Require Import List Arith Nat Lia Relations Bool.
From Undecidability.Shared.Libs.DLW.Utils
Require Import utils_tac utils_list utils_nat finite.
From Undecidability.Shared.Libs.DLW.Vec
Require Import pos vec.
From Undecidability.TRAKHTENBROT
Require Import notations decidable gfp fol_ops fo_sig fo_terms fo_logic fo_definable fo_sat.
Set Implicit Arguments.
Local Notation " e '#>' x " := (vec_pos e x).
Local Notation " e [ v / x ] " := (vec_change e x v).
Section discrete_quotient.
Variables (Σ : fo_signature) (ls : list (syms Σ)) (lr : list (rels Σ)).
Definition fo_bisimilar X M x y :=
forall A φ, incl (fol_syms A) ls
-> incl (fol_rels A) lr
-> @fol_sem Σ X M x·φ A <-> fol_sem M y·φ A.
Variables (X : Type)
(fin : finite_t X)
(M : fo_model Σ X)
(dec : fo_model_dec M).
Implicit Type (R T : X -> X -> Prop).
Local Definition fom_op1 R x y := forall s, In s ls
-> forall (v : vec _ (ar_syms Σ s)) p,
R (fom_syms M s (v[x/p])) (fom_syms M s (v[y/p])).
Local Definition fom_op2 x y := forall s, In s lr
-> forall (v : vec _ (ar_rels Σ s)) p,
fom_rels M s (v[x/p]) <-> fom_rels M s (v[y/p]).
Local Definition fom_op R x y := fom_op1 R x y /\ fom_op2 x y.
Hint Resolve finite_t_pos finite_t_vec : core.
Local Fact fom_op_mono R T : (forall x y, R x y -> T x y) -> (forall x y, fom_op R x y -> fom_op T x y).
Proof. unfold fom_op, fom_op1, fom_op2; intros ? ? ? []; split; intros; auto. Qed.
Local Fact fom_op_id x y : x = y -> fom_op (@eq _) x y.
Proof. unfold fom_op, fom_op1, fom_op2; intros []; split; auto; tauto. Qed.
Local Fact fom_op_sym R x y : fom_op R y x -> fom_op (fun x y => R y x) x y.
Proof. unfold fom_op, fom_op1, fom_op2; intros []; split; intros; auto; symmetry; auto. Qed.
Local Fact fom_op_trans R x z : (exists y, fom_op R x y /\ fom_op R y z)
-> fom_op (fun x z => exists y, R x y /\ R y z) x z.
Proof.
unfold fom_op, fom_op1, fom_op2.
intros (y & H1 & H2); split; intros s Hs v p.
+ exists (fom_syms M s (v[y/p])); split; [ apply H1 | apply H2 ]; auto.
+ transitivity (fom_rels M s (v[y/p])); [ apply H1 | apply H2 ]; auto.
Qed.
Local Fact fom_op_continuous : gfp_continuous fom_op.
Proof.
intros f Hf x y H; split; intros s Hs v p.
+ intros n.
generalize (H n); intros (H1 & H2).
apply H1; auto.
+ apply (H 0); auto.
Qed.
Let fom_op1_dec R : (forall x y, { R x y } + { ~ R x y })
-> (forall x y, { fom_op1 R x y } + { ~ fom_op1 R x y }).
Proof.
unfold fom_op1.
intros HR x y.
apply forall_list_sem_dec; intros.
do 2 (apply (fol_quant_sem_dec fol_fa); auto; intros).
Qed.
Let fom_op2_dec : (forall x y, { fom_op2 x y } + { ~ fom_op2 x y }).
Proof.
unfold fom_op2.
intros x y.
apply forall_list_sem_dec; intros.
do 2 (apply (fol_quant_sem_dec fol_fa); auto; intros).
apply (fol_bin_sem_dec fol_conj);
apply (fol_bin_sem_dec fol_imp); auto.
Qed.
Local Fact fom_op_dec R : (forall x y, { R x y } + { ~ R x y })
-> (forall x y, { fom_op R x y } + { ~ fom_op R x y }).
