version of Principia Logico-Metaphysica (PLM) implemented in this formalization
be found at @{url ‹http://mally.stanford.edu/principia-2021-10-13.pdf›}, while
latest version of PLM is available at @{url ‹hence "[α^bld>>🚫
›
‹Model for the Logic of AOT›
‹We introduce a primitive type for hyperintensional propositions.› o
‹To be able to model modal operators following Kripke semantics,
we introduce a primitive type for possible worlds and assert, by axiom,
that there is a surjective function mapping propositions to the
boolean-valued functions acting on possible worlds. We call the result
of applying this function to a proposition the Montague intension
of the proposition.›
w ―‹The primtive type of possible worlds.›
AOT_model_do :: ‹o==>(w==>bool)›
do_surj: ‹surj AOT_model_do›
‹The axioms of PLM require the existence of a non-actual world.›
w0 :: w ―‹The designated actual world.›
where AOT_model_nonactual_world: ‹∃w . w ≠ w0›
‹Validity of a proposition in a given world can now be modelled as the result
of applying that world to the Montague intension of the proposition.›
AOT_model_valid_in :: ‹w==>o==>bool› where ‹AOT_model_valid_in w φ ≡ AOT_model_do φ w›
‹By construction, we can choose a proposition for any given Montague intension,
s.t. the proposition is valid in a possible world iff the Montague intension
evaluates to true at that world.›
AOT_model_proposition_choice :: ‹(w==>bool) ==>o› (binder ‹ε\o› 8)
where ‹ε\o w. φ w ≡ (inv AOT_model_do) φ›
AOT_model_proposition_choice_simp: ‹
by (simp add: surj_f_inv_f[OF do_surj] AOT_model_valid_in_def
AOT_model_proposition_choice_def)
‹Nitpick can trivially show that there are models for the axioms above.› ‹True› nitpick[satisfy, user_axioms, expect = genuine] ..
ψ ―‹The primtive type of ordinary objects/urelements.›
‹Validating extended relation comprehension requires a large set of
special urelements. For simple models that do not validate extended
relation comprehension (and consequently the predecessor axiom in the
theory of natural numbers), it suffices to use a primitive type as @{text σ},
i.e. @{theory_text ‹typedecl σ›}.›
σ'
σ = ‹
‹Urrelations are proposition-valued functions on urelements.
Urrelations are required to evaluate to necessarily false propositions for
null-urelements (note that there may be several distinct necessarily false
propositions).›
urrel = ‹{ φ . ∀ x w . ¬AO lemma desc_nec_encode[PLM]
by (rule exI[where x=‹λ x . (ε\o w . ¬is_nullυ x)›
(auto simp: AOT_model_proposition_choice_simp)
‹A!,xP)& (\∀ F . {xP,F}\≡ φ F), G}\≡\A(φ G) in v]"
have to be mapped surjectively into the set of special urelements.
We show that any mapping from abstract objects to special urelements
has to involve at least one large set of collapsed abstract objects.
We will use this fact to extend arbitrary mappings from abstract objects
to special urelements to surjective mappings.›
ασ_pigeonhole: ―‹For any arbitrary mapping @{term ασ} from sets of urrelations to special
urelements, there exists an abstract object x, s.t. the cardinal of the set
of special urelements is strictly smaller than the cardinal of the set of
abstract objects that are mapped to the same urelement as x under @{term α -
java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
for ασ :: ‹urrel set ==> σ›
(ule ccontr
have card_σ_set_set_bound: ‹|UNIV::σ set set| ≤o |UNIV::urrel set|›
proof -
let ?pick = ‹λu s . ε\o w . case u of (συ s') ==> s' ∈ s | _ ==> False›
have ‹∃f :: σ set ==> urrel . inj f›
proof
show ‹inj (λs . Abs_urrel (λu . ?pick u s))›
proof(rule injI)
fix x y
assume ‹Abs_urrel (λu. ?pick u x) = Abs_urrel (λu. ?pick u y)›
hence ‹(λu. ?pick u x) = (λu. ?pick u y)›
by (auto intro!: Abs_urrel_inject[THEN iffD1]
simp: AOT_model_proposition_choice_simp)
hence ‹AOT_model_valid_in w0 (?pick (συ s) x) =
AOT_model_valid_in w0 (?pick (συ s) y)›
for s by metis
hence ‹(s ∈ x) = (s ∈ y)› for s
by (auto simp: AOT_model_proposition_choice_simp)
thus ‹x = y›
by blast
qed
qed
thus ?thesis
by (metis card_of_image inj_imp_surj_inv)
qed
text‹Assume, for a proof by contradiction, that there is no large collapsed set.› aa
assume ‹∄x . |UNIV::σ set| <o |{y . ασ x = ασ y}|›
hence A: ‹
by auto
have union_univ: ‹(∪x ∈ range(inv ασ) . {y . ασ x = ασ y}) = UNIV›
by auto (meson f_inv_into_f range_eqI)
text‹We refute by case distinction: there is either finitely many or
infinitely many special urelements and in both cases we can derive
a contradiction from the assumption above.›
{
text‹Finite case.›
assume finite_σ_set: ‹finite (UNIV::σ set)›
hence finite_collapsed: ‹finite {y . ασ x = ασ y}› for x
using A card_of_ordLeq_infinite by blast ‹
by (metis A finite_σ_set card_of_ordLeq inj_on_iff_card_le)
have 1: ‹card (range (inv ασ)) ≤ card (UNIV::σ set)›
using finite_σ_set card_image_le by blast
hence 2: ‹finite (range (inv ασ))›
using finite_σ_set by blast
define n where ‹n = card (UNIV::urrel set set)›
define m where ‹m = card (UNIV::σ set)›
have ‹n = card (∪x ∈ range(inv ασ) . {y . ασ x = ασ y})›
unfolding n_def using union_univ by argo
java.lang.NullPointerException
using card_UN_le 2 by blast
also have ‹…≤ (∑i∈range (inv ασ). card (UNIV::σ set))›
by (metis (no_types, lifting) 0 sum_mono)
also have ‹…≤ card (range (inv ασ)) * card (UNIV::σ set)›
using sum_bounded_above by auto
also have ‹…≤ card (UNIV::σ set) * card (UNIV::σ set)›
using 1 by force
also have ‹… = m*m›
unfolding m_def by blast
finally have n_upper: ‹n ≤ m*m›.
