Require Import Undecidability.FOL.Util.Syntax.
Require Import Undecidability.FOL.Util.Syntax_facts.
Require Import Undecidability.FOL.Util.FullTarski_facts.
Require Import Undecidability.FOL.ZF.
Require Import Lia.
From Undecidability.PCP Require Import PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb.
From Undecidability Require Import Shared.ListAutomation.
Import ListAutomationNotations.
Local Set Implicit Arguments.
Local Unset Strict Implicit.
Local Definition BSRS := list(card bool).
Fixpoint shift n x :=
match n with
| O => x
| S n => subst_term ↑ (shift n x)
end.
Definition sing a :=
{a; a}.
Definition opair a b :=
{{a; a}; {a; b}}.
Definition pairing f A :=
∀ $0 ∈ shift 1 f ~> ∃ ∃ $1 ∈ shift 3 A ∧ $2 ≡ opair $1 $0.
Definition function' f A :=
pairing f A ∧ ∀ ∃ $0 ∈ shift 2 A ∧ opair $0 $1 ∈ shift 2 f
∧ ∀ opair $1 $0 ∈ shift 2f ~> $2 ≡ $0.
Definition function f :=
∀ ∀ ∀ opair $2 $1 ∈ shift 3 f ~> opair $2 $0 ∈ shift 3 f ~> $1 ≡ $0.
Definition enc_bool (x : bool) :=
if x then {∅; ∅} else ∅.
Fixpoint prep_string (s : string bool) a :=
match s with
| nil => a
| b::s => opair (enc_bool b) (prep_string s a)
end.
Definition enc_string (s : string bool) :=
prep_string s ∅.
Fixpoint enc_stack (B : BSRS) :=
match B with
| nil => ∅
| (s,t)::B => enc_stack B ∪ sing (opair (enc_string s) (enc_string t))
end.
Definition is_rep phi a b :=
∀ $0 ∈ shift 1 b <~> ∃ $0 ∈ shift 2 a ∧ phi.
Definition comb_rel s t :=
∃ ∃ $2 ≡ opair $0 $1 ∧ $3 ≡ opair (prep_string s $0) (prep_string t $1).
Fixpoint combinations (B : BSRS) a b :=
match B with
| nil => b ≡ ∅
| (s,t)::B => ∃ ∃ shift 2 b ≡ $0 ∪ $1 ∧ combinations B (shift 2 a) $1
∧ is_rep (comb_rel s t) (shift 2 a) $0
end.
Definition solutions (B : BSRS) f n :=
opair ∅ (enc_stack B) ∈ f ∧ ∀ ∀ ∀ $2 ∈ shift 3 n ~> opair $2 $1 ∈ shift 3 f
~> combinations B $1 $0 ~> opair (σ $2) $0 ∈ shift 3 f.
Definition solvable (B : BSRS) :=
∃ ∃ ∃ ∃ $3 ∈ ω ∧ function $2 ∧ solutions B $2 $3 ∧ opair $3 $0 ∈ $2 ∧ opair $1 $1 ∈ $0.
Lemma bool_bound b n :
bounded_t n (enc_bool b).
Proof.
destruct b; repeat solve_bounds.
Qed.
Lemma prep_string_bound s t n :
bounded_t n t -> bounded_t n (prep_string s t).
Proof.
intros Ht. induction s; cbn; trivial.
repeat solve_bounds; trivial.
all: apply bool_bound.
Qed.
Lemma string_bound s n :
bounded_t n (enc_string s).
Proof.
apply prep_string_bound. solve_bounds.
Qed.
Lemma stack_bound B n :
bounded_t n (enc_stack B).
Proof.
induction B as [|[s t] B IH]; cbn; repeat solve_bounds; trivial.
all: apply string_bound.
Qed.
Lemma combinations_bound (B : BSRS) n k l :
n > 2 + l -> n > 4 + k -> bounded n (combinations B $k $l).
Proof.
revert n k l. induction B as [|[s t] B IH]; cbn; repeat solve_bounds; trivial.
apply IH; lia. all: apply prep_string_bound; solve_bounds.
Qed.
Lemma solvabe_bound B :
bounded 0 (solvable B).
Proof.
repeat solve_bounds.
- apply stack_bound.
- apply combinations_bound; lia.
Qed.
Declare Scope sem.
Open Scope sem.
Arguments Vector.nil {_}, _.
Arguments Vector.cons {_} _ {_} _, _ _ _ _.
