G] \<Rightarrow> [F])\<close>
AOT_theorem ConceptOfOrdinaryProperty: \<open>([H] \<Rightarrow> O!) \<rightarrow> [\<lambda>x ConceptOf(x,H)]\<down>\<close>
proof(rule "\<rightarrow>I")
AOT_assume \<open>[H] \<Rightarrow> O!\<close>
AOT_hence \<open>\<box>\<forall>x([H]x \<rightarrow> O!x)\<close>
using "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E" by blast
AOT_hence \<open>\<box>\<box>\<forall>x([H]x \<rightarrow> O!x)\<close>
using "S5Basic:6"[THEN "\<equiv>E"(1)] by blast
moreover AOT_have \<open>\<box>\<box>\<forall>x([H]x \<rightarrow> O!x) \<rightarrow>
\<box>\<forall>F\<forall>G(\<box>(G \<equiv>\<^sub>E F) \<rightarrow> ([H] \<Rightarrow> [F] \<equiv> [H] \<Rightarrow> [G]))\<close>
proof(rule RM; safe intro!: "\<rightarrow>I" GEN "\<equiv>I")
AOT_modally_strict {
fix F G
AOT_assume 0: \<open>\<box>\<forall>x([H]x \<rightarrow> O!x)\<close>
AOT_assume \<open>\<box>G \<equiv>\<^sub>E F\<close>
AOT_hence 1: \<open>\<box>\<forall>u([G]u \<equiv> [F]u)\<close>
by (AOT_subst_thm eqE[THEN "\<equiv>Df", THEN "\<equiv>S"(1), OF "&I",
OF "cqt:2[const_var]"[axiom_inst],
OF "cqt:2[const_var]"[axiom_inst], symmetric])
{
AOT_assume \<open>[H] \<Rightarrow> [F]\<close>
AOT_hence \<open>\<box>\<forall>x([H]x \<rightarrow> [F]x)\<close>
using "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E" by blast
moreover AOT_modally_strict {
AOT_assume \<open>\<forall>x([H]x \<rightarrow> O!x)\<close>
moreover AOT_assume \<open>\<forall>u([G]u \<equiv> [F]u)\<close>
moreover AOT_assume \<open>\<forall>x([H]x \<rightarrow> [F]x)\<close>
ultimately AOT_have \<open>[H]x \<rightarrow> [G]x\<close> for x
by (auto intro!: "\<rightarrow>I" dest!: "\<forall>E"(2) dest: "\<rightarrow>E" "\<equiv>E")
AOT_hence \<open>\<forall>x([H]x \<rightarrow> [G]x)\<close>
by (rule GEN)
}
ultimately AOT_have \<open>\<box>\<forall>x([H]x \<rightarrow> [G]x)\<close>
using "RN[prem]"[where
\<Gamma>="{\<guillemotleft>\<forall>x([H]x \<rightarrow> O!x)\<guillemotright>, \<guillemotleft>\<forall>u([G]u \<equiv> [F]u)\<guillemotright>, \<guillemotleft>\<forall>x([H]x \<rightarrow> [F]x)\<guillemotright>}"]
using 0 1 by fast
AOT_thus \<open>[H] \<Rightarrow> [G]\<close>
by (AOT_subst_def "F-imp-G")
(safe intro!: "cqt:2" "&I")
}
{
AOT_assume \<open>[H] \<Rightarrow> [G]\<close>
AOT_hence \<open>\<box>\<forall>x([H]x \<rightarrow> [G]x)\<close>
using "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E" by blast
moreover AOT_modally_strict {
AOT_assume \<open>\<forall>x([H]x \<rightarrow> O!x)\<close>
moreover AOT_assume \<open>\<forall>u([G]u \<equiv> [F]u)\<close>
moreover AOT_assume \<open>\<forall>x([H]x \<rightarrow> [G]x)\<close>
ultimately AOT_have \<open>[H]x \<rightarrow> [F]x\<close> for x
by (auto intro!: "\<rightarrow>I" dest!: "\<forall>E"(2) dest: "\<rightarrow>E" "\<equiv>E")
AOT_hence \<open>\<forall>x([H]x \<rightarrow> [F]x)\<close>
by (rule GEN)
}