have ‹
using 2 finite_collapsed by blast
hence finite_αset: ‹finite (UNIV::urrel set set)›
using union_univ by argo
have ‹2^2^m = (2::nat)^(card (UNIV::σ set set))›
by (metis Pow_UNIV card_Pow finite_σ_set m_def)
moreover have ‹card (UNIV::σ set set) ≤ (card (UNIV::urrel set))›
using card_σ_set_set_bound
by (meson Finite_Set.finite_set card_of_ordLeq finite_αset
finite_σ_set inj_on_iff_card_le)
ultimately have ‹2^2^m ≤ (2::nat)^(card (UNIV:: urrel set))›
by simp
also have ‹… = n›
unfolding n_def
by (metis Finite_Set.finite_set Pow_UNIV card_Pow finite_αset)
finally have ‹
hence ‹2^2^m ≤ m*m› using n_upper by linarith
moreover {
have ‹(2::nat)^(2^m) ≥ (2^(m + 1))›
by (metis Suc_eq_plus1 Suc_leI less_exp one_le_numeral power_increasing)
also have ‹(2^(m + 1)) = (2::nat) * 2^m›
by auto
have ‹m < 2^m›
by (simp add: less_exp)
hence ‹m*m < (2^m)*(2^m)›
by (simp add: mult_strict_mono)
moreover have ‹
by (simp add: power_add)
ultimately have ‹m*m < 2 ^ (m + m)› by presburger
moreover have ‹m+m ≤ 2^m›
proof (induct m)
case 0
thus ?case by auto
next
case (Suc m)
thus ?case
by (metis Suc_leI less_exp mult_2 mult_le_mono2 power_Suc)
qed
ultimately have ‹m*m < 2^2^m›
by (meson less_le_trans one_le_numeral power_increasing)
}
ultimately have False by auto
}
moreover {
text‹Infinite case.›
assume ‹infinite (UNIV::σ set)›<^old\
hence Cinfσ: ‹Cinfinite |UNIV::σ set|›
by (simp add: cinfinite_def)
have 1: ‹|range (inv ασ)| ≤o |UNIV::σ set|›
by auto
have 2: ‹∀i∈range (inv ασ). |{y . ασ i = ασ y}| ≤o |UNIV::σ set|›
proof
fix i :: ‹urrel set›
assume ‹i ∈ range (inv ασ)›
show ‹|{y . ασ i = ασ y}| ≤o |UNIV::σ set|›[a= (A!,x\∀xP,F}\≡ φ F)) in v]"
using A by blast
qed
have ‹|∪ ((λi. {y. ασ i = ασ y}) ` (range (inv ασ)))| ≤o
|Sigma (range (inv ασ)) (λi. {y. ασ i = ασ y})|›CP)
using card_of_UNION_Sigma by blast
hence ‹
|Sigma (range (inv ασ)) (λi. {y. ασ i = ασ y})|›
using union_univ by argo
moreover have ‹|Sigma (range (inv \∃E")
using card_of_Sigma_ordLeq_Cinfinite[OF Cinfσ, OF 1, OF 2] by blast
ultimately have ‹|UNIV::urrel set set| ≤o |UNIV::σ set|›
using ordLeq_transitive by blast
moreover {
have ‹sigma> set| <<o |UNIV::σ set set|›
by auto
moreover have ‹
using card_σ_set_set_bound by blast
moreover have ‹|UNIV::urrel set| <o |UNIV::urrel set set|›
by auto
ultimately have ‹|UNIV::σ set| <o |UNIV::urrel set set|›
by (metis ordLess_imp_ordLeq ordLess_ordLeq_trans)
}
ultimately have False
using not_ordLeq_ordLess by blast
}
ultimately show False by blast
‹We introduce a mapping from abstract objects (i.e. sets of urrelations) to
special urelements @{text ‹ασ›} that is surjective and distinguishes all
abstract objects that are distinguished by a (not necessarily surjective)
mapping @{text ‹ασ'›}. @{text ‹ασ'›} will be used to model extended relation
comprehension.›
ασ' :: ‹urrel set ==> σ›
ασ :: \<open
(ασ)
ασ_surj: ‹surj ασ›
ασ_ασ': ‹ασ x = ασ y ==> ασ' x = ασ' y›
-
obtain x where x_prop: ‹|UNIV::σ set| <o |{y. ασ' x = ασ' y}|›
using ασ_pigeonhole by blast
java.lang.NullPointerException
proof -
have ‹∃f :: urrel set ==> σ . f ` {y. ασ' x = ασ' y} = UNIV›
by (simp add: x_prop card_of_ordLeq2 ordLess_imp_ordLeq)
then obtain f :: ‹urrel set ==> σ› where ‹f ` {y. ασ' x = ασ' y} = UNIV›
by presburger
moreover obtain a where ‹f a = ασ' x› and ‹ασ' a = ασ' x›
by (smt (verit, best) calculation UNIV_I image_iff mem_Collect_eq)
ultimately have ‹(f (a := f x, x := f a)) ` {y. ασ' x = ασ' y} = UNIV ∧
(f (a := f x, x := f a)) x = ασ' x›
by (auto simp: image_def)
thus ?thesis by blast
qed
then obtain f where fimage: ‹f ` {y. ασ' x = ασ' y} = UNIV›
and fx: ‹f x = ασ' x›
by blast
define ασ :: ‹urrel set ==> σ› where ‹ασ ≡ λ urrels . if ασ in: beta_C_mequiv_sy)
then f urrels
else ασ' urrels›
have ‹surj ασ›
proof -
{
fix s :: σ
{
assume ‹s ∈ range ασ'›
hence 0: ‹ασ' (inv ασ' s) = s›
by (meson f_inv_into_f)
{
assume ‹s = ασ' x›
hence ‹
using ασ_def fx by presburger
hence ‹
by auto
}
moreover {
assume ‹s ≠ ασ' x›
hence ‹ασ (inv ασ' s) = s›
unfolding ασ_def 0 by presburger
hence ‹sup>P\<>
by blast
}
ultimately have ‹∃f . ασ (f s) = s›
by blast
}
moreover {
assume ‹s ∉ range ασ'›
moreover obtain urrels where ‹f urrels = s› and ‹ασ' x = ασ' urrels›
by (smt (verit, best) UNIV_I fimage image_iff mem_Collect_eq)
ultimately have ‹ασ urrels = s›
using ασ_def by presburger
hence ‹∃f . ασ (f s) = s›
by (meson f_inv_into_f range_eqI)
}
ultimately have ‹∃f . ασ (f s) = s›
by blast
}
thus ?thesis
by (metis surj_def)
qed
moreover have ‹∀x y. ασ x = ασ y ⟶ ασ' x = ασ' y›
by (metis ασ_def rangeI)
ultimately show ?thesis
by blast
‹
(and consequently the predecessor axiom), we specify which
abstract objects are distinguished by @{const ασ'}.›
urrel_to_ψrel :: ‹urrel ==> (ψ ==> w ==> bool)› where ‹urrel_to_ψrel ≡ λ r u w . AOT_model_valid_in w (Rep_urrel r (ψυ u))›
ψrel_to_urrel :: ‹(ψ ==> w ==> bool) ==> urrel› where ‹^bo>🚫
(λ u . ε\o w . case u of ψυ x ==> φ x w | _ ==> False)›
AOT_urrel_ψequiv :: ‹urrel ==> urrel ==> bool› where ‹AOT_urrel_ψequiv ≡deduction])
AOT_model_valid_in v (Rep_urrel s (ψυ u))›
urrel_ψrel_quot: ‹Quotient3 AOT_urrel_ψequiv urrel_to_ψrel ψrel_to_urrel›
(rule Quotient3I)
show ‹urrel_to_ψrel (ψrel_to_urrel a) = a› for a
unfolding ψrel_to_urrel_def urrel_to_ψrel_def
apply (rule ext)
apply (subst Abs_urrel_inverse)
by (auto simp: AOT_model_proposition_choice_simp)
show ‹AOT_urrel_ψequiv (ψrel_to_urrel a) (ψrel_to_urrel a)› for a
unfolding ψrel_to_urrel_def AOT_urrel_ψequiv_def
apply (subst (1 2) Abs_urrel_inverse)
by (auto simp: AOT_model_proposition_choice_simp)
show ‹AOT_urrel_ψequiv r s = (AOT_urrel_ψequiv r r ∧
urrel_to_ψrel r = urrel_to_ψrel s)› for r s
proof
assume ‹AOT_urrel_ψequiv r s›
hence ‹AOT_model_valid_in v (Rep_urrel r (ψυ u)) =
AOT_model_valid_in v (Rep_urrel s (ψυ u))› for u v
using AOT_urrel_ψequiv_def by metis
hence ‹urrel_to_ψrel r = urrel_to_ψrel s›
unfolding urrel_to_ψrel_def
by simp
thus ‹AOT_urrel_ψequiv r r ∧ AOT_urrel_ψequiv s s ∧
urrel_to_ψrel r = urrel_to_ψrel s›
unfolding AOT_urrel_ψequiv_def
by auto
next
assume ‹AOT_urrel_ψequiv r r ∧ AOT_urrel_ψequiv s s ∧
urrel_to_ψrel r = urrel_to_ψrel s›
hence ‹
AOT_model_valid_in v (Rep_urrel s (ψυ u))› for u v
by (metis urrel_to_ψrel_def)
thus ‹AOT_urrel_ψequiv r s›
using AOT_urrel_ψequiv_def by presburger
qed
(ασ')
ασ_eq_ord_exts_all: ‹ασ' a = ασ' b ==> (∧s . urrel_to_ψrel s = urrel_to_ψrel r ==> s ∈ a)
\<Longrightarrow !\^ in v]"
ασ_eq_ord_exts_ex: ‹ασ' a = ασ' b ==> (∃ s . s ∈ a ∧ urrel_to_ψrel s = urrel_to_ψrel r) ==> (∃s . s ∈ b ∧ urrel_to_ψrel s = urrel_to_ψrel r)›
-
define ασ_wit_intersection where ‹ασ_wit_intersection ≡ λ urrels .