Notation "x ∈ y" := (@i_atom _ _ _ _ elem (Vector.cons x (Vector.cons y Vector.nil))) (at level 35) : sem.
Notation "x ≡ y" := (@i_atom _ _ _ _ equal (Vector.cons x (Vector.cons y Vector.nil))) (at level 35) : sem.
Notation "x ⊆ y" := (forall z, z ∈ x -> z ∈ y) (at level 34) : sem.
Notation "∅" := (@i_func ZF_func_sig ZF_pred_sig _ _ eset Vector.nil) : sem.
Notation "'ω'" := (@i_func ZF_func_sig _ _ _ om Vector.nil) : sem.
Notation "{ x ; y }" := (@i_func ZF_func_sig _ _ _ pair (Vector.cons x (Vector.cons y Vector.nil))) (at level 31) : sem.
Notation "⋃ x" := (@i_func ZF_func_sig _ _ _ union (Vector.cons x Vector.nil)) (at level 32) : sem.
Notation "'PP' x" := (@i_func ZF_func_sig _ _ _ power (Vector.cons x Vector.nil)) (at level 31) : sem.
Notation "x ∪ y" := (⋃ {x; y}) (at level 32) : sem.
Notation "'σ' x" := (x ∪ {x; x}) (at level 32) : sem.
Section ZF.
Context { V : Type }.
Context { M : interp V }.
Hypothesis M_ZF : forall rho, rho ⊫ ZF'.
Hypothesis VIEQ : extensional M.
Lemma M_ext x y :
x ⊆ y -> y ⊆ x -> x = y.
Proof.
rewrite <- VIEQ. apply (@M_ZF (fun _ => ∅) ax_ext). cbn; tauto.
Qed.
Lemma M_eset x :
~ x ∈ ∅.
Proof.
refine (@M_ZF (fun _ => ∅) ax_eset _ x). cbn; tauto.
Qed.
Lemma M_pair x y z :
x ∈ {y; z} <-> x = y \/ x = z.
Proof.
rewrite <- !VIEQ. apply (@M_ZF (fun _ => ∅) ax_pair). cbn; tauto.
Qed.
Lemma M_union x y :
x ∈ ⋃ y <-> exists z, z ∈ y /\ x ∈ z.
Proof.
apply (@M_ZF (fun _ => ∅) ax_union). cbn; tauto.
Qed.
Lemma M_power x y :
x ∈ PP y <-> x ⊆ y.
Proof.
apply (@M_ZF (fun _ => ∅) ax_power). cbn; tauto.
Qed.
Definition M_inductive x :=
∅ ∈ x /\ forall y, y ∈ x -> σ y ∈ x.
Lemma M_om1 :
M_inductive ω.
Proof.
apply (@M_ZF (fun _ => ∅) ax_om1). cbn; tauto.
Qed.
Lemma M_om2 x :
M_inductive x -> ω ⊆ x.
Proof.
apply (@M_ZF (fun _ => ∅) ax_om2). cbn; tauto.
Qed.
Definition agrees_fun phi (P : V -> Prop) :=
forall x rho, P x <-> (x.:rho) ⊨ phi.
Definition def_pred (P : V -> Prop) :=
exists phi rho, forall d, P d <-> (d.:rho) ⊨ phi.
Lemma M_sep P x :
(forall phi rho, rho ⊨ ax_sep phi) -> def_pred P -> exists y, forall z, z ∈ y <-> z ∈ x /\ P z.
Proof.
cbn. intros H [phi [rho Hp]].
destruct (H phi rho x) as [y H']; clear H.
exists y. intros z. specialize (H' z). setoid_rewrite sat_comp in H'.
rewrite (sat_ext _ _ (xi:=z.:rho)) in H'; try now intros [].
firstorder.
Qed.
Definition M_is_rep R x y :=
forall v, v ∈ y <-> exists u, u ∈ x /\ R u v.
Lemma is_rep_unique R x y y' :
M_is_rep R x y -> M_is_rep R x y' -> y = y'.
Proof.
intros H1 H2. apply M_ext; intros v.
- intros H % H1. now apply H2.
- intros H % H2. now apply H1.
Qed.
Definition functional (R : V -> V -> Prop) :=
forall x y y', R x y -> R x y' -> y = y'.
Definition def_rel (R : V -> V -> Prop) :=
exists phi rho, forall x y, R x y <-> (x.:y.:rho) ⊨ phi.
Lemma M_rep R x :
(forall phi rho, rho ⊨ ax_rep phi) -> def_rel R -> functional R -> exists y, M_is_rep R x y.