ultimately AOT_have \<open>\<box>\<forall>x([H]x \<rightarrow> [F]x)\<close>
using "RN[prem]"[where
\<Gamma>="{\<guillemotleft>\<forall>x([H]x \<rightarrow> O!x)\<guillemotright>, \<guillemotleft>\<forall>u([G]u \<equiv> [F]u)\<guillemotright>, \<guillemotleft>\<forall>x([H]x \<rightarrow> [G]x)\<guillemotright>}"]
using 0 1 by fast
AOT_thus \<open>[H] \<Rightarrow> [F]\<close>
by (AOT_subst_def "F-imp-G")
(safe intro!: "cqt:2" "&I")
}
}
qed
ultimately AOT_have \<open>\<box>\<forall>F\<forall>G(\<box>(G \<equiv>\<^sub>E F) \<rightarrow> ([H] \<Rightarrow> [F] \<equiv> [H] \<Rightarrow> [G]))\<close>
using "\<rightarrow>E" by blast
AOT_hence 0: \<open>[\<lambda>x \<forall>F(x[F] \<equiv> ([H] \<Rightarrow> [F]))]\<down>\<close>
using Comprehension_3[THEN "\<rightarrow>E"] by blast
AOT_show \<open>[\<lambda>x ConceptOf(x,H)]\<down>\<close>
proof (rule "safe-ext"[axiom_inst, THEN "\<rightarrow>E", OF "&I"])
AOT_show \<open>[\<lambda>x C!x & [\<lambda>x \<forall>F(x[F] \<equiv> ([H] \<Rightarrow> [F]))]x]\<down>\<close> by "cqt:2"
next
AOT_show \<open>\<box>\<forall>x (C!x & [\<lambda>x \<forall>F (x[F] \<equiv> [H] \<Rightarrow> [F])]x \<equiv> ConceptOf(x,H))\<close>
proof (rule "RN[prem]"[where \<Gamma>=\<open>{\<guillemotleft>[\<lambda>x \<forall>F(x[F] \<equiv> ([H] \<Rightarrow> [F]))]\<down>\<guillemotright>}\<close>, simplified])
AOT_modally_strict {
AOT_assume 0: \<open>[\<lambda>x \<forall>F (x[F] \<equiv> [H] \<Rightarrow> [F])]\<down>\<close>
AOT_show \<open>\<forall>x (C!x & [\<lambda>x \<forall>F (x[F] \<equiv> [H] \<Rightarrow> [F])]x \<equiv> ConceptOf(x,H))\<close>
proof(safe intro!: GEN "\<equiv>I" "\<rightarrow>I" "&I")
fix x
AOT_assume \<open>C!x & [\<lambda>x \<forall>F (x[F] \<equiv> [H] \<Rightarrow> [F])]x\<close>
AOT_thus \<open>ConceptOf(x,H)\<close>
by (AOT_subst_def "concept-of-G")
(auto intro!: "&I" "cqt:2" dest: "&E" "\<beta>\<rightarrow>C")
next
fix x
AOT_assume \<open>ConceptOf(x,H)\<close>
AOT_hence \<open>C!x & (H\<down> & \<forall>F(x[F] \<equiv> [H] \<Rightarrow> [F]))\<close>
by (AOT_subst_def (reverse) "concept-of-G")
AOT_thus \<open>C!x\<close> and \<open>[\<lambda>x \<forall>F(x[F] \<equiv> [H] \<Rightarrow> [F])]x\<close>
by (auto intro!: "\<beta>\<leftarrow>C" 0 "cqt:2" dest: "&E")
qed
}
next
AOT_show \<open>\<box>[\<lambda>x \<forall>F(x[F] \<equiv> ([H] \<Rightarrow> [F]))]\<down>\<close>
using "exist-nec"[THEN "\<rightarrow>E"] 0 by blast
qed
qed
qed
AOT_theorem "con-exists:1": \<open>\<exists>c ConceptOf(c,G)\<close>
proof -
AOT_obtain c where \<open>\<forall>F (c[F] \<equiv> [G] \<Rightarrow> [F])\<close>
using "concept-comp:1" "Concept.\<exists>E"[rotated] by meson
AOT_hence \<open>ConceptOf(c,G)\<close>
by (auto intro!: "concept-of-G"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "cqt:2" Concept.\<psi>)
thus ?thesis by (rule "Concept.\<exists>I")
qed
AOT_theorem "con-exists:2": \<open>\<exists>!c ConceptOf(c,G)\<close>
proof -