{ordext . ∀urrel . urrel_to_ψrel urrel = ordext ⟶ urrel ∈ urrels}›
define ασ_wit_union where ‹ασ_wit_union ≡ λ urrels .
{ordext . ∃urrel∈urrels . urrel_to_ψrel urrel = ordext}›
let ?ασ_wit = ‹λ urrels .
let ordexts = ασ_wit_intersection urrels in
let ordexts' = ασ_wit_union urrels in
(ordexts, ordexts', undefined)›
define ασ_wit :: ‹urrel set ==> σ› where ‹ασ_wit ≡ λ urrels . Abs_σ (?ασ_wit urrels)›
{
fix a b :: ‹urrel set› and r s
assume ‹ασ {
hence 0: ‹{ordext. ∀urrel. urrel_to_ψrel urrel = ordext ⟶ urrel ∈ a} =
{ordext. ∀urrel. urrel_to_ψrel urrel = ordext ⟶ urrel ∈ b}›
unfolding ασ_wit_def Let_def
apply (subst (asm) Abs_σ_inject)
by (auto simp: ασ_wit_intersection_def ασ_wit_union_def)
assume ‹urrel_to_ψrel s = urrel_to_ψrel r ==> s ∈ a› for s
hence ‹
{ordext. ∀urrel. urrel_to_ψrel urrel = ordext ⟶ urrel ∈ a}›
by auto
hence ‹urrel_to_ψrel r ∈
{ordext. ∀urrel. urrel_to_ψrel urrel = ordext ⟶ urrel ∈ b}›
using 0 by blast
moreover assume ‹urrel_to_ψrel s = urrel_to_ψrel r›
ultimately have ‹s ∈ b›
by blast
}
moreover {
fix a b :: ‹urrel set›h"[(🚫
assume ‹ασ_wit a = ασ_wit b›
hence 0: ‹{ordext. ∃urrel ∈ a. urrel_to_ψrel urrel = ordext} =
{ordext. ∃urrel ∈ b. urrel_to_ψrel urrel = ordext}›
unfolding ασ_wit_def
using Abs_σ_inject ασ_wit_union_def by auto
assume ‹s ∈ a›
hence ‹urrel_to_ψrel s ∈ {ordext. ∃urrel ∈ a. urrel_to_ψrel urrel = ordext}›
by blast
moreover assume ‹urrel_to_ψrel s = urrel_to_ψrel r›
ultimately have ‹urrel_to_ψrel r ∈
{ordext. ∃unfolding p propnot_def .
using "0" by argo
hence ‹∃s. s ∈ b ∧ urrel_to_ψrel s = urrel_to_ψrel r›
by blast
}
ultimately show ?thesis
by (safe intro!: exI[where x=ασ_wit]; metis)
‹We enable the extended model version.›
(input) AOT_ExtendedModel where ‹AOT_ExtendedModel ≡ True›
‹x\<^>\
abstract objects, modelled as sets of urrelations, or null objects, used to
represent non-denoting definite descriptions.›
κ = ψκ ψ | ακ ‹urrel set› | is_nullκ: nullκ null
‹The mapping from abstract objects to urelements can be naturally
lifted to a surjective mapping from individual terms to urelements.›
κυ :: ‹κ==>υ› where ‹κυ (ψκ x) = ψυ x› ‹κυ (ακ x) = συ (ασ x)› ‹κυ (nullκ x) = nullυ x›
κυ_surj: ‹surj κυ›
using ασ_surj by (metis κυ.simps(1) κυ.simps(2) κυ.simps(3) υ.exhaust surj_def)
‹By construction if the urelement of an individual term is exemplified by
an urrelation, it cannot be a null-object.›
urrel_null_false:
apply (rul(rule beta_C_meta_1)
shows ‹¬is_nullκ x›
by (metis (mono_tags, lifting) assms Rep_urrel κ.collapse(3) κυ.simps(3)
mem_Collect_eq)
‹AOT requires any ordinary object to be @{emph ‹possibly concrete›} and that
there is an object that is not actually, but possibly concrete.›
AOT_model_concreteψ :: ‹ψ ==> w ==> bool›
(AOT_model_concreteψ)
AOT_model_ψ_concrete_in_some_world: ‹∃ w . AOT_model_concreteψ x w›
AOT_model_contingent_object: ‹∃ x w . AOT_model_concreteψ x w ∧¬AOT_model_concreteψ x w0›
by (rule exI[where x=‹λ_ w. w ≠ w0›]) (auto simp: AOT_model_nonactual_world)
‹We define a type class for AOT's terms specifying the conditions under which
objects of that type denote and require the set of denoting terms to be
non-empty.›
AOT_Term =
fixes AOT_model_denotes :: ‹'a ==> bool›
assumes AOT_model_denoting_ex: ‹∃ x . AOT_model_denotes x›
‹All types except the type of propositions involve non-denoting terms. We
define a refined type class for those.›
AOT_IncompleteTerm = AOT_Term +
assumes AOT_model_nondenoting_ex: ‹∃ x . ¬AOT_model_denotes x›
‹Generic non-denoting term.›
AOT_model_nondenoting :: ‹ hav"[\lparr^bold>\<>. ‹AOT_model_nondenoting ≡ SOME τ . ¬AOT_model_denotes τ›
AOT_model_nondenoing: ‹¬AOT_model_denotes (AOT_model_nondenoting)›
using someI_ex[OF AOT_model_nondenoting_ex]
unfolding AOT_model_nondenoting_def by blast
‹@{const AOT_model_denotes} can trivially be extended to products of types.›
prod :: (AOT_Term, AOT_Term) AOT_Term
AOT_model_denotes_prod :: ‹'a×'b ==> bool› where ‹AOT_model_denotes_prod ≡ λ(x,y) . AOT_model_denotes x ∧ AOT_model_denotes y›
proof
show ‹∃x::'a×'b. AOT_model_denotes x›
by (simp add: AOT_model_denotes_prod_def AOT_model_denoting_ex)
‹We specify a transformation of proposition-valued functions on terms, s.t.
the result is fully determined by @{emph ‹regular›} terms. This will be required
for modelling n-ary relations as functions on tuples while preserving AOT's
definition of n-ary relation identity.›
using l_idenl_identity[axiom eduction, deduction]
fixes AOT_model_irregular :: ‹('a ==>o) ==> 'a ==>o›
and AOT_model_regular :: ‹'a ==> bool›
and AOT_model_term_equiv :: ‹'a ==> 'a ==> bool›
assumes AOT_model_irregular_false: ‹¬AOT_model_valid_in w (AOT_model_irregular φ x)›
assumes AOT_model_irregular_equiv: ‹AOT_model_term_equiv x y ==>
AOT_model_irregular φ x = AOT_model_irregular φ y›
assumes AOT_model_irregular_eqI: ‹(∧ x . AOT_model_regular x ==> φ x = ψ x) ==>
AOT_model_irregular φ fas
‹We introduce a type class for individual terms that specifies being regular,
being equivalent (i.e. conceptually @{emph ‹sharing urelements›}) and the
transformation on proposition-valued functions as specified above.›
AOT_IndividualTerm = AOT_IncompleteTerm +
fixes AOT_model_regular :: ‹'a ==> bool›
fixes AOT_model_term_equiv :: ‹'a ==> 'a ==> bool›
fixes AOT_model_irregular :: ‹('a ==>o) ==> 'a ==>o›>\lparrin
assumes AOT_model_irregular_nondenoting: ‹¬AOT_model_regular x ==>¬AOT_model_denotes x›
assumes AOT_model_term_equiv_part_equivp: ‹equivp AOT_model_term_equiv›
assumes AOT_model_term_equiv_denotes: ‹AOT_model_term_equiv x y ==> (AOT_model_denotes x = AOT_model_denotes y)›
assumes AOT_model_term_equiv_regular: ‹AOT_model_term_equiv x y ==> (AOT_model_regular x = AOT_model_regular y)›
assumes AOT_model_irregular: ‹AOT_model_irregular_spec AOT_model_irregular AOT_model_regular
AOT_model_term_equiv›
AOT_model_irregular_spec AOT_model_irregular AOT_model_regular
AOT_model_term_equiv
using AOT_model_irregular .