Proof. intros; apply (fol_bin_sem_dec fol_conj); auto. Qed.
Tactic Notation "solve" "with" "proj" constr(t) :=
apply fot_def_equiv with (f := fun φ => φ t); fol def; intros; rew vec.
Let fol_def_fom_op1 R : fol_definable ls lr M (fun ψ => R (ψ 0) (ψ 1))
-> fol_definable ls lr M (fun ψ => fom_op1 R (ψ 0) (ψ 1)).
Proof.
intros H.
apply fol_def_list_fa; intros s Hs.
apply fol_def_vec_fa.
apply fol_def_finite_fa; auto; intro p.
apply fol_def_subst2; auto.
* apply fot_def_comp; auto; intro q.
destruct (pos_eq_dec p q); subst.
- solve with proj (ar_syms Σ s).
- solve with proj (pos2nat q).
* apply fot_def_comp; auto; intro q.
destruct (pos_eq_dec p q); subst.
- solve with proj (ar_syms Σ s+1).
- solve with proj (pos2nat q).
Qed.
Let fol_def_fom_op2 : fol_definable ls lr M (fun ψ => fom_op2 (ψ 0) (ψ 1)).
Proof.
apply fol_def_list_fa; intros r Hr.
apply fol_def_vec_fa.
apply fol_def_finite_fa; auto; intro p.
apply fol_def_iff.
* apply fol_def_atom; auto; intro q.
destruct (pos_eq_dec p q); subst.
- solve with proj (ar_rels Σ r).
- solve with proj (pos2nat q).
* apply fol_def_atom; auto; intro q.
destruct (pos_eq_dec p q); subst.
- solve with proj (ar_rels Σ r+1).
- solve with proj (pos2nat q).
Qed.
Let fol_def_fom_op R : fol_definable ls lr M (fun ψ => R (ψ 0) (ψ 1))
-> fol_definable ls lr M (fun ψ => fom_op R (ψ 0) (ψ 1)).
Proof. intro; apply fol_def_conj; auto. Qed.
Reserved Notation "x ≡ y" (at level 70, no associativity).
Definition fom_eq := gfp fom_op.
Infix "≡" := fom_eq.
Hint Resolve fom_op_mono fom_op_id fom_op_sym fom_op_trans
fom_op_continuous fom_op_dec : core.
Let fom_eq_equiv : equiv _ fom_eq.
Proof. apply gfp_equiv; eauto. Qed.
Fact fom_eq_fix x y : fom_op fom_eq x y <-> x ≡ y.
Proof. apply gfp_fix; eauto. Qed.
Fact fom_eq_incl R : (forall x y, R x y -> fom_op R x y)
-> (forall x y, R x y -> x ≡ y).
Proof. apply gfp_greatest; eauto. Qed.
Fact fom_eq_refl x : x ≡ x.
Proof. apply (proj1 fom_eq_equiv). Qed.
Fact fom_eq_sym x y : x ≡ y -> y ≡ x.
Proof. apply fom_eq_equiv. Qed.
Fact fom_eq_trans x y z : x ≡ y -> y ≡ z -> x ≡ z.
Proof. apply fom_eq_equiv. Qed.
Fact fom_eq_syms x y s v p : In s ls -> x ≡ y -> fom_syms M s (v[x/p]) ≡ fom_syms M s (v[y/p]).
Proof. intros; apply fom_eq_fix; auto. Qed.
Fact fom_eq_rels x y s v p : In s lr -> x ≡ y -> fom_rels M s (v[x/p]) <-> fom_rels M s (v[y/p]).
Proof. intros; apply fom_eq_fix; auto. Qed.
Hint Resolve fom_eq_refl fom_eq_sym fom_eq_trans fom_eq_syms fom_eq_rels : core.
Theorem fom_eq_syms_full s v w : In s ls -> (forall p, v#>p ≡ w#>p) -> fom_syms M s v ≡ fom_syms M s w.
Proof. intro; apply map_vec_pos_equiv; eauto. Qed.