Proof.
intros H1 [phi [rho Hp]]. intros H2.
cbn in H1. specialize (H1 phi rho). destruct H1 with x as [y Hy].
- intros a b b'. setoid_rewrite sat_comp.
erewrite sat_ext. rewrite <- (Hp a b). 2: now intros [|[]].
erewrite sat_ext. rewrite <- (Hp a b'). 2: now intros [|[]].
rewrite VIEQ. apply H2.
- exists y. intros v. split.
+ intros [u[U1 U2]] % Hy. exists u. split; trivial.
setoid_rewrite sat_comp in U2. rewrite sat_ext in U2. rewrite (Hp u v). apply U2. now intros [|[]]; cbn.
+ intros [u[U1 U2]]. apply Hy. exists u. split; trivial.
setoid_rewrite sat_comp. rewrite sat_ext. rewrite <- (Hp u v). apply U2. now intros [|[]]; cbn.
Qed.
Definition M_sing x :=
{x; x}.
Definition M_opair x y := ({{x; x}; {x; y}}).
Lemma binunion_el x y z :
x ∈ y ∪ z <-> x ∈ y \/ x ∈ z.
Proof.
split.
- intros [u [H1 H2]] % M_union.
apply M_pair in H1 as [->| ->]; auto.
- intros [H|H].
+ apply M_union. exists y. rewrite M_pair. auto.
+ apply M_union. exists z. rewrite M_pair. auto.
Qed.
Lemma sing_el x y :
x ∈ M_sing y <-> x = y.
Proof.
split.
- now intros [H|H] % M_pair.
- intros ->. apply M_pair. now left.
Qed.
Lemma M_pair1 x y :
x ∈ {x; y}.
Proof.
apply M_pair. now left.
Qed.
Lemma M_pair2 x y :
y ∈ {x; y}.
Proof.
apply M_pair. now right.
Qed.
Lemma sing_pair x y z :
{x; x} = {y; z} -> x = y /\ x = z.
Proof.
intros He. split.
- assert (H : y ∈ {y; z}) by apply M_pair1.
rewrite <- He in H. apply M_pair in H. intuition.
- assert (H : z ∈ {y; z}) by apply M_pair2.
rewrite <- He in H. apply M_pair in H. intuition.
Qed.
Lemma opair_inj1 x x' y y' :
M_opair x y = M_opair x' y' -> x = x'.
Proof.
intros He. assert (H : {x; x} ∈ M_opair x y) by apply M_pair1.
rewrite He in H. apply M_pair in H as [H|H]; apply (sing_pair H).
Qed.
Lemma opair_inj2 x x' y y' :
M_opair x y = M_opair x' y' -> y = y'.
Proof.
intros He. assert (y = x' \/ y = y') as [->| ->]; trivial.
- assert (H : {x; y} ∈ M_opair x y) by apply M_pair2.
rewrite He in H. apply M_pair in H as [H|H].
+ symmetry in H. apply sing_pair in H. intuition.
+ assert (H' : y ∈ {x; y}) by apply M_pair2.
rewrite H in H'. now apply M_pair in H'.
- assert (x = x') as -> by now apply opair_inj1 in He.
assert (H : {x'; y'} ∈ M_opair x' y') by apply M_pair2.
rewrite <- He in H. apply M_pair in H as [H|H]; apply (sing_pair (eq_sym H)).
Qed.
Lemma opair_inj x x' y y' :
M_opair x y = M_opair x' y' -> x = x' /\ y = y'.
Proof.
intros H. split.
- eapply opair_inj1; eassumption.
- eapply opair_inj2; eassumption.
Qed.
Lemma sigma_el x y :
x ∈ σ y <-> x ∈ y \/ x = y.
Proof.
split.
- intros [H|H] % binunion_el; auto.
apply sing_el in H. now right.
- intros [H| ->]; apply binunion_el; auto.
right. now apply sing_el.
Qed.
Lemma sigma_eq x :
x ∈ σ x.
Proof.
apply sigma_el. now right.
Qed.
Lemma sigma_sub x :
x ⊆ σ x.
Proof.
intros y H. apply sigma_el. now left.
Qed.
Lemma binunion_eset x :
x = ∅ ∪ x.
Proof.
apply M_ext.
- intros y H. apply binunion_el. now right.
- intros y [H|H] % binunion_el.
+ now apply M_eset in H.
+ assumption.
Qed.
Lemma pair_com x y :
{x; y} = {y; x}.
