(* ** Operations & Properties of FOL *)
Require Import Equations.Equations Equations.Prop.DepElim Arith Undecidability.Shared.Libs.PSL.Numbers List Setoid.
From Undecidability.Synthetic Require Export DecidabilityFacts EnumerabilityFacts ListEnumerabilityFacts ReducibilityFacts.
From Undecidability.FOLP Require Export FullSyntax unscoped.
From Undecidability.Shared Require Import ListAutomation.
Require Export Lia.
Import ListAutomationNotations.
(* Coercion var_term : fin >-> term. *)
Notation "phi --> psi" := (Impl phi psi) (right associativity, at level 55).
Notation "phi ∧ psi" := (Conj phi psi) (right associativity, at level 55).
Notation "phi ∨ psi" := (Disj phi psi) (right associativity, at level 55).
Notation "∀ phi" := (All phi) (at level 56, right associativity).
Notation "∃ phi" := (Ex phi) (at level 56, right associativity).
Notation "⊥" := (Fal).
Notation "¬ phi" := (phi --> ⊥) (at level 20).
Notation vector := Vector.t.
Import Vector.
Arguments nil {A}.
Arguments cons {A} _ {n}.
Derive Signature for vector.
(* **** Tactics *)
Ltac capply H := eapply H; try eassumption.
Ltac comp := repeat (progress (cbn in *; autounfold in *; asimpl in *)).
#[export] Hint Unfold idsRen : core.
Ltac resolve_existT :=
match goal with
| [ H2 : existT _ _ _ = existT _ _ _ |- _ ] => rewrite (Eqdep.EqdepTheory.inj_pair2 _ _ _ _ _ H2) in *
| _ => idtac
end.
Ltac inv H :=
inversion H; subst; repeat (progress resolve_existT).
Section FullFOL.
Context {Sigma : Signature}.
Lemma var_subst phi :
phi [var_term] = phi.
Proof.
now asimpl.
Qed.
Lemma var_subst_L A :
[phi[var_term] | phi ∈ A] = A.
Proof.
induction A; cbn; trivial.
now rewrite var_subst, IHA.
Qed.
Definition form_shift n := var_term (S n).
Notation "↑" := form_shift.
(* **** Subformula *)
Inductive sf : form -> form -> Prop :=
| SImplL phi psi : sf phi (phi --> psi)
| SImplR phi psi : sf psi (phi --> psi)
(* | SEq s t s' t' : sf (Pr s' t') (Eq s t) *)
| SDisjL phi psi : sf phi (phi ∨ psi)
| SDisjR phi psi : sf psi (phi ∨ psi)
| SConjL phi psi : sf phi (phi ∧ psi)
| SConjR phi psi : sf psi (phi ∧ psi)
| SAll phi t : sf (phi [t .: ids]) (∀ phi)
| SEx phi t : sf (phi [t .: ids]) (∃ phi).
Lemma sf_acc phi rho :
Acc sf (phi [rho]).
Proof.
revert rho; induction phi; intros rho; constructor; intros psi Hpsi; asimpl in *; inversion Hpsi;
subst; asimpl in *; eauto.
Qed.
Lemma sf_well_founded :
well_founded sf.
Proof.
intros phi. pose proof (sf_acc phi ids) as H. comp. erewrite -> idSubst_form in H; firstorder.
Qed.
(* ** Formula size *)
Fixpoint size (phi : form ) :=
match phi with
| Pred _ _ => 0
| Fal => 0
| Impl phi psi
| Conj phi psi
| Disj phi psi => S (size phi + size psi)
| All phi
| Ex phi => S (size phi)
end.
Lemma size_ind {X : Type} (f : X -> nat) (P : X -> Prop) :
(forall x, (forall y, f y < f x -> P y) -> P x) -> forall x, P x.
Proof.
intros H x. apply H.
induction (f x).
- intros y. lia.
- intros y. intros [] % le_lt_or_eq.
+ apply IHn; lia.
+ apply H. injection H0. now intros ->.
Qed.
Lemma subst_size rho phi :
size (subst_form rho phi) = size phi.
Proof.
revert rho; induction phi; intros rho; cbn; asimpl; try congruence.
Qed.
Lemma form_ind_subst (P : form -> Prop) :
P ⊥ -> (forall p v, P (Pred p v)) ->
(forall phi psi, P phi -> P psi -> P (phi --> psi)) ->
(forall phi psi, P phi -> P psi -> P (phi ∧ psi)) ->
(forall phi psi, P phi -> P psi -> P (phi ∨ psi)) ->
(forall phi, (forall t, P phi[t..]) -> P (∀ phi)) ->
(forall phi, (forall t, P phi[t..]) -> P (∃ phi)) ->
forall phi, P phi.