Theorem fom_eq_rels_full s v w : In s lr -> (forall p, v#>p ≡ w#>p) -> fom_rels M s v <-> fom_rels M s w.
Proof. intro; apply map_vec_pos_equiv; eauto; tauto. Qed.
Section fol_characterization.
Hint Resolve fom_eq_syms_full fom_eq_rels_full : core.
Let f : fo_simulation ls lr M M.
Proof. exists fom_eq; auto; intros a; exists a; auto. Defined.
Let fom_eq_fol_charac1 A phi psi :
(forall n, In n (fol_vars A) -> phi n ≡ psi n)
-> incl (fol_syms A) ls
-> incl (fol_rels A) lr
-> fol_sem M phi A <-> fol_sem M psi A.
Proof. intros; apply fo_model_simulation with (R := f); auto. Qed.
Local Fact fom_eq_fo_bisimilar x y : x ≡ y -> fo_bisimilar M x y.
Proof.
intros H A phi.
apply fom_eq_fol_charac1.
intros [ | n ] _; simpl; auto.
Qed.
Local Fact fo_bisimilar_fom_eq x y : fo_bisimilar M x y -> x ≡ y.
Proof.
revert x y; apply gfp_greatest; eauto.
intros x y H; split.
* intros s Hs v p A phi H1 H2.
destruct (fot_vec_env Σ p) as (w & Hw1 & Hw2).
set (B := fol_subst (fun n =>
match n with
| 0 => in_fot s w
| S n => £ (S n + ar_syms Σ s)
end) A).
assert (HB : forall z, fol_sem M (z·(env_vlift phi v)) B
<-> fol_sem M (fom_syms M s (v[z/p]))·phi A).
{ intros z; unfold B; rewrite fol_sem_subst; apply fol_sem_ext.
intros [ | n] _; rew fot; simpl; f_equal.
* apply vec_pos_ext; intros q; rewrite vec_pos_map; apply Hw1.
* rewrite env_vlift_fix1; auto. }
rewrite <- !HB; apply H.
- red; apply Forall_forall, fol_syms_subst.
intros [ | n ]; rew fot.
+ intros _; apply Forall_forall.
intros s' [ <- | Hs' ]; auto; apply H1; revert Hs'.
rewrite in_flat_map; intros (z & H3 & H4).
apply vec_list_inv in H3; destruct H3 as (q & ->).
rewrite Hw2 in H4; destruct H4.
+ constructor.
+ apply Forall_forall, H1.
- unfold B; rewrite fol_rels_subst; auto.
* intros r Hr v p; red in H.
destruct (fot_vec_env Σ p) as (w & Hw1 & Hw2).
set (B := fol_atom r w).
assert (HB : forall z, fol_sem M (z·(env_vlift (fun _ => x) v)) B
<-> fom_rels M r (v[z/p])).
{ intros z; unfold B; simpl; apply fol_equiv_ext; f_equal.
apply vec_pos_ext; intros q; rewrite vec_pos_map; apply Hw1. }
rewrite <- !HB; apply H.
- unfold B; simpl; intros z; rewrite in_flat_map.
intros (t & H3 & H4).
apply vec_list_inv in H3.
destruct H3 as (q & ->).
rewrite Hw2 in H4; destruct H4.
- unfold B; simpl; intros ? [ <- | [] ]; auto.
Qed.
Theorem fom_eq_fol_characterization x y :
x ≡ y <-> fo_bisimilar M x y.
Proof.
split.
+ apply fom_eq_fo_bisimilar.
+ apply fo_bisimilar_fom_eq.
Qed.
End fol_characterization.
Fact fom_eq_dec : forall x y, { x ≡ y } + { ~ x ≡ y }.
Proof. apply gfp_decidable; eauto. Qed.
Definition fo_congruence_upto R :=
( (equivalence _ R)
* (forall s v w, In s ls -> (forall p, R (v#>p) (w#>p)) -> R (fom_syms M s v) (fom_syms M s w))
* (forall r v w, In r lr -> (forall p, R (v#>p) (w#>p)) -> fom_rels M r v <-> fom_rels M r w) )%type.