‹Our concrete type for individual terms satisfies the type class of
individual terms.
Note that all unary individuals are regular. In general, an individual term
may be a tuple and is regular, if at most one tuple element does not denote.›
κ :: AOT_IndividualTerm
AOT_model_term_equiv_κ :: ‹κ ==> κ ==> bool› where ‹AOT_model_term_equiv_κ ≡
AOT_model_denotes_κ :: ‹κ ==> bool› where ‹AOT_model_denotes_κ ≡ λ x . ¬is_nullκ x›
AOT_model_regular_κ :: ‹κ ==> bool› where ‹AOT_model_regular_κ ≡ λ x . True›
AOT_model_irregular_κ :: ‹(κ ==>o) ==> κ ==>o›><^old\ ‹AOT_model_irregular_κ ≡ SOME φ . AOT_model_irregular_spec φ
AOT_model_regular AOT_model_term_equiv›
proof
show ‹∃x :: κ. AOT_model_denotes x›
by (rule exI[where x=‹ψκ undefined›])
(simp add: AOT_model_denotes_κ_def)
show ‹∃x :: κ. ¬AOT_model_denotes x›
by (rule exI[where x=‹nullκ undefined›])
(simp add: AOT_model_denotes_κ_def AOT_model_regular_κ_def)
show "¬AOT_model_regular x ==>¬ AOT_model_denotes x" for x :: κ
by (simp add: AOT_model_regular_κ_def)
show ‹
by (rule equivpI; rule reflpI exI sympI transpI)
(simp_all add: AOT_model_term_equiv_κ_def)
fix x y :: κ
show ‹AOT_model_term_equiv x y ==> AOT_model_denotes x = AOT_model_denotes y›
by (metis AOT_model_denotes_κ_def AOT_model_term_equiv_κ_def κ.exhaust_disc
κυ.simps υ.disc(1,3,5,6) is_ακ_def is_ψκ_def is_nullκ_def)
fix x y :: κ
show ‹AOT_model_term_equiv x y ==> AOT_model_regular x = AOT_model_regular y›
by (simp add: AOT_model_regular_κ_def)
have "AOT_model_irregular_spec (λ φ (x::κ) . ε\o w . False)
AOT_model_regular AOT_model_term_equiv"
by standard (auto simp: AOT_model_proposition_choice_simp)
thus ‹AOT_model_irregular_spec (AOT_model_irregular::(κ==>o) ==> κ ==>o)
AOT_model_regular AOT_model_term_equiv›
unfolding AOT_model_irregular_κ_def by (metis (no_types, lifting) someI_ex)
‹We define relations among individuals as proposition valued functions.
@{emph ‹Denoting›} unary relations (among @{typ κ}) will match the
urrelations introduced above.›
'a rel (‹🪙›) = ‹
type_definition_rel
‹We will use the transformation specified above to "fix" the behaviour of
functions on irregular terms when defining @{text ‹λ›}-expressions.›
fix_irregular :: ‹('a::AOT_IndividualTerm ==>o) ==> ('a ==>o)› where ‹fix_irregular ≡ λ φ x . if AOT_model_regular x
then φ x else AOT_model_irregular φ x›
fix_irregular_denoting: ‹AOT_model_denotes x ==> fix_irregular φ x = φ x›
by (meson AOT_model_irregular_nondenoting fix_irregular_def)
fix_irregular_regular: ‹AOT_model_regular x ==> fix_irregular φ oa_contingenapply - by PLM_s
by (meson AOT_model_irregular_nondenoting fix_irregular_def)
fix_irregular_irregular: ‹¬AOT_model_regular x ==> fix_irregular φ x = AOT_model_irregular φ x›
by (simp add: fix_irregular_def)
‹Relations among individual terms are (potentially non-denoting) terms.
A relation denotes, if it agrees on all equivalent terms (i.e. terms sharing
urelements), is necessarily false on all non-denoting terms and is
well-behaved on irregular terms.›
rel :: (AOT_IndividualTerm) AOT_IncompleteTerm
‹\linelabel{AOT_model_denotes_rel}›
AOT_model_denotes_rel :: ‹<'a> ==> bool›} ‹λ φ . (∀ x y . AOT_model_term_equiv x y ⟶ φ x = φ y) ∧
(∀ w x . AOT_model_valid_in w (φ x) ⟶ AOT_model_denotes x) ∧
(∀ x . ¬AOT_model_regular x ⟶ φ x = AOT_model_irregular φ x)› .
proof
have ‹AOT_model_irregular (fix_irregular φ) x = AOT_model_irregular φ x›
for φ and x :: 'a
by (rule AOT_model_irregular_eqI) (simp add: fix_irregular_def)
thus ‹∃ x :: <'a> . AOT_model_denotes x›?th
by (safe intro!: exI[where x=‹Abs_rel (fix_irregular (λx. ε\o w . False))›])
(transfer; auto simp: AOT_model_proposition_choice_simp fix_irregular_def
AOT_model_irregular_equiv AOT_model_term_equiv_regular
AOT_model_irregular_false)
show ‹∃f :: <'a> . ¬AOT_model_denotes f›
by (rule exI[where x=‹Abs_rel (λx. ε\o w . True)›];
auto simp: AOT_model_denotes_rel.abs_eq AOT_model_nondenoting_ex
AOT_model_proposition_choice_simp)
AOT_model_denotes_Abs_rel_fix_irregular:
assumes ‹∧ x y . AOT_model_term_equiv x y ==> φ x = φ y›
and ‹∧ w x . AOT_model_valid_in w (φ x) ==> AOT_model_denotes x›
shows ‹AOT_model_denotes (Abs_rel (fix_irregular φ))›
-
have ‹AOT_model_irregular φ x = AOT_model_irregular
(λx. if AOT_model_regular x then φ x else AOT_model_irregular φ x) x›
if ‹¬ AOT_model_regular x›
for x
by (rule AOT_model_irregular_eqI) auto
thus ?thesis
unfolding AOT_model_denotes_rel.rep_eq
using assms by (auto simp: AOT_model_irregular_false Abs_rel_inverse
AOT_model_irregular_equiv fix_irregular_def
AOT_model_term_equiv_regular)
AOT_model_term_equiv_rel_equiv:
assumes ‹AOT_model_denotes x›
and ‹AOT_model_denotes y›
shows ‹AOT_model_term_equiv x y = (∀ Π w . AOT_model_denotes Π ⟶
AOT_model_valid_in w (Rep_rel Π x) = AOT_model_valid_in w (Rep_rel Π y))›
assume ‹AOT_model_term_equiv x y›
thus ‹∀ Π w . AOT_model_denotes Π qed
AOT_model_valid_in w (Rep_rel Π y)›
by (simp add: AOT_model_denot
have 0: ‹(AOT_model_denotes x' ∧ AOT_model_term_equiv x' y) =
(AOT_model_denotes y' ∧ AOT_model_term_equiv y' y)›
open>AOT_model_term_equiv x' y'› x' y'
by (metis that AOT_model_term_equiv_denotes AOT_model_term_equiv_part_equivp
equivp_def)
assume ‹∀ Π w . AOT_model_denotes Π ⟶ AOT_model_valid_in w (Rep_rel Π x) =
AOT_model_valid_in w (Rep_rel Π y)›
moreover have ‹
java.lang.NullPointerException
(is "AOT_model_denotes ?r")
by (rule AOT_model_denotes_Abs_rel_fix_irregularI)
(auto simp: 0 AOT_model_denotes_rel.rep_eq Abs_rel_inverse fix_irregular_def
