From Undecidability.Synthetic Require Import Definitions DecidabilityFacts EnumerabilityFacts ListEnumerabilityFacts ReducibilityFacts.
From Undecidability Require Import FOL.Syntax.Facts FOL.Syntax.Theories.
From Undecidability Require Export FOL.Deduction.FullND.
From Undecidability Require Import Shared.ListAutomation.
Import ListAutomationNotations.
Local Set Implicit Arguments.
Local Unset Strict Implicit.
Require Import Lia.
Set Default Proof Using "Type".
#[local] Ltac comp := repeat (progress (cbn in *; autounfold in *)).
Section ND_def.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Context {ff : falsity_flag}.
Context {p : peirce}.
Theorem Weak A B phi :
A ⊢ phi -> A <<= B -> B ⊢ phi.
Proof.
intros H. revert B.
induction H; intros B HB; try unshelve (solve [econstructor; intuition]); try now econstructor.
Qed.
Hint Constructors prv : core.
Theorem subst_Weak A phi xi :
A ⊢ phi -> [phi[xi] | phi ∈ A] ⊢ phi[xi].
Proof.
induction 1 in xi |-*; comp.
1-2,7-15: eauto using in_map.
- apply AllI. setoid_rewrite map_map in IHprv. erewrite map_map, map_ext.
apply IHprv. intros ?. cbn. now rewrite up_form.
- specialize (IHprv xi). apply AllE with (t := t`[xi]) in IHprv. rewrite subst_comp in *.
erewrite subst_ext; try apply IHprv. intros [|]; cbn; trivial.
unfold funcomp. now setoid_rewrite subst_term_shift.
- specialize (IHprv xi). eapply ExI with (t := t`[xi]). rewrite subst_comp in *.
erewrite subst_ext; try apply IHprv. intros [|]; cbn; trivial.
unfold funcomp. now setoid_rewrite subst_term_shift.
- eapply ExE in IHprv1. eassumption. rewrite map_map.
specialize (IHprv2 (up xi)). setoid_rewrite up_form in IHprv2.
erewrite map_map, map_ext in IHprv2; try apply IHprv2. apply up_form.
Qed.
Definition cycle_shift n x :=
if Dec (n = x) then $0 else $(S x).
Lemma cycle_shift_shift n phi :
bounded n phi -> phi[cycle_shift n] = phi[↑].
Proof.
intros H. apply (bounded_subst H). intros k. unfold cycle_shift. decide _; trivial; lia.
Qed.
Lemma cycle_shift_subject n phi :
bounded (S n) phi -> phi[$n..][cycle_shift n] = phi.
Proof.
intros H. erewrite subst_comp, (bounded_subst H), subst_id; trivial.
intros []; cbn; unfold cycle_shift; decide _; trivial; lia.
Qed.
Lemma nameless_equiv_all' A phi n :
bounded_L n A -> bounded (S n) phi -> [p[↑] | p ∈ A] ⊢ phi <-> A ⊢ phi[$n..].
Proof.
intros H1 H2. split; intros H.
- apply (subst_Weak ($n..)) in H. rewrite map_map in *.
erewrite map_ext, map_id in H; try apply H. intros. apply subst_shift.
- apply (subst_Weak (cycle_shift n)) in H. rewrite (map_ext_in _ (subst_form ↑)) in H.
+ now rewrite cycle_shift_subject in H.
+ intros psi HP. now apply cycle_shift_shift, H1.
Qed.
Lemma nameless_equiv_ex' A phi psi n :
bounded_L n A -> bounded n phi -> bounded (S n) psi -> (psi::[p0[↑] | p0 ∈ A]) ⊢ phi[↑] <-> (psi[$n..]::A) ⊢ phi.
Proof.
intros HL Hphi Hpsi. split.
- intros H % (subst_Weak ($n..)). cbn in *.
rewrite map_map, (map_ext _ id), map_id in H.
+ now rewrite subst_shift in H.
+ intros. apply subst_shift.
- intros H % (subst_Weak (cycle_shift n)). cbn in *.
rewrite (map_ext_in _ (subst_form ↑)) in H.
+ now rewrite cycle_shift_subject, cycle_shift_shift in H.