Proof.
intros H1 H2 H3 H4 H5 H6 H7 phi.
induction phi using (@size_induction _ size).
destruct phi; trivial.
- apply H3; apply H; cbn; lia.
- apply H4; apply H; cbn; lia.
- apply H5; apply H; cbn; lia.
- apply H6. intros t. apply H.
cbn. unfold subst1. rewrite subst_size. lia.
- apply H7. intros t. apply H.
cbn. unfold subst1. rewrite subst_size. lia.
Qed.
(* **** Forall and Vector.t technology **)
Inductive Forall (A : Type) (P : A -> Type) : forall n, vector A n -> Type :=
| Forall_nil : Forall P (@Vector.nil A)
| Forall_cons : forall n (x : A) (l : vector A n), P x -> Forall P l -> Forall P (@Vector.cons A x n l).
Inductive vec_in (A : Type) (a : A) : forall n, vector A n -> Type :=
| vec_inB n (v : vector A n) : vec_in a (cons a v)
| vec_inS a' n (v :vector A n) : vec_in a v -> vec_in a (cons a' v).
Hint Constructors vec_in : core.
Lemma strong_term_ind' (p : term -> Type) :
(forall x, p (var_term x)) -> (forall F v, (Forall p v) -> p (Func F v)) -> forall (t : term), p t.
Proof.
intros f1 f2. fix strong_term_ind' 1. destruct t as [n|F v].
- apply f1.
- apply f2. induction v.
+ econstructor.
+ econstructor. now eapply strong_term_ind'. eauto.
Qed.
Lemma strong_term_ind (p : term -> Type) :
(forall x, p (var_term x)) -> (forall F v, (forall t, vec_in t v -> p t) -> p (Func F v)) -> forall (t : term), p t.
Proof.
intros f1 f2. eapply strong_term_ind'.
- apply f1.
- intros. apply f2. intros t. induction 1; inv X; eauto.
Qed.
(* **** Free variables *)
Inductive unused_term (n : nat) : term -> Prop :=
| uft_var m : n <> m -> unused_term n (var_term m)
| uft_Func F v : (forall t, vec_in t v -> unused_term n t) -> unused_term n (Func F v).
Inductive unused (n : nat) : form -> Prop :=
| uf_Fal : unused n Fal
| uf_Pred P v : (forall t, vec_in t v -> unused_term n t) -> unused n (Pred P v)
| uf_I phi psi : unused n phi -> unused n psi -> unused n (Impl phi psi)
| uf_A phi psi : unused n phi -> unused n psi -> unused n (Conj phi psi)
| uf_O phi psi : unused n phi -> unused n psi -> unused n (Disj phi psi)
| uf_All phi : unused (S n) phi -> unused n (All phi)
| uf_Ex phi : unused (S n) phi -> unused n (Ex phi).
Definition unused_L n A := forall phi, phi el A -> unused n phi.
Definition closed phi := forall n, unused n phi.
Lemma vec_unused n (v : vector term n) :
(forall t, vec_in t v -> { n | forall m, n <= m -> unused_term m t }) ->
{ k | forall t, vec_in t v -> forall m, k <= m -> unused_term m t }.
Proof.
intros Hun. induction v in Hun |-*.
- exists 0. intros n H. inv H.
- destruct IHv as [k H]. 1: eauto. destruct (Hun h (vec_inB h v)) as [k' H'].
exists (k + k'). intros t H2. inv H2; intros m Hm; [apply H' | apply H]; now try lia.
Qed.
Lemma find_unused_term t :
{ n | forall m, n <= m -> unused_term m t }.
Proof.
induction t using strong_term_ind.
- exists (S x). intros m Hm. constructor. lia.
- destruct (vec_unused X) as [k H]. exists k. eauto using unused_term.
Qed.
Lemma find_unused phi :
{ n | forall m, n <= m -> unused m phi }.
Proof with eauto using unused.
induction phi.
- exists 0...
- destruct (@vec_unused _ t) as [k H]. 1: eauto using find_unused_term. exists k. eauto using unused.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi. exists x. intros m Hm. constructor. apply u. lia.
- destruct IHphi. exists x. intros m Hm. constructor. apply u. lia.
Qed.
Lemma find_unused_L A :
{ n | forall m, n <= m -> unused_L m A }.
Proof.
induction A.
- exists 0. unfold unused_L. firstorder.
- destruct IHA. destruct (find_unused a).
exists (x + x0). intros m Hm. intros phi []; subst.
+ apply u0. lia.
+ apply u. lia. auto.
Qed.
Definition capture n phi := nat_rect _ phi (fun _ => All) n.
Lemma capture_captures n m phi :
(forall i, n <= i -> unused i phi) -> (forall i, n - m <= i -> unused i (capture m phi)).