AOT_model_proposition_choice_simp AOT_model_irregular_false)
ultimately have ‹AOT_model_valid_in w (Rep_rel ?r x) =
AOT_model_valid_in w (Rep_rel ?r y)›AOT_urrel_ψequiv (rel_to_urrel Π') (rel_to_urrel Π
by blast
us <>AOT_model_term_equiv
‹Denoting relations among terms of type @{typ κ} correspond to urrelations.›
rel_to_urrel :: ‹'_den) ‹rel_to_urrel Π' ∈
urrel_to_rel :: ‹urrel_to_urr <>' ‹urrel_to_rel ≡ λ φ . Abs_rel (λ x . Rep_urrel φ (κυ x))›
java.lang.NullPointerException ‹∃s. s ∈ b ∧ urrel_to_ψrel s = urrel_to_ψrel (rel_to_urrel Π)›
urrel_quotient3: ‹
have ‹(λu. Rep_urrel a_esing α_eq_ord_exts_ex α_eq ext α_α' by blast
by (rule ext) (metis (mono_tags, lifting) κ
thus ‹s ∈ b ∧ urrel_to_ψrel s = urre_o\omega (rel_to_urrel Π)›
by (simp add: Abs_rel_inverse rel_to_urrel_def urrel_to_rel_def
Rep_urrel_inverse)
show ‹ for a
unfolding AOT_rel_equiv_def urrel_to_rel_def
by transfer (simp add: AOT_model_regular_κ_def AOT_model_denotes_κ_def
AOT_model_term_equiv_κ_def urrel_null_false)
{
fix a
assume ‹w x. AOT_model_valid_in w (a x) ⟶¬ is_nullκ x›
hence ‹υ
{φ. ∀x w. ¬ AOT_model_valid_in w (φ (nullυ x))}›
by (simp; metis (mono_tags, lifting) κ.exhaust_disc κυ.simps υ.disc(1,3,5)
υ.disc(6) is_ακ_def is_ψκ_def someI_ex)
} note 1 = this
{
r s :::: \open\<kappakappa
assume A: ‹∀x y. AOT_model_term_equiv x y ⟶ r x = r y›
assume ‹∀w x. AOT_model_valid_in w (r x) ⟶ AOT_model_denotes x›AOT_model_valid_in v (Rep_rel Π'' x) =
hence 2: ‹(λu. r (SOME x. κυ (Rp_rl\Pi> x)\)›
{φ. ∀x w. ¬ AOT_model_valid_in w (φ (nullυ x))}›
using 1 AOT_model_denotes_κ_def by meson
assume B: ‹∀x y. AOT_model_term_equiv x y ⟶ s x = s y›if \<openw for v x
assume pproof(insert that)
hence 3: ‹∃w. AOT_model_concrete w x›
{φ. ∀x w. ¬ AOT_model_valid_in w (φ (nullυ x))}›
using 1 AOT_model_denotes_κ_def by meson
assume ‹Abs_urrel (λu. r (SOME x. κυ x = u)) =
Abs_urrel (λu. s (SOME x. κυ x = u))›
hence 4: ‹r (SOME x. κυ x = u) = s (SOME x::κ. κυ x = u)› for u
unfolding Abs_urrel_inject[OF 2 3] by metis
have ‹
using 4[of ‹κυ x›]
by (metis (mono_tags, lifting) A B AOT_model_term_equiv_κ_def someI_ex)
hence ‹r = s› by auto
}
thus ‹AOT_rel_equiv r s = (AOT_rel_equiv r r ∧ AOT_rel_equiv s s ∧
rel_to_urrel r = rel_to_urrel s)› for r s
unfolding AOT_rel_equiv_def rel_to_urrel_def
by transfer auto
urrel_quotient: ‹
(λx y. AOT_rel_equiv x x ∧ rel_to_urrel x = y)›
using Quotient3_to_Quotient[OF urrel_quotient3] by auto
‹Unary individual terms are always regular and equipped with encoding and
concreteness. The specification of the type class anticipates the required
properties for deriving the axiom system.›
AOT_UnaryIndividualTerm =
fixes AOT_model_enc :: ‹'a ==> <'a::AOT_IndividualTerm> ==> bool›
and AOT_model_concrete :: ‹w ==> 'a ==> bool›🚫
assumes AOT_model_unary_regular: ‹AOT_model_regular x›―‹All unary individual terms are regular.›
and AOT_model_enc_relid: ‹AOT_model_denotes F ==>
AOT_model_denotes G ==>
(∧ x . AOT_model_enc x F ⟷ AOT_model_enc x G) ==> F = G›
and AOT_model_A_objects: ‹∃x . AOT_model_denotes x ∧
(∀w. ¬ AOT_model_concrete w x) ∧
(∀F. AOT_model_denotes F ⟶ AOT_model_enc x F = φ F)›
and AOT_model_contingent: ‹x = ψκ u›
and AOT_model_nocoder: ‹.xhust)
and AOT_model_concrete_equiv: ‹
AOT_model_concrete w x = AOT_model_concrete w y› OT_ v Rp_r \Picl>
and AOT_model_concrete_denotes: ‹AOT_model_concrete w x ==> AOT_model_denotes x› ―‹The following are properties that will only hold in the extended models.›
and AOT_model_enc_indistinguishable_all: ‹AOT_ExtendedModel ==>
AOT_model_denotes a ==>¬(∃ w . AOT_model_concrete w a) ==>
AOT_model_denotes b ==>¬(∃ w . AOT_model_concrete w b) ==>
AOT_model_denotes Π ==> ==>
(\<And
AOT_model_valid_in v (Rep_rel Π' b))) ==>
(∧ Π' . AOT_model_denotes Π' ==>
(∧ v x . ∃ w . AOT_model_concrete w x ==>
AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x)) ==>
AOT_model_enc a Π') ==>
(∧<\<
(∧ v x . ∃ w . AOT_model_concrete w x \qed
AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x)) ==>
AOT_model_enc b Π')›
and AOT_model_enc_indistinguishable_ex: ‹AOT_ExtendedModel ==>
AOT_model_denotes a ==>¬(∃ w . AOT_model_concrete w a) ==> show ?thesis
<\<& (1) \λ y . q)) in v]"
AOT_model_denotes Π ==>
(∧ Π' . AOT_model_denotes Π' ==>
(∧equivp (AOT_model_term_equiv :: 'a×'b ==> 'a×]
AOT_model_valid_in v (Rep_rel Π
(∃
(∀ v x . (∃ w . AOT_model_concrete w x) ⟶
AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x))) ==>
(∃ Π' . AOT_model_denotes Π' ∧ AOT_model_enc b Π' ∧
(∀ v x . (∃ w . AOT_model_concrete w x) ⟶
AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x)))›
‹Instantiate the class of unary individual terms for our concrete type of
individual terms @{typ κ}.›
κ :: AOT_UnaryIndividualTerm
AOT_model_enc_κ :: ‹κ ==> <\<kappa>> ==> bool› where ‹
case x of α¬AOT_model_regular x ==>¬ for x :: ‹
| _ ==> False›
AOT_model_concrete_κ :: ‹AOT_model_term_equiv x y ==> AOT_model_denotes x = AOT_model_denotes y› ‹ ‹'a×'b› ‹
AOT_meta_A_objects_κ: ‹
<>. ¬ AOT_model_concrete w x) ∧
(∀F. AOT_model_denotes F ⟶ AOT_model_enc x F = φ F)› for φ
apply (rule exI[where x=‹])
apply (simp add: AOT_model_enc_κ_def AOT_model_denotes_κ_def)
by (metis (no_types, lifting) AOT_rel_equiv_def urrel_quotient
Quotient_rep_abs_fold_unmap)
proof
show ‹
by (simp add: AOT_model_regular_κ_def)
fix F G :: ‹kappa>>›
assume ‹
moreover assume ‹AOT_model_irregular_spec
moreover assume ‹'b ==>o a×b <>\o .
moreover obtain x where ‹G. AOT_model_denotes G 🚫
using AOT_meta_A_objects_κ by blast
ultimately show ‹
show ‹
(∀w. ¬SOME φ :: ('a×'b ==>o) ==> 'a×o .