+ intros theta HT. now apply cycle_shift_shift, HL.
Qed.
Lemma nameless_equiv_all A phi :
{ t : term | map (subst_form ↑) A ⊢ phi <-> A ⊢ phi[t..] }.
Proof.
destruct (find_bounded_L (phi::A)) as [n H].
exists $n. apply nameless_equiv_all'.
- intros ? ?. apply H. auto.
- eapply bounded_up; try apply H; auto.
Qed.
Lemma nameless_equiv_ex A phi psi :
{ t : term | (phi :: map (subst_form ↑) A) ⊢ psi[↑] <-> (phi[t..]::A) ⊢ psi }.
Proof.
destruct (find_bounded_L (phi::psi::A)) as [n H].
exists $n. apply nameless_equiv_ex'.
- intros ? ?. apply H. auto.
- apply H. auto.
- eapply bounded_up; try apply H; auto.
Qed.
Lemma AllI_named (A : list form) (phi : form) :
(forall t, A ⊢ phi[t..]) -> A ⊢ ∀phi.
Proof.
intros H. apply AllI.
destruct (nameless_equiv_all A phi) as [t Ht].
apply Ht, H.
Qed.
Lemma ExE_named A χ ψ :
A ⊢ ∃ χ -> (forall t, (χ[t..]::A) ⊢ ψ) -> A ⊢ ψ.
Proof.
intros H1 H2. destruct (nameless_equiv_ex A χ ψ) as [t Ht].
eapply ExE.
- eassumption.
- apply Ht, H2.
Qed.
Lemma imps T phi psi :
T ⊢ phi → psi <-> (phi :: T) ⊢ psi.
Proof.
split; try apply II.
intros H. apply IE with phi; auto. apply (Weak H). auto.
Qed.
Lemma CE T phi psi :
T ⊢ phi ∧ psi -> T ⊢ phi /\ T ⊢ psi.
Proof.
intros H. split.
- apply (CE1 H).
- apply (CE2 H).
Qed.
Lemma DE' A phi :
A ⊢ phi ∨ phi -> A ⊢ phi.
Proof.
intros H. apply (DE H); auto.
Qed.
Lemma switch_conj_imp alpha beta phi A :
A ⊢ alpha ∧ beta → phi <-> A ⊢ alpha → beta → phi.
Proof.
split; intros H.
- apply II, II. eapply IE.
apply (@Weak A). apply H. firstorder.
apply CI; apply Ctx; firstorder.
- apply II. eapply IE. eapply IE.
eapply Weak. apply H.
firstorder.
eapply CE1, Ctx; firstorder.
eapply CE2, Ctx; firstorder.
Qed.
Lemma impl_prv A B phi :
(rev B ++ A) ⊢ phi <-> A ⊢ (B ==> phi).
Proof.
revert A; induction B; intros A; cbn; simpl_list; intros.
- firstorder.
- split; intros.
+ eapply II. now eapply IHB.
+ now apply imps, IHB in H.
Qed.
Lemma subst_exist_prv {sigma N Gamma} phi :
Gamma ⊢ phi[sigma] -> bounded N phi -> Gamma ⊢ exist_times N phi.
Proof.
induction N in phi, sigma |-*; intros; cbn.
- erewrite <- (subst_bounded 0); eassumption.
- rewrite iter_switch. eapply (IHN (S >> sigma)).
cbn. eapply (ExI (sigma 0)).
now rewrite up_decompose.
now apply bounded_S_quant.
Qed.
Lemma subst_forall_prv phi {N Gamma} :
Gamma ⊢ (forall_times N phi) -> bounded N phi -> forall sigma, Gamma ⊢ phi[sigma].
Proof.
induction N in phi |-*; intros ?? sigma; cbn in *.
- change sigma with (iter up 0 sigma).
now rewrite (subst_bounded 0).
- specialize (IHN (∀ phi) ).
rewrite <-up_decompose.
apply AllE. apply IHN. unfold forall_times.
now rewrite <-iter_switch.
now apply bounded_S_quant.
Qed.
Lemma prv_cut A B phi :
A ⊢ phi -> (forall psi, psi el A -> B ⊢ psi) -> B ⊢ phi.