Proof.
intros H. induction m; cbn; intros i Hi.
- rewrite <- minus_n_O in *. intuition.
- constructor. apply IHm. lia.
Qed.
Definition close phi := capture (proj1_sig (find_unused phi)) phi.
Lemma close_closed phi :
closed (close phi).
Proof.
intros n. unfold close. destruct (find_unused phi) as [m Hm]; cbn.
apply (capture_captures Hm). lia.
Qed.
Fixpoint big_imp A phi :=
match A with
| List.nil => phi
| a :: A' => a --> (big_imp A' phi)
end.
(* **** Substituting unused variables *)
Definition shift_P P n :=
match n with
| O => False
| S n' => P n'
end.
Lemma vec_map_ext X Y (f g : X -> Y) n (v : vector X n) :
(forall x, vec_in x v -> f x = g x) -> map f v = map g v.
Proof.
intros Hext; induction v in Hext |-*; cbn.
- reflexivity.
- rewrite IHv, (Hext h). 1: reflexivity. all: eauto.
Qed.
Lemma subst_unused_term xi sigma P t :
(forall x, dec (P x)) -> (forall m, ~ P m -> xi m = sigma m) -> (forall m, P m -> unused_term m t) ->
subst_term xi t = subst_term sigma t.
Proof.
intros Hdec Hext Hunused. induction t using strong_term_ind; cbn; asimpl.
- destruct (Hdec x) as [H % Hunused | H % Hext].
+ inversion H; subst; congruence.
+ congruence.
- f_equal. apply vec_map_ext. intros t H'. apply (H t H'). intros n H2 % Hunused. inv H2. eauto.
Qed.
Lemma subst_unused_form xi sigma P phi :
(forall x, dec (P x)) -> (forall m, ~ P m -> xi m = sigma m) -> (forall m, P m -> unused m phi) ->
subst_form xi phi = subst_form sigma phi.
Proof.
induction phi in xi,sigma,P |-*; intros Hdec Hext Hunused; cbn; asimpl.
- reflexivity.
- f_equal. apply vec_map_ext. intros s H. apply (subst_unused_term Hdec Hext).
intros m H' % Hunused. inv H'. eauto.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- erewrite IHphi with (P := shift_P P). 1: reflexivity.
+ intros [| x]; [now right| now apply Hdec].
+ intros [| m]; [reflexivity|]. intros Heq % Hext; unfold ">>"; cbn; congruence.
+ intros [| m]; [destruct 1| ]. intros H % Hunused; now inversion H.
- erewrite IHphi with (P := shift_P P). 1: reflexivity.
+ intros [| x]; [now right| now apply Hdec].
+ intros [| m]; [reflexivity|]. intros Heq % Hext; unfold ">>"; cbn; congruence.
+ intros [| m]; [destruct 1| ]. intros H % Hunused; now inversion H.
Qed.
Lemma subst_unused_single xi sigma n phi :
unused n phi -> (forall m, n <> m -> xi m = sigma m) -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hext Hunused. apply subst_unused_form with (P := fun m => n = m). all: intuition.
now subst.
Qed.
Lemma subst_unused_range xi sigma phi n :
(forall m, n <= m -> unused m phi) -> (forall x, x < n -> xi x = sigma x) -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hle Hext. apply subst_unused_form with (P := fun x => n <= x); [apply le_dec| |assumption].
intros ? ? % not_le; intuition.
Qed.
Lemma subst_unused_closed xi sigma phi :
closed phi -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hcl. apply subst_unused_range with (n := 0); intuition. lia.
Qed.
Lemma subst_unused_closed' xi phi :
closed phi -> subst_form xi phi = phi.
Proof.
intros Hcl. rewrite <- idSubst_form with (sigmaterm := ids).
apply subst_unused_range with (n := 0). all: intuition; lia.
Qed.
Lemma vec_forall_map X Y (f : X -> Y) n (v : vector X n) (p : Y -> Type) :
(forall x, vec_in x v -> p (f x)) -> forall y, vec_in y (map f v) -> p y.
Proof.
intros H y Hmap. induction v; cbn; inv Hmap; eauto.
Qed.
(* **** Theories *)
Definition theory := form -> Prop.
Definition contains phi (T : theory) := T phi.
Definition contains_L (A : list form) (T : theory) := forall f, f el A -> contains f T.
Definition subset_T (T1 T2 : theory) := forall (phi : form), contains phi T1 -> contains phi T2.
Definition list_T A : theory := fun phi => phi el A.
Infix "⊏" := contains_L (at level 20).
Infix "⊑" := subset_T (at level 20).
Infix "∈" := contains (at level 70).
Hint Unfold contains : core.
Hint Unfold contains_L : core.
Hint Unfold subset_T : core.
Global Instance subset_T_trans : Transitive subset_T.