Proof.
apply M_ext; intros z [->| ->] % M_pair; apply M_pair; auto.
Qed.
Lemma binunion_com x y :
x ∪ y = y ∪ x.
Proof.
now rewrite pair_com.
Qed.
Lemma binunionl a x y :
a ∈ x -> a ∈ x ∪ y.
Proof.
intros H. apply binunion_el. now left.
Qed.
Lemma binunionr a x y :
a ∈ y -> a ∈ x ∪ y.
Proof.
intros H. apply binunion_el. now right.
Qed.
Hint Resolve binunionl binunionr : core.
Lemma binunion_assoc x y z :
(x ∪ y) ∪ z = x ∪ (y ∪ z).
Proof.
apply M_ext; intros a [H|H] % binunion_el; eauto.
- apply binunion_el in H as [H|H]; eauto.
- apply binunion_el in H as [H|H]; eauto.
Qed.
Fixpoint numeral n :=
match n with
| O => ∅
| S n => σ (numeral n)
end.
Lemma numeral_omega n :
numeral n ∈ ω.
Proof.
induction n; cbn; now apply M_om1.
Qed.
Definition trans x :=
forall y, y ∈ x -> y ⊆ x.
Lemma numeral_trans n :
trans (numeral n).
Proof.
induction n; cbn.
- intros x H. now apply M_eset in H.
- intros x [H| ->] % sigma_el; try apply sigma_sub.
apply IHn in H. intuition eauto using sigma_sub.
Qed.
Lemma numeral_wf n :
~ numeral n ∈ numeral n.
Proof.
induction n.
- apply M_eset.
- intros [H|H] % sigma_el; fold numeral in *.
+ apply IHn. eapply numeral_trans; eauto. apply sigma_eq.
+ assert (numeral n ∈ numeral (S n)) by apply sigma_eq.
now rewrite H in H0.
Qed.
Lemma numeral_lt k l :
k < l -> numeral k ∈ numeral l.
Proof.
induction 1; cbn; apply sigma_el; auto.
Qed.
Lemma numeral_inj k l :
numeral k = numeral l -> k = l.
Proof.
intros Hk. assert (k = l \/ k < l \/ l < k) as [H|[H|H]] by lia; trivial.
all: apply numeral_lt in H; rewrite Hk in H; now apply numeral_wf in H.
Qed.
Definition M_enc_bool (x : bool) :=
if x then {∅; ∅} else ∅.
Fixpoint M_prep_string (s : string bool) x :=
match s with
| nil => x
| b::s => M_opair (M_enc_bool b) (M_prep_string s x)
end.
Definition M_enc_string (s : string bool) :=
M_prep_string s ∅.
Definition M_enc_card s t :=
M_opair (M_enc_string s) (M_enc_string t).
Fixpoint M_enc_stack (B : BSRS) :=
match B with
| nil => ∅
| (s,t)::B => M_enc_stack B ∪ M_sing (M_enc_card s t)
end.
Lemma enc_bool_inj b c :
M_enc_bool b = M_enc_bool c -> b = c.
Proof.
destruct b, c; trivial; cbn.
- intros H. contradiction (@M_eset ∅).
pattern ∅ at 2. rewrite <- H. apply M_pair; auto.
- intros H. contradiction (@M_eset ∅).
pattern ∅ at 2. rewrite H. apply M_pair; auto.
Qed.
Lemma enc_string_inj s t :
M_enc_string s = M_enc_string t -> s = t.
Proof.
induction s in t|-*; destruct t as [|b t]; cbn; trivial.
- intros H. contradiction (M_eset (x:=M_sing (M_enc_bool b))).
rewrite H. apply M_pair. now left.
- intros H. contradiction (M_eset (x:=M_sing (M_enc_bool a))).
rewrite <- H. apply M_pair. now left.
- intros [H1 H2] % opair_inj. apply IHs in H2 as ->.
apply enc_bool_inj in H1 as ->. reflexivity.
Qed.
Lemma eval_opair rho x y :
eval rho (opair x y) = M_opair (eval rho x) (eval rho y).
Proof.
reflexivity.
Qed.
Lemma eval_enc_bool rho b :
eval rho (enc_bool b) = M_enc_bool b.
Proof.
destruct b; reflexivity.
Qed.
Lemma eval_prep_string rho s x :
eval rho (prep_string s x) = M_prep_string s (eval rho x).
Proof.
induction s; trivial. cbn [prep_string].
now rewrite eval_opair, IHs, eval_enc_bool.
Qed.