Proof.
induction A in phi |- *; intros H1 H2.
- eapply Weak; eauto.
- rewrite <- imps in H1. apply IHA in H1; auto. apply IE with a; trivial. now apply H2.
Qed.
Lemma prv_intu_peirce T φ : T ⊢I φ -> T ⊢ φ.
Proof using.
remember intu. induction 1. all: now eauto.
Qed.
End ND_def.
Local Hint Constructors prv : core.
Lemma prv_ind_full {Σ_funcs : funcs_signature} {Σ_preds : preds_signature} :
forall P : peirce -> list (form falsity_on) -> (form falsity_on) -> Prop,
(forall (p : peirce) (A : list form) (phi psi : form),
(phi :: A) ⊢ psi -> P p (phi :: A) psi -> P p A (phi → psi)) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ phi → psi -> P p A (phi → psi) -> A ⊢ phi -> P p A phi -> P p A psi) ->
(forall (p : peirce) (A : list form) (phi : form),
(map (subst_form ↑) A) ⊢ phi -> P p (map (subst_form ↑) A) phi -> P p A (∀ phi)) ->
(forall (p : peirce) (A : list form) (t : term) (phi : form),
A ⊢ ∀ phi -> P p A (∀ phi) -> P p A phi[t..]) ->
(forall (p : peirce) (A : list form) (t : term) (phi : form),
A ⊢ phi[t..] -> P p A phi[t..] -> P p A (∃ phi)) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ ∃ phi ->
P p A (∃ phi) ->
(phi :: [p[↑] | p ∈ A]) ⊢ psi[↑] -> P p (phi :: [p[↑] | p ∈ A]) psi[↑] -> P p A psi) ->
(forall (p : peirce) (A : list form) (phi : form), A ⊢ ⊥ -> P p A ⊥ -> P p A phi) ->
(forall (p : peirce) (A : list form) (phi : form), phi el A -> P p A phi) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ phi -> P p A phi -> A ⊢ psi -> P p A psi -> P p A (phi ∧ psi)) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ phi ∧ psi -> P p A (phi ∧ psi) -> P p A phi) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ phi ∧ psi -> P p A (phi ∧ psi) -> P p A psi) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ phi -> P p A phi -> P p A (phi ∨ psi)) ->
(forall (p : peirce) (A : list form) (phi psi : form),
A ⊢ psi -> P p A psi -> P p A (phi ∨ psi)) ->
(forall (p : peirce) (A : list form) (phi psi theta : form),
A ⊢ phi ∨ psi ->
P p A (phi ∨ psi) ->
(phi :: A) ⊢ theta ->
P p (phi :: A) theta -> (psi :: A) ⊢ theta -> P p (psi :: A) theta -> P p A theta) ->
(forall (A : list form) (phi psi : form), P class A (((phi → psi) → phi) → phi)) ->
forall (p : peirce) (l : list form) (f14 : form), l ⊢ f14 -> P p l f14.
Proof.
intros. specialize (@prv_ind _ _ (fun ff => match ff with falsity_on => P | _ => fun _ _ _ => True end)). intros H'.
apply H' with (ff := falsity_on); clear H'. all: intros; try destruct ff; trivial. all: intuition eauto 2.
Qed.
Ltac subsimpl_in H :=
rewrite ?up_term, ?subst_term_shift in H.
Ltac subsimpl :=
rewrite ?up_term, ?subst_term_shift.
Ltac assert1 H :=
match goal with |- (?phi :: ?T) ⊢ _ => assert (H : (phi :: T) ⊢ phi) by auto end.
Ltac assert2 H :=
match goal with |- (?phi :: ?psi :: ?T) ⊢ _ => assert (H : (phi :: psi :: T) ⊢ psi) by auto end.
Ltac assert3 H :=
match goal with |- (?phi :: ?psi :: ?theta :: ?T) ⊢ _ => assert (H : (phi :: psi :: theta :: T) ⊢ theta) by auto end.
Ltac assert4 H :=
match goal with |- (?f :: ?phi :: ?psi :: ?theta :: ?T) ⊢ _ => assert (H : (f :: phi :: psi :: theta :: T) ⊢ theta) by auto end.