Proof.
intros T1 T2 T3. intuition.
Qed.
Definition extend T (phi : form) := fun psi => T psi \/ psi = phi.
Infix "⋄" := extend (at level 20).
Definition closed_T (T : theory) := forall phi n, contains phi T -> unused n phi.
Lemma closed_T_extend T phi :
closed_T T -> closed phi -> closed_T (T ⋄ phi).
Proof.
intros ? ? ? ? []; subst; intuition.
Qed.
Section ContainsAutomation.
Lemma contains_nil T :
List.nil ⊏ T.
Proof. firstorder. Qed.
Lemma contains_cons a A T :
a ∈ T -> A ⊏ T -> (a :: A) ⊏ T.
Proof. intros ? ? ? []; subst; intuition. Qed.
Lemma contains_cons2 a A T :
(a :: A) ⊏ T -> A ⊏ T.
Proof. firstorder. Qed.
Lemma contains_app A B T :
A ⊏ T -> B ⊏ T -> (A ++ B) ⊏ T.
Proof. intros ? ? ? [] % in_app_or; intuition. Qed.
Lemma contains_extend1 phi T :
phi ∈ (T ⋄ phi).
Proof. now right. Qed.
Lemma contains_extend2 phi psi T :
phi ∈ T -> phi ∈ (T ⋄ psi).
Proof. intros ?. now left. Qed.
Lemma contains_extend3 A T phi :
A ⊏ T -> A ⊏ (T ⋄ phi).
Proof.
intros ? ? ?. left. intuition.
Qed.
End ContainsAutomation.
End FullFOL.
Definition tmap {S1 S2 : Signature} (f : @form S1 -> @form S2) (T : @theory S1) : @theory S2 :=
fun phi => exists psi, T psi /\ f psi = phi.
Lemma enum_tmap {S1 S2 : Signature} (f : @form S1 -> @form S2) (T : @theory S1) L :
list_enumerator L T -> list_enumerator(L >> List.map f) (tmap f T).
Proof.
intros H0. unfold ">>".
intros x; split.
+ intros (phi & [m Hin] % H0 & <-). exists m. apply in_map_iff. firstorder.
+ intros (m & (phi & <- & Hphi) % in_map_iff). firstorder.
Qed.
Lemma tmap_contains_L {S1 S2 : Signature} (f : @form S1 -> @form S2) T A :
contains_L A (tmap f T) -> exists B, A = List.map f B /\ contains_L B T.
Proof.
induction A.
- intros. now exists List.nil.
- intros H. destruct IHA as (B & -> & HB). 1: firstorder.
destruct (H a (or_introl eq_refl)) as (b & Hb & <-).
exists (b :: B). split. 1: auto. intros ? []; subst; auto.
Qed.
Local Hint Constructors vec_in : core.
Infix "⊏" := contains_L (at level 20).
Infix "⊑" := subset_T (at level 20).
Infix "∈" := contains (at level 70).
Infix "⋄" := extend (at level 20).
Local Hint Resolve contains_nil contains_cons contains_cons2 contains_app : contains_theory.
Local Hint Resolve contains_extend1 contains_extend2 contains_extend3 : contains_theory.
Ltac use_theory A := exists A; split; [eauto 15 with contains_theory|].
(* **** Equality deciders *)
Lemma dec_vec_in X n (v : vector X n) :
(forall x, vec_in x v -> forall y, dec (x = y)) -> forall v', dec (v = v').
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros Hv. induction v; intros v'; dependent destruction v'...
destruct (Hv h (vec_inB h v) h0)... destruct (IHv (fun x H => Hv x (vec_inS h0 H)) v')...
Qed.
Instance dec_vec X {HX : eq_dec X} n : eq_dec (vector X n).
Proof.
intros v. refine (dec_vec_in _).
Qed.
Section EqDec.
Context {Sigma : Signature}.
Hypothesis eq_dec_Funcs : eq_dec Funcs.
Hypothesis eq_dec_Preds : eq_dec Preds.
Global Instance dec_term : eq_dec term.
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros t. induction t using strong_term_ind; intros []...
- decide (x = n)...
- decide (F = f)... destruct (dec_vec_in X t)...
Qed.
Global Instance dec_form : eq_dec form.
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros phi. induction phi; intros []...
- decide (P = P0)... decide (t = t0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi = f)...
- decide (phi = f)...
Qed.
End EqDec.
Notation "↑" := form_shift.
Notation "A ⟹ phi" := (big_imp A phi) (at level 60).
Require Import Equations.Equations Equations.Prop.DepElim Arith Undecidability.Shared.Libs.PSL.Numbers List Setoid.
From Undecidability.Synthetic Require Export DecidabilityFacts EnumerabilityFacts ListEnumerabilityFacts ReducibilityFacts.