AOT_have \<open>\<exists>!c \<forall>F (c[F] \<equiv> [G] \<Rightarrow> [F])\<close>
using "concept-comp:2" by simp
moreover {
AOT_modally_strict {
fix x
AOT_assume \<open>\<forall>F (x[F] \<equiv> [G] \<Rightarrow> [F])\<close>
moreover AOT_have \<open>[G] \<Rightarrow> [G]\<close>
by (safe intro!: "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "cqt:2" RN GEN "\<rightarrow>I")
ultimately AOT_have \<open>x[G]\<close>
using "\<forall>E"(2) "\<equiv>E" by blast
AOT_hence \<open>A!x\<close>
using "encoders-are-abstract"[THEN "\<rightarrow>E", OF "\<exists>I"(2)] by simp
AOT_hence \<open>C!x\<close>
using concepts[THEN "rule-id-df:1[zero]", OF "oa-exist:2", symmetric]
"rule=E"[rotated]
by fast
}
}
ultimately show ?thesis
by (AOT_subst \<open>ConceptOf(c,G)\<close> \<open>\<forall>F (c[F] \<equiv> [G] \<Rightarrow> [F])\<close> for: c;
AOT_subst_def "concept-of-G")
(auto intro!: "\<equiv>I" "\<rightarrow>I" "&I" "cqt:2" Concept.\<psi> dest: "&E")
qed
AOT_theorem "con-exists:3": \<open>\<^bold>\<iota>c ConceptOf(c,G)\<down>\<close>
by (safe intro!: "A-Exists:2"[THEN "\<equiv>E"(2)] "con-exists:2"[THEN "RA[2]"])
AOT_define theConceptOfG :: \<open>\<tau> \<Rightarrow> \<kappa>\<^sub>s\<close> (\<open>\<^bold>c\<^sub>_\<close>)
"concept-G": \<open>\<^bold>c\<^sub>G =\<^sub>d\<^sub>f \<^bold>\<iota>c ConceptOf(c, G)\<close>
AOT_theorem "concept-G[den]": \<open>\<^bold>c\<^sub>G\<down>\<close>
by (auto intro!: "rule-id-df:1"[OF "concept-G"]
"t=t-proper:1"[THEN "\<rightarrow>E"]
"con-exists:3")
AOT_theorem "concept-G[concept]": \<open>C!\<^bold>c\<^sub>G\<close>
proof -
AOT_have \<open>\<^bold>\<A>(C!\<^bold>c\<^sub>G & ConceptOf(\<^bold>c\<^sub>G, G))\<close>
by (auto intro!: "actual-desc:2"[unvarify x, THEN "\<rightarrow>E"]
"rule-id-df:1"[OF "concept-G"]
"concept-G[den]"
"con-exists:3")
AOT_hence \<open>\<^bold>\<A>C!\<^bold>c\<^sub>G\<close>
by (metis "Act-Basic:2" "con-dis-i-e:2:a" "intro-elim:3:a")
AOT_hence \<open>\<^bold>\<A>A!\<^bold>c\<^sub>G\<close>
using "rule-id-df:1[zero]"[OF concepts, OF "oa-exist:2"]
"rule=E" by fast
AOT_hence \<open>A!\<^bold>c\<^sub>G\<close>
using "oa-facts:8"[unvarify x, THEN "\<equiv>E"(2)] "concept-G[den]" by blast
thus ?thesis
using "rule-id-df:1[zero]"[OF concepts, OF "oa-exist:2", symmetric]
"rule=E" by fast
qed
AOT_theorem "conG-strict": \<open>\<^bold>c\<^sub>G = \<^bold>\<iota>c \<forall>F(c[F] \<equiv> [G] \<Rightarrow> [F])\<close>
proof (rule "id-eq:3"[unvarify \<alpha> \<beta> \<gamma>, THEN "\<rightarrow>E"])
AOT_have \<open>\<box>\<forall>x (C!x & ConceptOf(x,G) \<equiv> C!x & \<forall>F (x[F] \<equiv> [G] \<Rightarrow> [F]))\<close>
by (auto intro!: "concept-of-G"[THEN "\<equiv>\<^sub>d\<^sub>fI"] RN GEN "\<equiv>I" "\<rightarrow>I" "&I" "cqt:2"
dest: "&E";
auto dest: "\<forall>E"(2) "\<equiv>E"(1,2) dest!: "&E"(2) "concept-of-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"])