Lemma eval_enc_string rho s :
eval rho (enc_string s) = M_enc_string s.
Proof.
unfold enc_string. now rewrite eval_prep_string.
Qed.
Lemma eval_enc_stack rho B :
eval rho (enc_stack B) = M_enc_stack B.
Proof.
induction B; cbn; trivial. destruct a. unfold M_enc_card.
now rewrite <- IHB, <- !eval_enc_string with (rho:=rho), <- eval_opair.
Qed.
Lemma M_enc_stack_app A B :
M_enc_stack (A ++ B) = M_enc_stack A ∪ M_enc_stack B.
Proof.
induction A as [|[s t] A IH]; cbn.
- apply binunion_eset.
- rewrite IH. rewrite !binunion_assoc.
now rewrite (binunion_com (M_enc_stack B) (M_sing (M_enc_card s t))).
Qed.
Lemma enc_stack_el' x A :
x ∈ M_enc_stack A -> exists s t, (s, t) el A /\ x = M_enc_card s t.
Proof.
induction A as [|[s t] A IH]; cbn.
- now intros H % M_eset.
- intros [H|H] % binunion_el.
+ destruct (IH H) as (u&v&H'&->). exists u, v. intuition.
+ apply sing_el in H as ->. exists s, t. intuition.
Qed.
Lemma enc_stack_el B s t :
(s, t) el B -> M_enc_card s t ∈ M_enc_stack B.
Proof.
induction B as [|[u b] B IH]; cbn; auto.
intros [H|H]; apply binunion_el.
- right. apply sing_el. congruence.
- left. apply IH, H.
Qed.
Lemma M_prep_enc s s' :
M_prep_string s (M_enc_string s') = M_enc_string (s ++ s').
Proof.
induction s; cbn; trivial.
now rewrite IHs.
Qed.
Definition append_all A (c : card bool) :=
map (fun p => (fst c ++ fst p, snd c ++ snd p)) A.
Definition derivation_step B C :=
flat_map (append_all C) B.
Fixpoint derivations B n :=
match n with
| O => B
| S n => derivation_step B (derivations B n)
end.
Lemma derivable_derivations B s t :
derivable B s t -> exists n, (s, t) el derivations B n.
Proof.
induction 1.
- exists 0. apply H.
- destruct IHderivable as [n Hn]. exists (S n).
apply in_flat_map. exists (x, y). split; trivial.
apply in_map_iff. exists (u,v). cbn. split; trivial.
Qed.
Fixpoint M_enc_derivations B n :=
match n with
| O => M_sing (M_opair ∅ (M_enc_stack B))
| S n => M_enc_derivations B n ∪
M_sing (M_opair (numeral (S n)) (M_enc_stack (derivations B (S n))))
end.
Lemma enc_derivations_base B n :
M_opair ∅ (M_enc_stack B) ∈ M_enc_derivations B n.
Proof.
induction n; cbn.
- now apply sing_el.
- apply binunion_el. now left.
Qed.
Lemma enc_derivations_bound B n k x :
M_opair k x ∈ M_enc_derivations B n -> k ∈ σ (numeral n).
Proof.
induction n; cbn.
- intros H % sing_el. apply opair_inj in H as [-> _].
apply sigma_el. now right.
- intros [H|H] % binunion_el.
+ apply sigma_el. left. apply IHn, H.
+ apply sing_el in H. apply opair_inj in H as [-> _]. apply sigma_eq.
Qed.
Lemma enc_derivations_fun B n :
forall k x y, M_opair k x ∈ M_enc_derivations B n -> M_opair k y ∈ M_enc_derivations B n -> x = y.
Proof.
induction n; cbn -[derivations]; intros k x y.
- intros H1 % sing_el H2 % sing_el.
rewrite <- H2 in H1. now apply opair_inj in H1.
- intros [H1|H1 % sing_el] % binunion_el [H2|H2 % sing_el] % binunion_el.
+ now apply (IHn k x y).
+ exfalso. apply enc_derivations_bound in H1.
destruct (opair_inj H2) as [-> _]. now apply (@numeral_wf (S n)).
+ exfalso. apply enc_derivations_bound in H2.
destruct (opair_inj H1) as [-> _]. now apply (@numeral_wf (S n)).
+ rewrite <- H2 in H1. now apply opair_inj in H1.
Qed.
Lemma enc_derivations_el B n k x :
M_opair k x ∈ M_enc_derivations B n -> exists l, k = numeral l /\ x = M_enc_stack (derivations B l).
Proof.
induction n; cbn.