From Undecidability.FOLP Require Export FullSyntax unscoped.
From Undecidability.Shared Require Import ListAutomation.
Require Export Lia.
Import ListAutomationNotations.
(* Coercion var_term : fin >-> term. *)
Notation "phi --> psi" := (Impl phi psi) (right associativity, at level 55).
Notation "phi ∧ psi" := (Conj phi psi) (right associativity, at level 55).
Notation "phi ∨ psi" := (Disj phi psi) (right associativity, at level 55).
Notation "∀ phi" := (All phi) (at level 56, right associativity).
Notation "∃ phi" := (Ex phi) (at level 56, right associativity).
Notation "⊥" := (Fal).
Notation "¬ phi" := (phi --> ⊥) (at level 20).
Notation vector := Vector.t.
Import Vector.
Arguments nil {A}.
Arguments cons {A} _ {n}.
Derive Signature for vector.
(* **** Tactics *)
Ltac capply H := eapply H; try eassumption.
Ltac comp := repeat (progress (cbn in *; autounfold in *; asimpl in *)).
#[export] Hint Unfold idsRen : core.
Ltac resolve_existT :=
match goal with
| [ H2 : existT _ _ _ = existT _ _ _ |- _ ] => rewrite (Eqdep.EqdepTheory.inj_pair2 _ _ _ _ _ H2) in *
| _ => idtac
end.
Ltac inv H :=
inversion H; subst; repeat (progress resolve_existT).
Section FullFOL.
Context {Sigma : Signature}.
Lemma var_subst phi :
phi [var_term] = phi.
Proof.
now asimpl.
Qed.
Lemma var_subst_L A :
[phi[var_term] | phi ∈ A] = A.
Proof.
induction A; cbn; trivial.
now rewrite var_subst, IHA.
Qed.
Definition form_shift n := var_term (S n).
Notation "↑" := form_shift.
(* **** Subformula *)
Inductive sf : form -> form -> Prop :=
| SImplL phi psi : sf phi (phi --> psi)
| SImplR phi psi : sf psi (phi --> psi)
(* | SEq s t s' t' : sf (Pr s' t') (Eq s t) *)
| SDisjL phi psi : sf phi (phi ∨ psi)
| SDisjR phi psi : sf psi (phi ∨ psi)
| SConjL phi psi : sf phi (phi ∧ psi)
| SConjR phi psi : sf psi (phi ∧ psi)
| SAll phi t : sf (phi [t .: ids]) (∀ phi)
| SEx phi t : sf (phi [t .: ids]) (∃ phi).
Lemma sf_acc phi rho :
Acc sf (phi [rho]).
Proof.
revert rho; induction phi; intros rho; constructor; intros psi Hpsi; asimpl in *; inversion Hpsi;
subst; asimpl in *; eauto.
Qed.
Lemma sf_well_founded :
well_founded sf.
Proof.
intros phi. pose proof (sf_acc phi ids) as H. comp. erewrite -> idSubst_form in H; firstorder.
Qed.
(* ** Formula size *)
Fixpoint size (phi : form ) :=
match phi with
| Pred _ _ => 0
| Fal => 0
| Impl phi psi
| Conj phi psi
| Disj phi psi => S (size phi + size psi)
| All phi
| Ex phi => S (size phi)
end.
Lemma size_ind {X : Type} (f : X -> nat) (P : X -> Prop) :
(forall x, (forall y, f y < f x -> P y) -> P x) -> forall x, P x.
Proof.
intros H x. apply H.
induction (f x).
- intros y. lia.
- intros y. intros [] % le_lt_or_eq.
+ apply IHn; lia.
+ apply H. injection H0. now intros ->.
Qed.
Lemma subst_size rho phi :
size (subst_form rho phi) = size phi.
Proof.
revert rho; induction phi; intros rho; cbn; asimpl; try congruence.
Qed.
Lemma form_ind_subst (P : form -> Prop) :
P ⊥ -> (forall p v, P (Pred p v)) ->
(forall phi psi, P phi -> P psi -> P (phi --> psi)) ->
(forall phi psi, P phi -> P psi -> P (phi ∧ psi)) ->
(forall phi psi, P phi -> P psi -> P (phi ∨ psi)) ->
(forall phi, (forall t, P phi[t..]) -> P (∀ phi)) ->
(forall phi, (forall t, P phi[t..]) -> P (∃ phi)) ->
forall phi, P phi.
Proof.
intros H1 H2 H3 H4 H5 H6 H7 phi.
induction phi using (@size_induction _ size).
destruct phi; trivial.
- apply H3; apply H; cbn; lia.
- apply H4; apply H; cbn; lia.
- apply H5; apply H; cbn; lia.