AOT_thus \<open>\<^bold>c\<^sub>G = \<^bold>\<iota>c ConceptOf(c, G) & \<^bold>\<iota>c ConceptOf(c, G) = \<^bold>\<iota>c \<forall>F(c[F] \<equiv> [G] \<Rightarrow> [F])\<close>
by (auto intro!: "&I" "rule-id-df:1"[OF "concept-G"] "con-exists:3"
"equiv-desc-eq:3"[THEN "\<rightarrow>E"])
qed(auto simp: "concept-G[den]" "con-exists:3" "concept-comp:3")
AOT_theorem "conG-lemma:1": \<open>\<forall>F(\<^bold>c\<^sub>G[F] \<equiv> [G] \<Rightarrow> [F])\<close>
proof(safe intro!: GEN "\<equiv>I" "\<rightarrow>I")
fix F
AOT_have \<open>\<^bold>\<A>\<forall>F(\<^bold>c\<^sub>G[F] \<equiv> [G] \<Rightarrow> [F])\<close>
using "actual-desc:4"[THEN "\<rightarrow>E", OF "concept-comp:3",
THEN "Act-Basic:2"[THEN "\<equiv>E"(1)],
THEN "&E"(2)]
"conG-strict"[symmetric] "rule=E" by fast
AOT_hence \<open>\<^bold>\<A>(\<^bold>c\<^sub>G[F] \<equiv> [G] \<Rightarrow> [F])\<close>
using "logic-actual-nec:3"[axiom_inst, THEN "\<equiv>E"(1)] "\<forall>E"(2)
by blast
AOT_hence 0: \<open>\<^bold>\<A>\<^bold>c\<^sub>G[F] \<equiv> \<^bold>\<A>[G] \<Rightarrow> [F]\<close>
using "Act-Basic:5"[THEN "\<equiv>E"(1)] by blast
{
AOT_assume \<open>\<^bold>c\<^sub>G[F]\<close>
AOT_hence \<open>\<^bold>\<A>\<^bold>c\<^sub>G[F]\<close>
by(safe intro!: "en-eq:10[1]"[unvarify x\<^sub>1, THEN "\<equiv>E"(2)]
"concept-G[den]")
AOT_hence \<open>\<^bold>\<A>[G] \<Rightarrow> [F]\<close>
using 0[THEN "\<equiv>E"(1)] by blast
AOT_hence \<open>\<^bold>\<A>(F\<down> & G\<down> & \<box>\<forall>x([G]x \<rightarrow> [F]x))\<close>
by (AOT_subst_def (reverse) "F-imp-G")
AOT_hence \<open>\<^bold>\<A>\<box>\<forall>x([G]x \<rightarrow> [F]x)\<close>
using "Act-Basic:2"[THEN "\<equiv>E"(1)] "&E" by blast
AOT_hence \<open>\<box>\<forall>x([G]x \<rightarrow> [F]x)\<close>
using "qml-act:2"[axiom_inst, THEN "\<equiv>E"(2)] by simp
AOT_thus \<open>[G] \<Rightarrow> [F]\<close>
by (AOT_subst_def "F-imp-G"; auto intro!: "&I" "cqt:2")
}
{
AOT_assume \<open>[G] \<Rightarrow> [F]\<close>
AOT_hence \<open>\<box>\<forall>x([G]x \<rightarrow> [F]x)\<close>
by (safe dest!: "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E"(2))
AOT_hence \<open>\<^bold>\<A>\<box>\<forall>x([G]x \<rightarrow> [F]x)\<close>
using "qml-act:2"[axiom_inst, THEN "\<equiv>E"(1)] by simp
AOT_hence \<open>\<^bold>\<A>(F\<down> & G\<down> & \<box>\<forall>x([G]x \<rightarrow> [F]x))\<close>
by (auto intro!: "Act-Basic:2"[THEN "\<equiv>E"(2)] "&I" "cqt:2"
intro: "RA[2]")
AOT_hence \<open>\<^bold>\<A>([G] \<Rightarrow> [F])\<close>
by (AOT_subst_def "F-imp-G")
AOT_hence \<open>\<^bold>\<A>\<^bold>c\<^sub>G[F]\<close>
using 0[THEN "\<equiv>E"(2)] by blast
AOT_thus \<open>\<^bold>c\<^sub>G[F]\<close>
by(safe intro!: "en-eq:10[1]"[unvarify x\<^sub>1, THEN "\<equiv>E"(1)]
"concept-G[den]")
}
qed
AOT_theorem conH_enc_ord:
\<open>([H] \<Rightarrow> O!) \<rightarrow> \<box>\<forall>F \<forall>G (\<box>G \<equiv>\<^sub>E F \<rightarrow> (\<^bold>c\<^sub>H[F] \<equiv> \<^bold>c\<^sub>H[G]))\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>[H] \<Rightarrow> O!\<close>