- intros H % sing_el. exists 0. apply (opair_inj H).
- intros [H|H] % binunion_el.
+ apply IHn, H.
+ apply sing_el in H. exists (S n). apply (opair_inj H).
Qed.
Lemma enc_derivations_step B n l :
numeral l ∈ numeral n
-> M_opair (σ (numeral l)) (M_enc_stack (derivations B (S l))) ∈ M_enc_derivations B n.
Proof.
induction n; cbn -[derivations].
- now intros H % M_eset.
- intros [H|H % sing_el] % binunion_el; apply binunion_el.
+ left. apply IHn, H.
+ right. apply numeral_inj in H as ->. now apply sing_el.
Qed.
Lemma enc_stack_combinations B rho C x X Y :
rho ⊨ combinations B X Y -> eval rho X = M_enc_stack C -> eval rho Y = x -> x = M_enc_stack (derivation_step B C).
Proof.
induction B as [|[s t] B IH] in rho,C,x,X,Y |-*.
- cbn. rewrite VIEQ. now intros -> _ <-.
- intros [x1[x2[[H1 H2]H3]]] R1 R2; fold sat in *.
assert (x = x2 ∪ x1) as ->. { rewrite <- R2. cbn in H1. rewrite !eval_comp in H1. apply VIEQ, H1. } clear H1.
cbn. fold (derivation_step B C). rewrite M_enc_stack_app.
enough (x1 = M_enc_stack (derivation_step B C)) as E1.
+ enough (x2 = M_enc_stack (append_all C (s, t))) as E2 by now rewrite E1, E2.
apply M_ext; intros u Hu.
* apply H3 in Hu as [v [Hv[a [b Ha]]]].
cbn in Hv. erewrite !eval_comp, eval_ext, R1 in Hv; trivial.
apply enc_stack_el' in Hv as (s'&t'&H&H').
enough (u = M_enc_card (s++s') (t++t')) as ->.
{ apply enc_stack_el. apply in_map_iff. now exists (s', t'). }
cbn in Ha. rewrite !VIEQ in Ha. destruct Ha as [D1 D2].
rewrite D1 in H'. unfold M_enc_card in H'. apply opair_inj in H' as [-> ->].
rewrite D2; unfold M_enc_card, M_opair; repeat f_equal.
all: rewrite eval_prep_string; cbn. all: apply M_prep_enc.
* apply enc_stack_el' in Hu as (s'&t'&H&->).
unfold append_all in H. eapply in_map_iff in H as [[a b][H H']].
cbn in H. apply H3. exists (M_enc_card a b). split.
{ cbn. erewrite !eval_comp, eval_ext, R1; trivial. now apply enc_stack_el. }
exists (M_enc_string b), (M_enc_string a). split.
-- cbn. apply VIEQ. reflexivity.
-- cbn. apply VIEQ. rewrite !eval_prep_string. cbn.
rewrite !M_prep_enc. injection H. intros -> ->. reflexivity.
+ eapply IH; eauto. unfold shift. now erewrite !eval_comp, eval_ext, R1.
Qed.
Lemma enc_derivations_solutions B n rho a b :
(a .: b .: M_enc_derivations B n .: numeral n .: rho) ⊨ solutions B $2 $3.
Proof.
cbn. split.
- rewrite eval_enc_stack. apply enc_derivations_base.
- intros k x x' H1 H2 H3.
destruct (enc_derivations_el H2) as [l[-> ->]].
specialize (enc_derivations_step B H1).
replace (M_enc_stack (derivations B (S l))) with x'; trivial.
apply (enc_stack_combinations H3); trivial.
Qed.
Lemma derivations_enc_derivations B n :
M_opair (numeral n) (M_enc_stack (derivations B n)) ∈ M_enc_derivations B n.
Proof.
induction n; cbn -[derivations].
- now apply sing_el.
- apply binunion_el. right.
now apply sing_el.
Qed.
Lemma derivations_el B n s t :
(s, t) el derivations B n -> M_enc_card s t ∈ M_enc_stack (derivations B n).
Proof.
apply enc_stack_el.
Qed.
Theorem PCP_ZF1 B s :
derivable B s s -> forall rho, rho ⊨ solvable B.
Proof.
intros H rho. destruct (derivable_derivations H) as [n Hn]. unfold solvable.
exists (numeral n), (M_enc_derivations B n), (M_enc_string s), (M_enc_stack (derivations B n)).
split; [split; [split; [split |] |] |].
- apply numeral_omega.