(∀⟶ F)›
using AOT_meta_A_objects_κ .
proo
by (metis AOT_model_concrete_κ¬AOT_model_valid_in w (AOT_model_irregular φ (a, b))›
κ.collapse(2))
show ‹AOT_model_valid_in w (AOT_model_irregular φ x)" for w φ'a×'b›
if ‹
for x y :: κ and w
using that by (induct x; induct y; auto simp: AOT_model_term_equiv_κ
show ‹
java.lang.NullPointerException
(* Extended models only *) next fix κ κ assume ext: ‹1› assume‹
moreover assume ‹
ultimately obtain a where a_def: ‹›
by meiATmel\omegaconcrete_in_some_world AOT_model_concrete_κ.simps(1)
java.lang.NullPointerException
assume ‹
moreover assume ‹
ultimately obtain b where b_def: ‹
by (metis AOT_model_ψ_concrete_in_some_world AOT_model_conc\>
AOT_model_denotes_κ_def κ.discI(3) κ.exhaust_sel)
assume ‹>_equiv x2_equiv)
java.lang.NullPointerException
hence ‹ κ
AOT_model_valid_in w (Rep_urrel r (κυ κ'))›
by (metis AOT_rel_equiv_def Abs_rel_inverse Quotient3_rel_rep
iso_tuple_UNIV_I urrel_quotient3 urrel_to_rel_def)
hence ‹
AOT_model_valid_in w (Rep_urrel r (κυ κ)) =
java.lang.NullPointerException
by presburger
hence ασ_eq: ‹
unfolding Let_def
s_urrel_inverse)
using AOT_model_proposition_choice_simp a_def b_def by force+
assume Π_den: ‹
have ‹
by (metis (mono_tags, lifting) AOT_model_denotes_κ_def
AOT_model_denotes_rel.rep_eq κ.exhaust_disc κ
java.lang.NullPointerException
is_ακ_def is_ψey ‹1 ∧ AOT_model_de y<^>2
java.lang.NullPointerException
by (metis (mono_tags, lifting) AOT_model_denotes_rel.rep_eq
AOT_model_term_equiv_κ_def κυ.simps(1) Π_den verit_sko_ex')
ultimately have ‹2y2_equiv)
unfolding rel_to_urrel_def
by (subst Abs_urrel_inverse) auto
hence ‹∃r . ∀ x . Rep_rel Π (ψκ x) = Rep_urrel r (ψυ x)›
by (auto intro!: exI[where x=‹
then obtain r where r_prop: ‹
by blast
assume ‹
java.lang.NullPointerException
AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Πusing AOT_model_ter AOTmdlte_eqivregar
java.lang.NullPointerException
(∧by b
AOT_model_valid_in v (Rep_rel Π
by (metis AOT_model_co sho\openAOT_ode_tem_equv y\Longrightarrow
κ.exhaust_disc is_α <>
hence ‹'a×'b›
T_model_valid_invR_rel \Pi> \omegaκ x)) \Longrightarrow r ∈ for r
unfolding a_def[symmetric] AOT_model_enc_κ_def apply simp
by (smt (verit, best) AOT_rel_equiv_def Abs_rfix \>ψ'a×o
κυAOT_model_regular x ==> φ x = ψ x›
hence ‹
AOT_model_valid_in v (Rep_urrel r (ψυAOT_model_denotes x ∧ AOT_model_regular y ∨ \noto_eote x and AOOT_model_denotes y› for x y
hence ‹AOT_model_irregular φ (x,y) = AOT_model_irregular ψ
by (metis urrel_to_ψ
<><And>s. urrel_to_ψrel s = urrel_to_ψrel r ==> s ∈ b›
using ασ1φ
assume Π'_den: ‹AOT_model_denotes Π'›
assume ‹∃w. AOT_model_concrete w x ==> AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x)› for v x
hence ‹AOT_model_valid_in v (Rep_rel Π' (ψκ x)) =
AOT_model_valid_in v (Rep_rel Π (ψκ x))› for v x
using AOT_model_ψ_concrete_in_some_world AOT_model_concrete_κ.simps(1)
by presburger
hence ‹
AOT_model_valid_in v (Rep_urrel r (ψυ x))› fo AOT_model_term_equiv_denotes AOTAOT_mode_tem_equvep(1)
by (smt (verit, best) AOT_rel_equv_ef bsrelinvese Quoen3def
κ\<case
hence ‹
by (metis (full_types) AOT_urrel_ψ
hence 🚫 Π' ∈ b›
thus ‹
unfolding b_def[symmetric] AOT_model_enc_κ_def by (auto simp: ΠeglarI rjpi)
fix κ κ' :: \thusopen>AOT_model_irregular φ x = AOT_model_irregular ψ x› for x :: ‹
assume e \opennO_ExenddMde\close>
assume ‹
moreover assume ‹
ultimately obtain a where a_def: ‹
by (metis AOT_model_ψ_concrete_in_some_world AOT_model_concrete_κ.simps(1)
AOT_model_denotes_κs "∧(a::':AO_ayndvdale) ( :'b:OTInidulrm)
assume ‹
moreover assume ‹
ultimately obtain b where b_def: ‹_euvxb yb)
by (metis AOT_model_ψ
AOT_model_denotes_κT_oeem_eqipaeip equi_reflp+
assume ‹The type of propositions are trivial instances of terms.›
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
hence ‹
AOT_model_valid_in w (Rep_urrel r (κυ κshow \<penen
by (metis AOT_rel_equiv_def Abs_rel_inverse Quotient3_rel_rep
_ue_UII urequtet3urrel_t_l_def)
hence ‹
AOT_model_valid_in w (Rew (Rep_urrel rr (κ κ)) =
AOT_model_valid_in w (Rep_urrel r (κυ κ'))›AOT's variables are modelled by restricting the type of terms to those terms
by presburger
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
unfolding Let_def
apply (subst (asm) (1 2) Abs_urrel_inverse)
using AOT_model_proposition_choice_simp a_def b_def by force+
assume Π_den: ‹AOT_model_denotes Π›
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
by (metis (mono_tags, lifting) AOT_model_denotes_κ_def
AOT_model_denotes_rel.rep_eq κ.exhaust_disc κυ.simps(1,2,3) ‹AOT_model_denotes Π› υ.disc(8) υ.disc(9) υ.distinct(3)
is_ακ_def is_ψκ_def verit_sko_ex')
moreover have ‹Rep_rel Π (ψκ x) = Rep_rel Π (SOME xa. κυ xa = ψυ x)› for x
by (metis (mono_tags, lifting) AOT_model_denotes_rel.rep_eq
AOT_model_term_equiv_κ_def κυ.simps(1) Π_den verit_sko_ex')