- apply H6. intros t. apply H.
cbn. unfold subst1. rewrite subst_size. lia.
- apply H7. intros t. apply H.
cbn. unfold subst1. rewrite subst_size. lia.
Qed.
(* **** Forall and Vector.t technology **)
Inductive Forall (A : Type) (P : A -> Type) : forall n, vector A n -> Type :=
| Forall_nil : Forall P (@Vector.nil A)
| Forall_cons : forall n (x : A) (l : vector A n), P x -> Forall P l -> Forall P (@Vector.cons A x n l).
Inductive vec_in (A : Type) (a : A) : forall n, vector A n -> Type :=
| vec_inB n (v : vector A n) : vec_in a (cons a v)
| vec_inS a' n (v :vector A n) : vec_in a v -> vec_in a (cons a' v).
Hint Constructors vec_in : core.
Lemma strong_term_ind' (p : term -> Type) :
(forall x, p (var_term x)) -> (forall F v, (Forall p v) -> p (Func F v)) -> forall (t : term), p t.
Proof.
intros f1 f2. fix strong_term_ind' 1. destruct t as [n|F v].
- apply f1.
- apply f2. induction v.
+ econstructor.
+ econstructor. now eapply strong_term_ind'. eauto.
Qed.
Lemma strong_term_ind (p : term -> Type) :
(forall x, p (var_term x)) -> (forall F v, (forall t, vec_in t v -> p t) -> p (Func F v)) -> forall (t : term), p t.
Proof.
intros f1 f2. eapply strong_term_ind'.
- apply f1.
- intros. apply f2. intros t. induction 1; inv X; eauto.
Qed.
(* **** Free variables *)
Inductive unused_term (n : nat) : term -> Prop :=
| uft_var m : n <> m -> unused_term n (var_term m)
| uft_Func F v : (forall t, vec_in t v -> unused_term n t) -> unused_term n (Func F v).
Inductive unused (n : nat) : form -> Prop :=
| uf_Fal : unused n Fal
| uf_Pred P v : (forall t, vec_in t v -> unused_term n t) -> unused n (Pred P v)
| uf_I phi psi : unused n phi -> unused n psi -> unused n (Impl phi psi)
| uf_A phi psi : unused n phi -> unused n psi -> unused n (Conj phi psi)
| uf_O phi psi : unused n phi -> unused n psi -> unused n (Disj phi psi)
| uf_All phi : unused (S n) phi -> unused n (All phi)
| uf_Ex phi : unused (S n) phi -> unused n (Ex phi).
Definition unused_L n A := forall phi, phi el A -> unused n phi.
Definition closed phi := forall n, unused n phi.
Lemma vec_unused n (v : vector term n) :
(forall t, vec_in t v -> { n | forall m, n <= m -> unused_term m t }) ->
{ k | forall t, vec_in t v -> forall m, k <= m -> unused_term m t }.
Proof.
intros Hun. induction v in Hun |-*.
- exists 0. intros n H. inv H.
- destruct IHv as [k H]. 1: eauto. destruct (Hun h (vec_inB h v)) as [k' H'].
exists (k + k'). intros t H2. inv H2; intros m Hm; [apply H' | apply H]; now try lia.
Qed.
Lemma find_unused_term t :
{ n | forall m, n <= m -> unused_term m t }.
Proof.
induction t using strong_term_ind.
- exists (S x). intros m Hm. constructor. lia.
- destruct (vec_unused X) as [k H]. exists k. eauto using unused_term.
Qed.
Lemma find_unused phi :
{ n | forall m, n <= m -> unused m phi }.
Proof with eauto using unused.
induction phi.
- exists 0...
- destruct (@vec_unused _ t) as [k H]. 1: eauto using find_unused_term. exists k. eauto using unused.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi1, IHphi2. exists (x + x0). intros m Hm. constructor; [ apply u | apply u0 ]; lia.
- destruct IHphi. exists x. intros m Hm. constructor. apply u. lia.
- destruct IHphi. exists x. intros m Hm. constructor. apply u. lia.
Qed.
Lemma find_unused_L A :
{ n | forall m, n <= m -> unused_L m A }.
Proof.
induction A.
- exists 0. unfold unused_L. firstorder.
- destruct IHA. destruct (find_unused a).
exists (x + x0). intros m Hm. intros phi []; subst.
+ apply u0. lia.
+ apply u. lia. auto.
Qed.
Definition capture n phi := nat_rect _ phi (fun _ => All) n.
Lemma capture_captures n m phi :
(forall i, n <= i -> unused i phi) -> (forall i, n - m <= i -> unused i (capture m phi)).
Proof.
intros H. induction m; cbn; intros i Hi.
- rewrite <- minus_n_O in *. intuition.
- constructor. apply IHm. lia.
Qed.