- unfold function'. intros k x y H1 H2. apply VIEQ. apply (enc_derivations_fun H1 H2).
- apply enc_derivations_solutions.
- cbn. apply derivations_enc_derivations.
- now apply enc_stack_el.
Qed.
Definition M_comb_rel s t :=
fun u v => exists u1 u2, u = M_opair u1 u2 /\ v = M_opair (M_prep_string s u1) (M_prep_string t u2).
Fixpoint M_combinations B x y :=
match B with
| nil => y = ∅
| (s,t)::B => exists y1 y2, y = y2 ∪ y1 /\ M_combinations B x y1 /\ M_is_rep (M_comb_rel s t) x y2
end.
Lemma M_combinations_spec B rho x y a b :
M_combinations B x y -> eval rho a = x -> eval rho b = y -> rho ⊨ combinations B a b.
Proof.
induction B in y,a,b,rho|-*; cbn.
- rewrite VIEQ. now intros -> _ ->.
- destruct a0 as [s t]. intros (y1&y2&H1&H2&H3) Ha Hb. exists y1, y2. repeat split.
+ cbn. apply VIEQ. erewrite !eval_comp. unfold funcomp. cbn.
change (eval (fun x => rho x) b) with (eval rho b). now rewrite Hb.
+ eapply (IHB _ y1); trivial. erewrite !eval_comp. unfold funcomp. cbn.
change (eval (fun x => rho x) a) with (eval rho a). now rewrite Ha.
+ intros (u & Hu & c & d' & H) % H3. exists u. split.
* cbn. erewrite !eval_comp. erewrite eval_ext, Ha; trivial.
* exists d', c. cbn. rewrite !VIEQ, !eval_prep_string. apply H.
+ intros (u & Hu & c & d' & H). apply H3. exists u. split.
* cbn in Hu. erewrite !eval_comp in Hu. rewrite <- Ha. apply Hu.
* exists d', c. cbn in H. rewrite !VIEQ, !eval_prep_string in H. apply H.
Qed.
Definition M_solutions B f n :=
M_opair ∅ (M_enc_stack B) ∈ f /\ forall k x y, k ∈ n -> M_opair k x ∈ f -> M_combinations B x y -> M_opair (σ k) y ∈ f.
Lemma comb_rel_rep C s t :
M_is_rep (M_comb_rel s t) (M_enc_stack C) (M_enc_stack (append_all C (s, t))).
Proof.
intros y. split.
- intros (u&v&H&->) % enc_stack_el'.
unfold append_all in H. apply in_map_iff in H as [[a b][H1 H2]]. cbn in H1.
exists (M_enc_card a b). split; try now apply enc_stack_el.
exists (M_enc_string a), (M_enc_string b). split; trivial.
assert (u = s++a) as -> by congruence. assert (v = t++b) as -> by congruence.
now rewrite !M_prep_enc.
- intros (u&H&a&b&->&->). apply enc_stack_el' in H as [u[v[H1 H2]]].
apply opair_inj in H2 as [-> ->]. rewrite !M_prep_enc. apply enc_stack_el.
apply in_map_iff. now exists (u, v).
Qed.
Lemma M_combinations_step B C :
M_combinations B (M_enc_stack C) (M_enc_stack (derivation_step B C)).
Proof.
induction B as [|[s t] B IH]; cbn; trivial.
exists (M_enc_stack (derivation_step B C)), (M_enc_stack (append_all C (s, t))).
rewrite M_enc_stack_app. split; trivial. split; trivial.
apply comb_rel_rep.
Qed.
Lemma solutions_derivations B f n k :
M_solutions B f (numeral n) -> k <= n -> M_opair (numeral k) (M_enc_stack (derivations B k)) ∈ f.
Proof.
intros H Hk; induction k; cbn.
- apply H.
- assert (Hk' : k <= n) by lia. specialize (IHk Hk').
destruct H as [_ H]. eapply H in IHk; eauto.
+ now apply numeral_lt.
+ apply M_combinations_step.
Qed.
Lemma derivations_derivable B n s t :
(s, t) el derivations B n -> derivable B s t.
Proof.
induction n in s,t|-*; cbn.
- now constructor.
- unfold derivation_step. intros [[u v][H1 H2]] % in_flat_map.
unfold append_all in H2. apply in_map_iff in H2 as [[a b][H2 H3]]. cbn in H2.
assert (s = u++a) as -> by congruence. assert (t = v++b) as -> by congruence.
constructor 2; trivial. apply IHn, H3.
Qed.