AOT_have 0: \<open>\<box>([H] \<Rightarrow> O!)\<close>
apply (AOT_subst_def "F-imp-G")
using 0[THEN "\<equiv>\<^sub>d\<^sub>fE"[OF "F-imp-G"]]
by (auto intro!: "KBasic:3"[THEN "\<equiv>E"(2)] "&I" "exist-nec"[THEN "\<rightarrow>E"]
dest: "&E" 4[THEN "\<rightarrow>E"])
moreover AOT_have \<open>\<box>([H] \<Rightarrow> O!) \<rightarrow> \<box>\<forall>F \<forall>G (\<box>G \<equiv>\<^sub>E F \<rightarrow> (\<^bold>c\<^sub>H[F] \<equiv> \<^bold>c\<^sub>H[G]))\<close>
proof(rule RM; safe intro!: "\<rightarrow>I" GEN)
AOT_modally_strict {
fix F G
AOT_assume \<open>[H] \<Rightarrow> O!\<close>
AOT_hence 0: \<open>\<box>\<forall>x ([H]x \<rightarrow> O!x)\<close>
by (safe dest!: "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E"(2))
AOT_assume 1: \<open>\<box>G \<equiv>\<^sub>E F\<close>
AOT_assume \<open>\<^bold>c\<^sub>H[F]\<close>
AOT_hence \<open>[H] \<Rightarrow> [F]\<close>
using "conG-lemma:1"[THEN "\<forall>E"(2), THEN "\<equiv>E"(1)] by simp
AOT_hence 2: \<open>\<box>\<forall>x ([H]x \<rightarrow> [F]x)\<close>
by (safe dest!: "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E"(2))
AOT_modally_strict {
AOT_assume 0: \<open>\<forall>x ([H]x \<rightarrow> O!x)\<close>
AOT_assume 1: \<open>\<forall>x ([H]x \<rightarrow> [F]x)\<close>
AOT_assume 2: \<open>G \<equiv>\<^sub>E F\<close>
AOT_have \<open>\<forall>x ([H]x \<rightarrow> [G]x)\<close>
proof(safe intro!: GEN "\<rightarrow>I")
fix x
AOT_assume \<open>[H]x\<close>
AOT_hence \<open>O!x\<close> and \<open>[F]x\<close>
using 0 1 "\<forall>E"(2) "\<rightarrow>E" by blast+
AOT_thus \<open>[G]x\<close>
using 2[THEN eqE[THEN "\<equiv>\<^sub>d\<^sub>fE"], THEN "&E"(2)]
"\<forall>E"(2) "\<rightarrow>E" "\<equiv>E"(2) calculation by blast
qed
}
AOT_hence \<open>\<box>\<forall>x ([H]x \<rightarrow> [G]x)\<close>
using "RN[prem]"[where \<Gamma>=\<open>{\<guillemotleft>\<forall>x ([H]x \<rightarrow> O!x)\<guillemotright>,
\<guillemotleft>\<forall>x ([H]x \<rightarrow> [F]x)\<guillemotright>,
\<guillemotleft>G \<equiv>\<^sub>E F\<guillemotright>}\<close>, simplified] 0 1 2 by fast
AOT_hence \<open>[H] \<Rightarrow> [G]\<close>
by (safe intro!: "F-imp-G"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "cqt:2")
AOT_hence \<open>\<^bold>c\<^sub>H[G]\<close>
using "conG-lemma:1"[THEN "\<forall>E"(2), THEN "\<equiv>E"(2)] by simp
} note 0 = this
AOT_modally_strict {
fix F G
AOT_assume \<open>[H] \<Rightarrow> O!\<close>
moreover AOT_assume \<open>\<box>G \<equiv>\<^sub>E F\<close>
moreover AOT_have \<open>\<box>F \<equiv>\<^sub>E G\<close>
by (AOT_subst \<open>F \<equiv>\<^sub>E G\<close> \<open>G \<equiv>\<^sub>E F\<close>)
(auto intro!: calculation(2)
eqE[THEN "\<equiv>\<^sub>d\<^sub>fI"]
"\<equiv>I" "\<rightarrow>I" "&I" "cqt:2" Ordinary.GEN
dest!: eqE[THEN "\<equiv>\<^sub>d\<^sub>fE"] "&E"(2)
dest: "\<equiv>E"(1,2) "Ordinary.\<forall>E")