Theorem fo_bisimilar_dec_congr : fo_congruence_upto (@fo_bisimilar X M)
* (forall x y, decidable (fo_bisimilar M x y)).
Proof.
split; [ split; [ split | ] | ].
+ split; red; [ intros ? | intros ? ? ? | intros ? ?]; rewrite <- !fom_eq_fol_characterization; auto.
apply fom_eq_trans.
+ intros ? ? ? ? ?; apply fom_eq_fol_characterization, fom_eq_syms_full; auto.
intro; apply fom_eq_fol_characterization; auto.
+ intros ? ? ? ? ?; apply fom_eq_rels_full; auto.
intro; apply fom_eq_fol_characterization; auto.
+ intros x y.
destruct (fom_eq_dec x y); [ left | right ]; rewrite <- fom_eq_fol_characterization; auto.
Qed.
Theorem fom_eq_finite : { n | forall x y, x ≡ y <-> iter fom_op (fun _ _ => True) n x y }.
Proof. apply gfp_finite_t; eauto. Qed.
Theorem fom_eq_fol_def : fol_definable ls lr M (fun φ => φ 0 ≡ φ 1).
Proof.
destruct fom_eq_finite as (n & Hn).
apply fol_def_equiv with (R := fun φ => iter fom_op (fun _ _ : X => True) n (φ 0) (φ 1)).
+ intro; rewrite <- Hn; tauto.
+ clear Hn; induction n as [ | n IHn ].
* simpl; fol def.
* rewrite iter_S; auto.
Qed.
Section fom_eq_form.
Let A := proj1_sig fom_eq_fol_def.
Definition fom_eq_form := fol_subst (fun n => match n with 0 => £1 | _ => £0 end) A.
Fact fom_eq_form_sem φ x y : fol_sem M y·x·φ fom_eq_form <-> x ≡ y.
Proof.
unfold fom_eq_form; rewrite fol_sem_subst.
apply (proj2_sig fom_eq_fol_def).
Qed.
Fact fom_eq_form_vars : incl (fol_vars fom_eq_form) (0::1::nil).
Proof.
unfold fom_eq_form; rewrite fol_vars_subst.
intros n; rewrite in_flat_map; intros (? & _ & H).
revert x H; intros [ | [] ]; simpl; tauto.
Qed.
Fact fom_eq_form_syms : incl (fol_syms fom_eq_form) ls.
Proof.
unfold fom_eq_form; red.
apply Forall_forall, fol_syms_subst.
+ intros [ | []]; rew fot; auto.
+ apply Forall_forall, (proj2_sig fom_eq_fol_def).
Qed.
Fact fom_eq_form_rels : incl (fol_rels fom_eq_form) lr.
Proof.
unfold fom_eq_form; rewrite fol_rels_subst.
apply (proj2_sig fom_eq_fol_def).
Qed.
End fom_eq_form.
Hint Resolve fom_eq_form_vars fom_eq_form_syms fom_eq_form_rels fom_eq_dec : core.
Section build_the_model.
Let l := proj1_sig fin.
Let Hl : forall x, In x l := proj2_sig fin.
Let Q : fin_quotient fom_eq.
Proof. apply decidable_EQUIV_fin_quotient with (l := l); eauto. Qed.
Let n := fq_size Q.
Let cls := fq_class Q.
Let repr := fq_repr Q.
Let E1 p : cls (repr p) = p. Proof. apply fq_surj. Qed.
Let E2 x y : x ≡ y <-> cls x = cls y. Proof. apply fq_equiv. Qed.
Let Md : fo_model Σ (pos n).
Proof.
exists.
+ intros s v; apply cls, (fom_syms M s), (vec_map repr v).
+ intros s v; apply (fom_rels M s), (vec_map repr v).
Defined.
Let H1 s v : In s ls -> cls (fom_syms M s v) = fom_syms Md s (vec_map cls v).
Proof.
intros Hs; simpl.
apply E2.
apply fom_eq_syms_full; auto.
intros p; rewrite vec_map_map, vec_pos_map.
apply E2; rewrite E1; auto.