Ltac prv_all x :=
apply AllI; edestruct (@nameless_equiv_all) as [x ->]; cbn; subsimpl.
Ltac use_exists H x :=
apply (ExE H); edestruct (@nameless_equiv_ex) as [x ->]; cbn; subsimpl.
Section Enumerability.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Variable list_Funcs : nat -> list syms.
Hypothesis enum_Funcs' : list_enumerator__T list_Funcs syms.
Variable list_Preds : nat -> list preds.
Hypothesis enum_Preds' : list_enumerator__T list_Preds preds.
Hypothesis eq_dec_Funcs : eq_dec syms.
Hypothesis eq_dec_Preds : eq_dec preds.
Instance eqdec_binop : eq_dec binop.
Proof.
intros x y. unfold dec. decide equality.
Qed.
Instance eqdec_quantop : eq_dec quantop.
Proof.
intros x y. unfold dec. decide equality.
Qed.
Definition list_binop (n : nat) := [Conj; Impl; Disj].
Instance enum_binop :
list_enumerator__T list_binop binop.
Proof.
intros []; exists 0; cbn; tauto.
Qed.
Definition list_quantop (n : nat) := [All; Ex].
Instance enum_quantop :
list_enumerator__T list_quantop quantop.
Proof.
intros []; exists 0; cbn; tauto.
Qed.
Lemma enumT_binop :
enumerable__T binop.
Proof.
apply enum_enumT. exists list_binop. apply enum_binop.
Qed.
Lemma enumT_quantop :
enumerable__T quantop.
Proof.
apply enum_enumT. exists list_quantop. apply enum_quantop.
Qed.
Instance enum_term' :
list_enumerator__T (L_term _) term :=
enum_term _.
Instance enum_form' {ff : falsity_flag} :
list_enumerator__T (L_form _ _ _ _) form :=
enum_form _ _ _ _.
Fixpoint L_ded {p : peirce} {b : falsity_flag} (A : list form) (n : nat) : list form :=
match n with
| 0 => A
| S n => L_ded A n ++
concat ([ [ phi → psi | psi ∈ L_ded (phi :: A) n ] | phi ∈ L_T form n ]) ++
[ psi | (phi, psi) ∈ (L_ded A n × L_T form n) , (phi → psi el L_ded A n) ] ++
[ ∀ phi | phi ∈ L_ded (map (subst_form ↑) A) n ] ++
[ phi[t..] | (phi, t) ∈ (L_T form n × L_T term n), (∀ phi) el L_ded A n ] ++
[ ∃ phi | (phi, t) ∈ (L_T form n × L_T term n), (phi[t..]) el L_ded A n ] ++
[ psi | (phi, psi) ∈ (L_T form n × L_T form n),
(∃ phi) el L_ded A n /\ psi[↑] el L_ded (phi::(map (subst_form ↑) A)) n ] ++
(match b with falsity_on => fun A =>
[ phi | phi ∈ L_T form n, ⊥ el @L_ded _ falsity_on A n ]
| _ => fun _ => nil end A) ++
(if p then
[ (((phi → psi) → phi) → phi) | (pair phi psi) ∈ (L_T form n × L_T form n)]
else nil) ++
[ phi ∧ psi | (phi, psi) ∈ (L_ded A n × L_ded A n) ] ++
[ phi | (phi, psi) ∈ (L_T form n × L_T form n), phi ∧ psi el L_ded A n] ++
[ psi | (phi, psi) ∈ (L_T form n × L_T form n), phi ∧ psi el L_ded A n] ++
[ phi ∨ psi | (phi, psi) ∈ (L_T form n × L_T form n), phi el L_ded A n] ++
[ phi ∨ psi | (phi, psi) ∈ (L_T form n × L_T form n), psi el L_ded A n] ++
[ theta | (phi, (psi, theta)) ∈ (L_T form n × (L_T form n × L_T form n)),
theta el L_ded (phi::A) n /\ theta el L_ded (psi::A) n /\ phi ∨ psi el L_ded A n]
end.
Opaque in_dec.
Lemma enum_prv {p : peirce} {b : falsity_flag} A :
list_enumerator (L_ded A) (prv A).