ultimately AOT_show \<open>(\<^bold>c\<^sub>H[F] \<equiv> \<^bold>c\<^sub>H[G])\<close>
using 0 "\<equiv>I" "\<rightarrow>I" by auto
}
qed
ultimately AOT_show \<open>\<box>\<forall>F \<forall>G (\<box>G \<equiv>\<^sub>E F \<rightarrow> (\<^bold>c\<^sub>H[F] \<equiv> \<^bold>c\<^sub>H[G]))\<close>
using "\<rightarrow>E" by blast
qed
AOT_theorem concept_inclusion_denotes_1:
\<open>([H] \<Rightarrow> O!) \<rightarrow> [\<lambda>x \<^bold>c\<^sub>H \<preceq> x]\<down>\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>[H] \<Rightarrow> O!\<close>
AOT_show \<open>[\<lambda>x \<^bold>c\<^sub>H \<preceq> x]\<down>\<close>
proof(rule "safe-ext"[axiom_inst, THEN "\<rightarrow>E", OF "&I"])
AOT_show \<open>[\<lambda>x C!x & \<forall>F(\<^bold>c\<^sub>H[F] \<rightarrow> x[F])]\<down>\<close>
by (safe intro!: conjunction_denotes[THEN "\<rightarrow>E", OF "&I"]
Comprehension_2'[THEN "\<rightarrow>E"]
conH_enc_ord[THEN "\<rightarrow>E", OF 0]) "cqt:2"
next
AOT_show \<open>\<box>\<forall>x (C!x & \<forall>F (\<^bold>c\<^sub>H[F] \<rightarrow> x[F]) \<equiv> \<^bold>c\<^sub>H \<preceq> x)\<close>
by (safe intro!: RN GEN; AOT_subst_def "con:1")
(auto intro!: "\<equiv>I" "\<rightarrow>I" "&I" "concept-G[concept]" dest: "&E")
qed
qed
AOT_theorem concept_inclusion_denotes_2:
\<open>([H] \<Rightarrow> O!) \<rightarrow> [\<lambda>x x \<preceq> \<^bold>c\<^sub>H]\<down>\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>[H] \<Rightarrow> O!\<close>
AOT_show \<open>[\<lambda>x x \<preceq> \<^bold>c\<^sub>H]\<down>\<close>
proof(rule "safe-ext"[axiom_inst, THEN "\<rightarrow>E", OF "&I"])
AOT_show \<open>[\<lambda>x C!x & \<forall>F(x[F] \<rightarrow> \<^bold>c\<^sub>H[F])]\<down>\<close>
by (safe intro!: conjunction_denotes[THEN "\<rightarrow>E", OF "&I"]
Comprehension_1'[THEN "\<rightarrow>E"]
conH_enc_ord[THEN "\<rightarrow>E", OF 0]) "cqt:2"
next
AOT_show \<open>\<box>\<forall>x (C!x & \<forall>F (x[F] \<rightarrow> \<^bold>c\<^sub>H[F]) \<equiv> x \<preceq> \<^bold>c\<^sub>H)\<close>
by (safe intro!: RN GEN; AOT_subst_def "con:1")
(auto intro!: "\<equiv>I" "\<rightarrow>I" "&I" "concept-G[concept]" dest: "&E")
qed
qed
AOT_define ThickForm :: \<open>\<tau> \<Rightarrow> \<tau> \<Rightarrow> \<phi>\<close> (\<open>FormOf'(_,_')\<close>)
"tform-of": \<open>FormOf(x,G) \<equiv>\<^sub>d\<^sub>f A!x & G\<down> & \<forall>F(x[F] \<equiv> [G] \<Rightarrow> [F])\<close>
AOT_theorem FormOfOrdinaryProperty: \<open>([H] \<Rightarrow> O!) \<rightarrow> [\<lambda>x FormOf(x,H)]\<down>\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>[H] \<Rightarrow> [O!]\<close>
AOT_show \<open>[\<lambda>x FormOf(x,H)]\<down>\<close>
proof (rule "safe-ext"[axiom_inst, THEN "\<rightarrow>E", OF "&I"])
AOT_show \<open>[\<lambda>x ConceptOf(x,H)]\<down>\<close>
using 0 ConceptOfOrdinaryProperty[THEN "\<rightarrow>E"] by blast
AOT_show \<open>\<box>\<forall>x (ConceptOf(x,H) \<equiv> FormOf(x,H))\<close>
proof(safe intro!: RN GEN)