Qed.
Let H2 s v : In s lr -> fom_rels M s v <-> fom_rels Md s (vec_map cls v).
Proof.
intros Hs; simpl.
apply fom_eq_rels_full; auto.
intros p; rewrite vec_map_map, vec_pos_map.
apply E2; rewrite E1; auto.
Qed.
Let f : fo_projection ls lr M Md.
Proof. exists cls repr; auto. Defined.
Let H3 A phi : incl (fol_syms A) ls
-> incl (fol_rels A) lr
-> fol_sem M phi A
<-> fol_sem Md (fun n => cls (phi n)) A.
Proof. intros; apply fo_model_projection with (p := f); auto. Qed.
Let H4 p q : fo_bisimilar Md p q <-> p = q.
Proof.
split.
+ intros H.
rewrite <- (E1 q), <- (E1 p).
apply E2, fom_eq_fol_characterization.
intros A phi Hs Hr.
specialize (H A (fun p => cls (phi p)) Hs Hr).
revert H; apply fol_equiv_impl.
all: rewrite H3; auto; apply fol_sem_ext; intros []; now simpl.
+ intros []; red; tauto.
Qed.
Theorem fo_fin_model_discretize :
{ n : nat &
{ Md : fo_model Σ (pos n) &
{ _ : fo_model_dec Md &
{ _ : fo_projection ls lr M Md &
(forall p q, fo_bisimilar Md p q <-> p = q) } } } }.
Proof.
exists n, Md.
exists; eauto.
intros x y; simpl; apply dec.
Qed.
End build_the_model.
End discrete_quotient.
Section counter_model_to_class_FO_definability.
Let Σ := Σrel 2.
Let M : fo_model Σ bool.
Proof.
exists.
+ intros [].
+ intros []; simpl.
exact (rel2_on_vec eq).
Defined.
Let M_dec : fo_model_dec M.
Proof. intros [] ?; apply bool_dec. Qed.
Let f : @fo_projection Σ nil (tt::nil) _ M _ M.
Proof.
exists negb negb.
+ intros []; auto.
+ intros [].
+ intros [] v _; simpl.
vec split v with x; vec split v with y; vec nil v; simpl.
revert x y; now intros [] [].
Defined.
Notation "⟪ A ⟫" := (fun φ => fol_sem M φ A).
Let homeomorphism (A : fol_form Σ) phi :
⟪A⟫ phi <-> ⟪A⟫ (fun x=> negb (phi x)).
Proof.
apply fo_model_projection with (p := f); auto.
all: intros []; simpl; auto.
Qed.
Infix "≡" := (fom_eq (Σ := Σ) nil (tt::nil) M) (at level 70, no associativity).
Hint Resolve finite_t_bool : core.
Let true_is_not_false : ~ true ≡ false.
Proof.
intros H.
apply fom_eq_fol_characterization in H; auto.
specialize (H (@fol_atom Σ tt (£0##£1##ø)) (fun n => match n with 0 => true | _ => false end)).
revert H; unfold M; simpl; rew fot; simpl.
intros [H _]; cbv; auto.
specialize (H eq_refl); discriminate.
Qed.
Let no_disctinct A phi : (forall n, In n (fol_vars A) -> n = 0)
-> ⟪A⟫ true·phi <-> ⟪A⟫ false·phi.
Proof.
intros H.
set (psi n := negb (phi n)).
rewrite homeomorphism with (phi := false·phi) at 1.
apply fol_sem_ext.
intros n Hn; apply H in Hn; subst; auto.
Qed.
Theorem FO_does_not_characterize_classes :
exists (M : fo_model Σ bool) (_ : fo_model_dec M) (x y : bool),
~ fom_eq (Σ := Σ) nil (tt::nil) M x y
/\ forall A φ, (forall n, In n (fol_vars A) -> n = 0)
-> fol_sem M x·φ A <-> fol_sem M y·φ A.
Proof. exists M, M_dec, true, false; auto. Qed.
End counter_model_to_class_FO_definability.