Definition close phi := capture (proj1_sig (find_unused phi)) phi.
Lemma close_closed phi :
closed (close phi).
Proof.
intros n. unfold close. destruct (find_unused phi) as [m Hm]; cbn.
apply (capture_captures Hm). lia.
Qed.
Fixpoint big_imp A phi :=
match A with
| List.nil => phi
| a :: A' => a --> (big_imp A' phi)
end.
(* **** Substituting unused variables *)
Definition shift_P P n :=
match n with
| O => False
| S n' => P n'
end.
Lemma vec_map_ext X Y (f g : X -> Y) n (v : vector X n) :
(forall x, vec_in x v -> f x = g x) -> map f v = map g v.
Proof.
intros Hext; induction v in Hext |-*; cbn.
- reflexivity.
- rewrite IHv, (Hext h). 1: reflexivity. all: eauto.
Qed.
Lemma subst_unused_term xi sigma P t :
(forall x, dec (P x)) -> (forall m, ~ P m -> xi m = sigma m) -> (forall m, P m -> unused_term m t) ->
subst_term xi t = subst_term sigma t.
Proof.
intros Hdec Hext Hunused. induction t using strong_term_ind; cbn; asimpl.
- destruct (Hdec x) as [H % Hunused | H % Hext].
+ inversion H; subst; congruence.
+ congruence.
- f_equal. apply vec_map_ext. intros t H'. apply (H t H'). intros n H2 % Hunused. inv H2. eauto.
Qed.
Lemma subst_unused_form xi sigma P phi :
(forall x, dec (P x)) -> (forall m, ~ P m -> xi m = sigma m) -> (forall m, P m -> unused m phi) ->
subst_form xi phi = subst_form sigma phi.
Proof.
induction phi in xi,sigma,P |-*; intros Hdec Hext Hunused; cbn; asimpl.
- reflexivity.
- f_equal. apply vec_map_ext. intros s H. apply (subst_unused_term Hdec Hext).
intros m H' % Hunused. inv H'. eauto.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- rewrite IHphi1 with (sigma := sigma) (P := P). rewrite IHphi2 with (sigma := sigma) (P := P).
all: try tauto. all: intros m H % Hunused; now inversion H.
- erewrite IHphi with (P := shift_P P). 1: reflexivity.
+ intros [| x]; [now right| now apply Hdec].
+ intros [| m]; [reflexivity|]. intros Heq % Hext; unfold ">>"; cbn; congruence.
+ intros [| m]; [destruct 1| ]. intros H % Hunused; now inversion H.
- erewrite IHphi with (P := shift_P P). 1: reflexivity.
+ intros [| x]; [now right| now apply Hdec].
+ intros [| m]; [reflexivity|]. intros Heq % Hext; unfold ">>"; cbn; congruence.
+ intros [| m]; [destruct 1| ]. intros H % Hunused; now inversion H.
Qed.
Lemma subst_unused_single xi sigma n phi :
unused n phi -> (forall m, n <> m -> xi m = sigma m) -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hext Hunused. apply subst_unused_form with (P := fun m => n = m). all: intuition.
now subst.
Qed.
Lemma subst_unused_range xi sigma phi n :
(forall m, n <= m -> unused m phi) -> (forall x, x < n -> xi x = sigma x) -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hle Hext. apply subst_unused_form with (P := fun x => n <= x); [apply le_dec| |assumption].
intros ? ? % not_le; intuition.
Qed.
Lemma subst_unused_closed xi sigma phi :
closed phi -> subst_form xi phi = subst_form sigma phi.
Proof.
intros Hcl. apply subst_unused_range with (n := 0); intuition. lia.
Qed.
Lemma subst_unused_closed' xi phi :
closed phi -> subst_form xi phi = phi.
Proof.
intros Hcl. rewrite <- idSubst_form with (sigmaterm := ids).
apply subst_unused_range with (n := 0). all: intuition; lia.
Qed.
Lemma vec_forall_map X Y (f : X -> Y) n (v : vector X n) (p : Y -> Type) :
(forall x, vec_in x v -> p (f x)) -> forall y, vec_in y (map f v) -> p y.
Proof.
intros H y Hmap. induction v; cbn; inv Hmap; eauto.
Qed.
(* **** Theories *)
Definition theory := form -> Prop.
Definition contains phi (T : theory) := T phi.
Definition contains_L (A : list form) (T : theory) := forall f, f el A -> contains f T.
Definition subset_T (T1 T2 : theory) := forall (phi : form), contains phi T1 -> contains phi T2.
Definition list_T A : theory := fun phi => phi el A.
Infix "⊏" := contains_L (at level 20).
Infix "⊑" := subset_T (at level 20).
Infix "∈" := contains (at level 70).
Hint Unfold contains : core.