AOT_modally_strict {
fix x
AOT_modally_strict {
AOT_have \<open>A!x \<equiv> A!x\<close>
by (simp add: "oth-class-taut:3:a")
AOT_hence \<open>C!x \<equiv> A!x\<close>
using "rule-id-df:1[zero]"[OF concepts, OF "oa-exist:2"]
"rule=E" id_sym by fast
}
AOT_thus \<open>ConceptOf(x,H) \<equiv> FormOf(x,H)\<close>
by (AOT_subst_def "tform-of";
AOT_subst_def "concept-of-G";
AOT_subst \<open>C!x\<close> \<open>A!x\<close>)
(auto intro!: "\<equiv>I" "\<rightarrow>I" "&I" dest: "&E")
}
qed
qed
qed
AOT_theorem equal_E_rigid_one_to_one: \<open>Rigid\<^sub>1\<^sub>-\<^sub>1((=\<^sub>E))\<close>
proof (safe intro!: "df-1-1:2"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "df-1-1:1"[THEN "\<equiv>\<^sub>d\<^sub>fI"]
GEN "\<rightarrow>I" "df-rigid-rel:1"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "=E[denotes]")
fix x y z
AOT_assume \<open>x =\<^sub>E z & y =\<^sub>E z\<close>
AOT_thus \<open>x = y\<close>
by (metis "rule=E" "&E"(1) "Conjunction Simplification"(2)
"=E-simple:2" id_sym "\<rightarrow>E")
next
AOT_have \<open>\<forall>x\<forall>y \<box>(x =\<^sub>E y \<rightarrow> \<box>x =\<^sub>E y)\<close>
proof(rule GEN; rule GEN)
AOT_show \<open>\<box>(x =\<^sub>E y \<rightarrow> \<box>x =\<^sub>E y)\<close> for x y
by (meson RN "deduction-theorem" "id-nec3:1" "\<equiv>E"(1))
qed
AOT_hence \<open>\<forall>x\<^sub>1...\<forall>x\<^sub>n \<box>([(=\<^sub>E)]x\<^sub>1...x\<^sub>n \<rightarrow> \<box>[(=\<^sub>E)]x\<^sub>1...x\<^sub>n)\<close>
by (rule tuple_forall[THEN "\<equiv>\<^sub>d\<^sub>fI"])
AOT_thus \<open>\<box>\<forall>x\<^sub>1...\<forall>x\<^sub>n ([(=\<^sub>E)]x\<^sub>1...x\<^sub>n \<rightarrow> \<box>[(=\<^sub>E)]x\<^sub>1...x\<^sub>n)\<close>
using BF[THEN "\<rightarrow>E"] by fast
qed
AOT_theorem equal_E_domain: \<open>InDomainOf(x,(=\<^sub>E)) \<equiv> O!x\<close>
proof(safe intro!: "\<equiv>I" "\<rightarrow>I")
AOT_assume \<open>InDomainOf(x,(=\<^sub>E))\<close>
AOT_hence \<open>\<exists>y x =\<^sub>E y\<close>
by (metis "\<equiv>\<^sub>d\<^sub>fE" "df-1-1:5")
then AOT_obtain y where \<open>x =\<^sub>E y\<close>
using "\<exists>E"[rotated] by blast
AOT_thus \<open>O!x\<close>
using "=E-simple:1"[THEN "\<equiv>E"(1)] "&E" by blast
next
AOT_assume \<open>O!x\<close>
AOT_hence \<open>x =\<^sub>E x\<close>
by (metis "ord=Eequiv:1"[THEN "\<rightarrow>E"])
AOT_hence \<open>\<exists>y x =\<^sub>E y\<close>
using "\<exists>I"(2) by fast
AOT_thus \<open>InDomainOf(x,(=\<^sub>E))\<close>
by (metis "\<equiv>\<^sub>d\<^sub>fI" "df-1-1:5")
qed
AOT_theorem shared_urelement_projection_identity:
assumes \<open>\<forall>y [\<lambda>x (y[\<lambda>z [R]zx])]\<down>\<close>
shows \<open>\<forall>F([F]a \<equiv> [F]b) \<rightarrow> [\<lambda>z [R]za] = [\<lambda>z [R]zb]\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>\<forall>F([F]a \<equiv> [F]b)\<close>
{
fix z
AOT_have \<open>[\<lambda>x (z[\<lambda>z [R]zx])]\<down>\<close>
using assms[THEN "\<forall>E"(2)].