ultimately have ‹Rep_rel Π (ψκ x) = Rep_urrel (rel_to_urrel Π) (ψυ x)› for x
unfolding rel_to_urrel_def
by (subst Abs_urrel_inverse) auto
hence ‹∃r . ∀ x . Rep_rel Π (ψκ x) = Rep_urrel r (ψυ x)›
by (auto intro!: exI[where x=‹rel_to_urrel Π›])
then ob }
by blast
assume ‹∃Πultimashow "[forall\lbrace^sup>P, F}>\<hi
AOT_model_enc κ Π' ∧
(∀v x. (∃w. AOT_model_concrete w x) ⟶ AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x))›
then obtain Π' where
Π'_den: ‹AOT_model_denotes Π'› and
κ_enc_Π': ‹AOT_model_enc κ Π'› and
Π'_prop: ‹∃w. AOT_model_concrete w x ==>
java.lang.NullPointerException
AOT_model_valid_in v (Rep_rel Π x)› for v x
by blast
have ‹AOT_model_valid_in v (Rep_rel Π' (ψκ x)) =
AOT_model_valid_in v (Rep_rel Π (ψκ x))› for x v
by (simp add: AOT_model_ψ_concrete_in_some_world Π'_prop)
hence 0: ‹AOT_urrel_ψequiv (rel_to_urrel Π') (rel_to_urrel Π)›
unfolding AOT_urrel_ψequiv_def
by (smt (verit) AOT_rel_equiv_def Abs_rel_inverse Quotient3_def
κυ.simps(1) iso_tuple_UNIV_I urrel_quotient3 urrel_to_rel_def
Π_den Π'_den)
have ‹rel_to_urrel Π' ∈ a›
and ‹urrel_to_ψrel (rel_to_urrel Π') = urrel_to_ψrel (rel_to_urrel Π)›
by (metis Quotient3_rel 0 urrel_ψrel_quot)
hence ‹∃s. s ∈ b ∧ urrel_to_ψrel s = urrel_to_ψrel (rel_to_urrel Π)›
using ασ_eq_ord_exts_ex ασ_eq ext ασ_ασ' by blast
then obtain s where
java.lang.NullPointerException
by blast
then obtain Π'' where
Π''_prop: ‹rel_to_urrel Π'' = s› and Π''_den: ‹AOT_model_denotes Π''›
AOT_rel Quotient3_def urrel_quoti)
moreover have ‹AOT_model_enc κ' Π''›
by (metis AOT_model_enc_κ_def Π''_den Π''_prop κ.simps(11) b_def s_prop)
moreover have ‹
AOT_model_valid_in v (Rep_rel Π x)›
open>exists>w. AOT_model_concrete w x\<>for
proof(insert that)
assume ‹∃w. AOT_model_concrete w x›
then obtain u where x_def: ‹x = ψκ u›
by (metis AOT_model_concrete_κ.simps(2,3) κ.exhaust)
show \<open
AOT_model_valid_in v (Rep_rel Π x)›
unfolding x_def
by (smt (verit, best) AOT_rel_equiv_def Abs_rel_inverse Quotient3_def
Π''_den Π''_prop Π_den κυ.simps(1) iso_tuple_UNIV_I s_prop
urrel_quotient3 urrel_to_ψ[conj] by auto
qed
java.lang.NullPointerException
(∀v x. (∃w. AOT_model_concrete w x) ⟶ AOT_model_valid_in v (Rep_rel Π' x) =
AOT_model_valid_in v (Rep_rel Π x))›
apply (safe intro!: exI[where x=Π''])
by auto
‹Products of unary individual terms and individual terms are individual terms.
A tuple is regular, if at most one element does not denote. I.e. a pair is
regular, if the first (unary) element denotes and the second is regular (i.e.
at most one of its rerecursive tuple elements does no denote),), oor the f firs does
not denote, but the second denotes (i.e. all its recursive tuple elements
denote).›
fix
AOT_model_regular_prod :: ‹'a×'b ==> bool›^bold>\<rightarrow v ‹AOT_model_regular_prod ≡ λ (x,y) . AOT_model_denotes x ∧ AOT_model_regular y ∨ ¬AOT_model_denotes x ∧ AOT_model_denotes y›
AOT_model_term_equiv_prod :: ‹'a×'b ==> 'a×'b ==> "\<^>< ‹AOT_model_term_equiv_prod ≡ λ (x1,y1) (x[🚫
AOT_model_term_equiv x1 x2∧ AOT_model_term_equiv y1 y2›
AOT_model_irregular_prod :: ‹('a×'b ==>o) ==> 'a×'b ==>o› where
AOT_model_irregular_proj2: ‹AOT_model_denotes x ==>
moreover have "[<^old<
AOT_model_irregular (λy. φ (SOME x' . AOT_model_term_equiv x x', y)) y›
AOT_model_irregular_proj1: ‹¬AOT_model_denotes x ∧ AOT_model_denotes y ==>
AOT_model_irregular φ (x,y) =
AOT_model_irregular (λx. φ (x, SOME y' . AOT_model_term_equiv y y')) x›
>\not x ∧
AOT_model_irregular φ (x,y) =
(SOME Φ . AOT_model_irregular_spec Φ AOT_model_regular AOT_model_term_equiv)
φ (x,y)›
by auto blast
using "termination" by blast
proof
obtain x :: 'a and y :: 'b where ‹¬AOT_model_denotes x› and ‹¬AOT_model_denotes y›
by (meson AOT_model_nondenoting_ex AOT_model_denoting_ex)
thus ‹∃x::'a×'b. ¬AOT_model_denotes x›
by (auto simp: AOT_model_denotes_prod_def AOT_model_regular_prod_def)
show ‹
by (metis (mono_tags, lifting) AOT_model_denotes_prod_def case_prod_unfold
AOT_model_irregular_nondenoting AOT_model_regular_prod_def)
java.lang.NullPointerException
show ‹AOT_model_term_equiv x y ==> AOT_model_denotes x = AOT_model_denotes y›
by (metis (mono_tags, lifting) AOT_model_denotes_prod_def case_prod_beta
AOT_model_term_equiv_denotes AOT_model_term_equiv_prod_def )
fix x y :: ‹'a×'b›
show \< thus
by (induct x; induct y;
simp add: AOT_model_term_equiv_prod_def AOT_model_regular_prod_def)
(meson AOT_model_term_equiv_denotes AOT_model_term_equiv_regular)
interpret sp: AOT_m \<open\
AOT_model_regular AOT_model_term_equiv
by (simp add: AOT_model_irregular_spec_def AOT_model_proposition_choice_simp)
have ex_spec: ‹∃ φ :: ('a×'b ==>o) ==> 'a×'b ==>o .
AOT_model_irregular_spec φ AOT_model_regular AOT_model_term_equiv›
using sp.AOT_model_irregular_spec_axioms by blast
have some_spec: ‹AOT_model_irregular_spec
(SOME φ :: ('a×'b ==>o) ==> 'a×'b ==>o .