Definition M_function f :=
forall x y y', M_opair x y ∈ f -> M_opair x y' ∈ f -> y = y'.
Definition standard :=
forall x, x ∈ ω -> exists n, x ≡ numeral n.
Lemma M_solutions_el B f k X p :
standard -> k ∈ ω -> M_function f -> M_solutions B f k -> M_opair k X ∈ f
-> p ∈ X -> exists u v, p = M_enc_card u v /\ derivable B u v.
Proof.
intros HS HO Hf Hk HX Hp. destruct (HS k HO) as [n -> % VIEQ].
pose proof (H := solutions_derivations Hk (le_n n)).
rewrite (Hf _ _ _ HX H) in Hp. apply enc_stack_el' in Hp as (s&t&H'&->).
exists s, t. split; trivial. eapply derivations_derivable; eauto.
Qed.
Theorem PCP_ZF2 B rho :
standard -> rho ⊨ solvable B -> exists s, derivable B s s.
Proof.
intros VIN (n & f & s & X & [[[[H1 H2] H3] H4] H5]).
assert (H1' : n ∈ ω) by apply H1. clear H1.
assert (H4' : M_opair n X ∈ f) by apply H4. clear H4.
assert (H5' : M_opair s s ∈ X) by apply H5. clear H5.
assert (H2' : M_function f).
{ intros x y y' H H'. apply VIEQ. eapply H2. apply H. apply H'. } clear H2.
assert (H3' : M_opair ∅ (M_enc_stack B) ∈ f).
{ erewrite <- eval_enc_stack. apply H3. } destruct H3 as [_ H3].
assert (H3'' : forall k x y, k ∈ n -> M_opair k x ∈ f -> M_combinations B x y -> M_opair (σ k) y ∈ f).
{ intros k x y Hn Hk Hy. apply (H3 k x y); auto. fold sat. eapply M_combinations_spec; eauto. } clear H3.
destruct (@M_solutions_el B f n X (M_opair s s)) as (u&v&H1&H2); trivial.
now split. exists u. apply opair_inj in H1 as [H ->]. apply enc_string_inj in H as ->. apply H2.
Qed.
End ZF.
Arguments standard {_} _.
Theorem PCP_ZF' B :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho, rho ⊫ ZF')
-> PCPb B <-> entailment_ZF' (solvable B).
Proof.
intros HZF. rewrite PCPb_iff_dPCPb. split; intros H.
- clear HZF. destruct H as [s H]. intros M HM rho H1 H2. eapply PCP_ZF1; eauto.
- destruct HZF as (M & H1 & H2 & H3 & H4).
specialize (H M H1 (fun _ => @i_func _ _ _ _ eset Vector.nil) H2 H4).
apply PCP_ZF2 in H as [s Hs]; trivial. now exists s.
Qed.
Theorem PCP_Z B :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, Z psi -> rho ⊨ psi)
-> PCPb B <-> entailment_Z (solvable B).
Proof.
intros HZF. rewrite PCPb_iff_dPCPb. split; intros H.
- clear HZF. destruct H as [s H]. intros M HM rho H1 H2.
eapply PCP_ZF1; eauto. intros sigma phi HP. apply H2. now constructor.
- destruct HZF as (M & H1 & H2 & H3 & H4).
specialize (H M H1 (fun _ => @i_func _ _ _ _ eset Vector.nil) H2 H4).
apply PCP_ZF2 in H as [s Hs]; trivial; try now exists s.
intros sigma phi HP. apply H4. now constructor.
Qed.
Theorem PCP_ZF B :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi)
-> PCPb B <-> entailment_ZF (solvable B).
Proof.
intros HZF. rewrite PCPb_iff_dPCPb. split; intros H.
- clear HZF. destruct H as [s H]. intros M HM rho H1 H2.
eapply PCP_ZF1; eauto. intros sigma phi HP. apply H2. now constructor.
- destruct HZF as (M & H1 & H2 & H3 & H4).
specialize (H M H1 (fun _ => @i_func _ _ _ _ eset Vector.nil) H2 H4).
apply PCP_ZF2 in H as [s Hs]; trivial; try now exists s.
intros sigma phi HP. apply H4. now constructor.
Qed.
Lemma extensional_eq V (M : interp V) rho :
extensional M -> rho ⊫ ZF' -> rho ⊫ ZFeq'.
Proof.
intros H1 H2 phi [<-|[<-|[<-|[<-|H]]]]; try now apply H2.
all: cbn; intros; rewrite !H1 in *; congruence.
Qed.