AOT_hence 1: \<open>\<forall>x \<forall>y (\<forall>F ([F]x \<equiv> [F]y) \<rightarrow> \<box>(z[\<lambda>z [R]zx] \<equiv> z[\<lambda>z [R]zy]))\<close>
using "kirchner-thm-cor:1"[THEN "\<rightarrow>E"]
by blast
AOT_have \<open>\<box>(z[\<lambda>z [R]za] \<equiv> z[\<lambda>z [R]zb])\<close>
using 1[THEN "\<forall>E"(2), THEN "\<forall>E"(2), THEN "\<rightarrow>E", OF 0] by blast
}
AOT_hence \<open>\<forall>z \<box>(z[\<lambda>z [R]za] \<equiv> z[\<lambda>z [R]zb])\<close>
by (rule GEN)
AOT_hence \<open>\<box>\<forall>z(z[\<lambda>z [R]za] \<equiv> z[\<lambda>z [R]zb])\<close>
by (rule BF[THEN "\<rightarrow>E"])
AOT_thus \<open>[\<lambda>z [R]za] = [\<lambda>z [R]zb]\<close>
by (AOT_subst_def "identity:2")
(auto intro!: "&I" "cqt:2")
qed
AOT_theorem shared_urelement_exemplification_identity:
assumes \<open>\<forall>y [\<lambda>x (y[\<lambda>z [G]x])]\<down>\<close>
shows \<open>\<forall>F([F]a \<equiv> [F]b) \<rightarrow> ([G]a) = ([G]b)\<close>
proof(rule "\<rightarrow>I")
AOT_assume 0: \<open>\<forall>F([F]a \<equiv> [F]b)\<close>
{
fix z
AOT_have \<open>[\<lambda>x (z[\<lambda>z [G]x])]\<down>\<close>
using assms[THEN "\<forall>E"(2)].
AOT_hence 1: \<open>\<forall>x \<forall>y (\<forall>F ([F]x \<equiv> [F]y) \<rightarrow> \<box>(z[\<lambda>z [G]x] \<equiv> z[\<lambda>z [G]y]))\<close>
using "kirchner-thm-cor:1"[THEN "\<rightarrow>E"]
by blast
AOT_have \<open>\<box>(z[\<lambda>z [G]a] \<equiv> z[\<lambda>z [G]b])\<close>
using 1[THEN "\<forall>E"(2), THEN "\<forall>E"(2), THEN "\<rightarrow>E", OF 0] by blast
}
AOT_hence \<open>\<forall>z \<box>(z[\<lambda>z [G]a] \<equiv> z[\<lambda>z [G]b])\<close>
by (rule GEN)
AOT_hence \<open>\<box>\<forall>z(z[\<lambda>z [G]a] \<equiv> z[\<lambda>z [G]b])\<close>
by (rule BF[THEN "\<rightarrow>E"])
AOT_hence \<open>[\<lambda>z [G]a] = [\<lambda>z [G]b]\<close>
by (AOT_subst_def "identity:2")
(auto intro!: "&I" "cqt:2")
AOT_thus \<open>([G]a) = ([G]b)\<close>
by (safe intro!: "identity:4"[THEN "\<equiv>\<^sub>d\<^sub>fI"] "&I" "log-prop-prop:2")
qed
text\<open>The assumptions of the theorems above are derivable, if the additional
introduction rules for the upcoming extension of @{thm "cqt:2[lambda]"}
are explicitly allowed (while they are currently not part of the
abstraction layer).\<close>
notepad
begin
AOT_modally_strict {
AOT_have \<open>\<forall>R\<forall>y [\<lambda>x (y[\<lambda>z [R]zx])]\<down>\<close>
by (safe intro!: GEN "cqt:2" AOT_instance_of_cqt_2_intro_next)
AOT_have \<open>\<forall>G\<forall>y [\<lambda>x (y[\<lambda>z [G]x])]\<down>\<close>
by (safe intro!: GEN "cqt:2" AOT_instance_of_cqt_2_intro_next)
}
end
end