AOT_model_irregular_spec φ AOT_model_regular AOT_model_term_equiv)
AOT_model_regular AOT_model_term_equiv›
using someI_ex[OF ex_spec] by argo
interpret sp_some: AOT_model_irregular_spec ‹SOME φ :: ('a×'b ==>bol>◻
AOT_model_irregular_spec φ AOT_model_regular AOT_model_term_equiv›
AOT_model_regular AOT_model_term_equiv
using some_spec by blast
show ‹
AOT_model_regular AOT_model_term_equiv›
proof
have ‹¬AOT_model_valid_in w (AOT_model_irregular φ (a, b))›
for w φ and a :: 'a and b :: 'b
by (induct arbitrary: φ rule: AOT_model_irregular_prod.induct)
(auto simp: AOT_model_irregular_false sp_some.AOT_model_irregular_false)
thus "¬AOT_model_valid_in w (AOT_model_irregular φ x)" for w φ and x :: ‹
by (induct x)
next
{
fix x1 y1 :: 'a and x2 y[onj2] "] "🚫
assume x1y1_equiv: ‹AOT_model_term_equiv x1 y1›
moreover assume x2y2_equiv: ‹AOT_model_term_equiv x2 y2›
ultimately have xy_equiv: ‹AOT_model_term_equiv (x1,x2) (yusing encoding[axiom_instan, deduction] by simp
by (simp add: AOT_model_term_equiv_prod_def)
{
assume ‹AOT_model_denotes x1›
moreover hence ‹AOT_model_denotes y1›
using AOT_model_term_equiv_denotes AOT_model_term_equiv_regular
java.lang.NullPointerException
ultimately have ‹AOT_model_irregular φ (x1,x2) =
java.lang.NullPointerException
using AOT_model_irregular_equiv AOT_model_term_equiv_eps(3)
x1y1_equiv x2y2_equiv by fastforce
}
moreover {
assume ‹~AOT_model_denotes x1∧ AOT_model_denotes x2›
moreover hence ‹~AOT_model_denotes y1∧ AOT_model_denotes y2›
java.lang.NullPointerException
ultimately have ‹AOT_model_irregular φ (x1,x2) =
AOT_model_irregular φ
using AOT_model_irregular_equiv AOT_model_term_equiv_eps(3)
x1y1_equiv x2y2_equiv by fastforce
}
moreover {
assume denotes_x: ‹(¬AOT_model_denotes xbold<>\
hence denotes_y: ‹(¬AOT_model_denotes y1∧¬AOT_model_denotes y2)›
by (meson AOT_model_term_equiv_denotes AOT_model_term_equiv_regular
x1y1_equiv x2y2_equiv)
have ep: ‹
by (simp add: AOT_model_term_equiv_eps(3) x1y1_equiv)
have ‹AOT_model_irregular φ (x1,x2) = AOT_model_irregular φ (y1,y2)›
using denotes_x denotes_y
using sp_some.AOT_model_irregular_equiv xy_equiv by auto
}
moreover {
assume denotes_x: ‹¬AOT_model_denotes x1∧ AOT_model_denotes x2›
hence deno: ‹
by (meson AOT_model_term_equiv_denotes x1y1_equiv x2y2_equiv)
have eps_eq: ‹Eps (AOT_model_term_equiv x2) = Eps (AOT_model_term_equiv y2)›
by (simp add: AOT_model_term_equiv_eps(3) x2y2_equiv)
have ‹AOT_model_irregular φ (x1,x2) = AOT_model_irregular φ (y1,y2)›
using denotes_x denotes_y
using AOT_model_irregular_nondenoting calculation(2) by blast
}
ultimately have ‹let?<psi ^supP\<>\
using AOT_model_term_equiv_denotes AOT_model_term_equiv_regular
sp_some.AOT_model_irregular_equiv x1y1_equiv x2y2_equiv xy_equiv
by blast
} note 0 = this
show ‹AOT_model_term_equiv x y ==>
AOT_model_irregular φ x = AOT_model_irregular φ y›
for x y :: ‹'a×'b› and φ
by (induct x; induct y; simp add: AOT_model_term_equiv_prod_def 0)
next
java.lang.NullPointerException
java.lang.NullPointerException
hence ‹φ (x, y) = ψ (x, y)›
if ‹AOT_model_denotes x ∧ AOT_model_regular y ∨
java.lang.NullPointerException
using that unfolding AOT_model_regular_prod_def by simp
hence ‹AOT_model_irregular φ (x,y) = AOT_model_irregular ψ (x,y)›
for x :: 'a and y
proof (induct arbitrary: ψ φ rule: AOT_model_irregular_prod.induct)
case (1 x y φ)
thus ?case
apply simp
by (meson AOT_model_irregular_eqI AOT_model_irregular_nondenoting
AOT_model_term_equiv_denotes AOT_model_term_equiv_eps(1))
next
case (2 x y φ)
thus ?case
apply simp
by (meson AOT_model_irregular_nondenoting AOT_model_term_equiv_denotes
AOT_model_term_equiv_eps(1))
next
case (3 x y φ)
thus ?case
apply simp
by (metis (mono_tags, lifting) AOT_model_regular_prod_def case_prod_conv
sp_some.AOT_model_irregular_eqI surj_pair)
qed
thus ‹AOT_model_irregular φ x = AOT_model_irregular ψ x› for x :: ‹'a×'b›
by (metis surjective_pairing)
qed
‹Introduction rules for term equivalence on tuple terms.›
AOT_meta_prod_equivI:
shows "∧ (a::'a::AOT_UnaryIndividualTerm) x (y :: 'b::AOT_IndividualTerm) .
AOT_model_term_equiv x y ==> AOT_model_term_equiv (a,x) (a,y)"
and "∧ (x::'a::AOT_UnaryIndividualTerm) y (b :: 'b::AOT_IndividualTerm) .
AOT_model_term_equiv x y ==> AOT_model_term_equiv (x,b) (y,b)"
unfolding AOT_model_term_equiv_prod_def
by (simp add: AOT_model_term_equiv_part_equivp equivp_reflp)+
‹
o :: AOT_Term
AOT_model_denotes_o :: ‹o==> bool› where ‹AOT_model_denotes_o≡ λ_. True›
proof
show ‹∃x::o. AOT_model_denotes x›
by (simp add: AOT_model_denotes_o_def)
‹AOT's variables are modelled by restricting the type of terms to those terms
that denote.›
'a AOT_var = ‹{ x :: 'a::AOT_Term . AOT_model_denotes x }›
morphisms AOT_term_of_var AOT_var_of_term
by (simp add: AOT_model_denoting_ex)
‹Simplify automatically generated theorems and rules.›
AOT_var_of_term_induct[induct del]
AOT_var_of_term_cases[cases del]
AOT_term_of_var_induct[induct del]
AOT_term_of_var_cases[cases del]
AOT_var_of_term_inverse = AOT_var_of_term_inverse[simplified]
and AOT_var_of_term_inject = AOT_var_of_term_inject[simplified]
and AOT_var_of_term_induct =
AOT_var_of_term_induct[simplified, induct type: AOT_var]
and AOT_var_of_term_cases =
AOT_var_of_term_cases[simplified, cases type: AOT_var]
and AOT_term_o = AOT_term_of_va[simplifi]
and AOT_term_of_var_cases =
AOT_term_of_var_cases[simplified, induct pred: AOT_term_of_var]
and AOT_term_of_var_induct =
AOT_term_of_var_induct[simplified, induct pred: AOT_term_of_var]
and AOT_term_of_var_inverse = AOT_term_of_var_inverse[simplified]
AOT_term = AOT_term_of_var_inject[simpli]
‹Equivalence by definition is modelled as necessary equivalence.›
AOT_model_equiv_def :: ‹o==>o==> bool›
(AOT_model_equiv_def)
AOT_model_equiv_def: : ‹
AOT_model_valid_in v ψ)›
by (rule exI[where x=‹λ φ ψ . ∀ v . AOT_model_valid_in v φ =
AOT_model_valid_in v ψ›]) simp
‹Identity by definition is modelled as identity for denoting terms plus
co-denoting.›
AOT_model_id_def :: ‹('b ==> 'a::AOT_Term) ==> ('b ==> 'a) ==> bool›
(AOT_model_id_def)
AOT_model_id_def: ‹(AOT_model_id_def τ σ) = (∀ α . if AOT_model_denotes (σ α)
then τ α = σ α
else ¬AOT_model_denotes (τ α))›
by (rule exI[where x="λ τ σ . ∀ α . if AOT_model_denotes (σ α)
then τ
else ¬AOT_model_denotes (τ α)"])
blast ‹To reduce definitions by identity without free variables to definitions
by identity with free variables acting on the unit type, we give the unit type
a trivial instantiation to @{class AOT_Term}.›
unit :: AOT_Term
‹Modally-strict and modally-fragile axioms are as necessary,
resp. actually valid propositions.›
AOT_model_axiom where ‹AOT_model_axiom ≡ λ φ . ∀ v . AOT_model_valid_in v φ›
AOT_model_act_axiom where ‹AOT_model_act_axiom ≡ λ φ . AOT_model_valid_in w0 φ›
AOT_model_axiomI:
assumes ‹∧v . AOT_model_valid_in v φ› \open <>\
unfolding AOT_model_axiom_def using assms ..
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.71Bemerkung:
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.
Suchen