Hint Unfold contains_L : core.
Hint Unfold subset_T : core.
Global Instance subset_T_trans : Transitive subset_T.
Proof.
intros T1 T2 T3. intuition.
Qed.
Definition extend T (phi : form) := fun psi => T psi \/ psi = phi.
Infix "⋄" := extend (at level 20).
Definition closed_T (T : theory) := forall phi n, contains phi T -> unused n phi.
Lemma closed_T_extend T phi :
closed_T T -> closed phi -> closed_T (T ⋄ phi).
Proof.
intros ? ? ? ? []; subst; intuition.
Qed.
Section ContainsAutomation.
Lemma contains_nil T :
List.nil ⊏ T.
Proof. firstorder. Qed.
Lemma contains_cons a A T :
a ∈ T -> A ⊏ T -> (a :: A) ⊏ T.
Proof. intros ? ? ? []; subst; intuition. Qed.
Lemma contains_cons2 a A T :
(a :: A) ⊏ T -> A ⊏ T.
Proof. firstorder. Qed.
Lemma contains_app A B T :
A ⊏ T -> B ⊏ T -> (A ++ B) ⊏ T.
Proof. intros ? ? ? [] % in_app_or; intuition. Qed.
Lemma contains_extend1 phi T :
phi ∈ (T ⋄ phi).
Proof. now right. Qed.
Lemma contains_extend2 phi psi T :
phi ∈ T -> phi ∈ (T ⋄ psi).
Proof. intros ?. now left. Qed.
Lemma contains_extend3 A T phi :
A ⊏ T -> A ⊏ (T ⋄ phi).
Proof.
intros ? ? ?. left. intuition.
Qed.
End ContainsAutomation.
End FullFOL.
Definition tmap {S1 S2 : Signature} (f : @form S1 -> @form S2) (T : @theory S1) : @theory S2 :=
fun phi => exists psi, T psi /\ f psi = phi.
Lemma enum_tmap {S1 S2 : Signature} (f : @form S1 -> @form S2) (T : @theory S1) L :
list_enumerator L T -> list_enumerator(L >> List.map f) (tmap f T).
Proof.
intros H0. unfold ">>".
intros x; split.
+ intros (phi & [m Hin] % H0 & <-). exists m. apply in_map_iff. firstorder.
+ intros (m & (phi & <- & Hphi) % in_map_iff). firstorder.
Qed.
Lemma tmap_contains_L {S1 S2 : Signature} (f : @form S1 -> @form S2) T A :
contains_L A (tmap f T) -> exists B, A = List.map f B /\ contains_L B T.
Proof.
induction A.
- intros. now exists List.nil.
- intros H. destruct IHA as (B & -> & HB). 1: firstorder.
destruct (H a (or_introl eq_refl)) as (b & Hb & <-).
exists (b :: B). split. 1: auto. intros ? []; subst; auto.
Qed.
Local Hint Constructors vec_in : core.
Infix "⊏" := contains_L (at level 20).
Infix "⊑" := subset_T (at level 20).
Infix "∈" := contains (at level 70).
Infix "⋄" := extend (at level 20).
Local Hint Resolve contains_nil contains_cons contains_cons2 contains_app : contains_theory.
Local Hint Resolve contains_extend1 contains_extend2 contains_extend3 : contains_theory.
Ltac use_theory A := exists A; split; [eauto 15 with contains_theory|].
(* **** Equality deciders *)
Lemma dec_vec_in X n (v : vector X n) :
(forall x, vec_in x v -> forall y, dec (x = y)) -> forall v', dec (v = v').
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros Hv. induction v; intros v'; dependent destruction v'...
destruct (Hv h (vec_inB h v) h0)... destruct (IHv (fun x H => Hv x (vec_inS h0 H)) v')...
Qed.
Instance dec_vec X {HX : eq_dec X} n : eq_dec (vector X n).
Proof.
intros v. refine (dec_vec_in _).
Qed.
Section EqDec.
Context {Sigma : Signature}.
Hypothesis eq_dec_Funcs : eq_dec Funcs.
Hypothesis eq_dec_Preds : eq_dec Preds.
Global Instance dec_term : eq_dec term.
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros t. induction t using strong_term_ind; intros []...
- decide (x = n)...
- decide (F = f)... destruct (dec_vec_in X t)...
Qed.
Global Instance dec_form : eq_dec form.
Proof with subst; try (now left + (right; intros[=]; resolve_existT; congruence)).
intros phi. induction phi; intros []...
- decide (P = P0)... decide (t = t0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi1 = f)... decide (phi2 = f0)...
- decide (phi = f)...
- decide (phi = f)...
Qed.
End EqDec.
Notation "↑" := form_shift.
Notation "A ⟹ phi" := (big_imp A phi) (at level 60).