Proof with try (eapply cum_ge'; eauto; lia).
split.
- rename x into phi. induction 1; try congruence; subst.
+ destruct IHprv as [m1], (el_T phi) as [m2]. exists (1 + m1 + m2). cbn. in_app 2.
eapply in_concat_iff. eexists. split. 2:in_collect phi... in_collect psi...
+ destruct IHprv1 as [m1], IHprv2 as [m2], (el_T psi) as [m3]; eauto.
exists (1 + m1 + m2 + m3).
cbn. in_app 3. in_collect (phi, psi)...
+ destruct IHprv as [m]. exists (1 + m). cbn. in_app 4. in_collect phi...
+ destruct IHprv as [m1], (el_T t) as [m2], (el_T phi) as [m3]. exists (1 + m1 + m2 + m3).
cbn. in_app 5. in_collect (phi, t)...
+ destruct IHprv as [m1], (el_T t) as [m2], (el_T phi) as [m3]. exists (1 + m1 + m2 + m3).
cbn. in_app 6. in_collect (phi, t)...
+ destruct IHprv1 as [m1], IHprv2 as [m2], (el_T phi) as [m4], (el_T psi) as [m5].
exists (1 + m1 + m2 + m4 + m5). cbn. in_app 7. cbn. in_collect (phi, psi)...
+ destruct IHprv as [m1], (el_T phi) as [m2]. exists (1 + m1 + m2). cbn. in_app 8. in_collect phi...
+ now exists 0.
+ destruct IHprv1 as [m1], IHprv2 as [m2]. exists (1 + m1 + m2). cbn. in_app 10. in_collect (phi, psi)...
+ destruct IHprv as [m1], (el_T phi) as [m2], (el_T psi) as [m3].
exists (1 + m1 + m2 + m3). cbn. in_app 11. in_collect (phi, psi)...
+ destruct IHprv as [m1], (el_T phi) as [m2], (el_T psi) as [m3].
exists (1 + m1 + m2 + m3). cbn. in_app 12. in_collect (phi, psi)...
+ destruct IHprv as [m1], (el_T phi) as [m2], (el_T psi) as [m3].
exists (1 + m1 + m2 + m3). cbn. in_app 13. in_collect (phi, psi)...
+ destruct IHprv as [m1], (el_T phi) as [m2], (el_T psi) as [m3].
exists (1 + m1 + m2 + m3). cbn. in_app 14. in_collect (phi, psi)...
+ destruct IHprv1 as [m1], IHprv2 as [m2], IHprv3 as [m3], (el_T phi) as [m4], (el_T psi) as [m5], (el_T theta) as [m6].
exists (1 + m1 + m2 + m3 + m4 + m5 + m6). cbn. in_app 15. cbn. in_collect (phi, (psi, theta))...
+ destruct (el_T phi) as [m1], (el_T psi) as [m2]. exists (1 + m1 + m2). cbn. in_app 9. in_collect (phi, psi)...
- intros [m]; induction m in A, x, H |-*; cbn in *.
+ now apply Ctx.
+ destruct p, b; inv_collect. all: eauto 3.
* eapply IE; apply IHm; eauto.
* eapply ExE; apply IHm; eauto.
* eapply DE; apply IHm; eauto.
* eapply IE; apply IHm; eauto.
* eapply ExE; apply IHm; eauto.
* eapply DE; apply IHm; eauto.
* eapply IE; apply IHm; eauto.
* eapply ExE; apply IHm; eauto.
* eapply DE; apply IHm; eauto.
* eapply IE; apply IHm; eauto.
* eapply ExE; apply IHm; eauto.
* eapply DE; apply IHm; eauto.
Qed.
Fixpoint L_con `{falsity_flag} (L : nat -> list form) (n : nat) : list (list form) :=
match n with
| 0 => [ nil ]
| S n => L_con L n ++ [ phi :: A | (pair phi A) ∈ (L n × L_con L n) ]
end.
Lemma enum_el X (p : X -> Prop) L x :
list_enumerator L p -> p x -> exists m, x el L m.
Proof.
firstorder.
Qed.
Arguments enum_el {X p L} x _ _.
Lemma enum_p X (p : X -> Prop) L x m :
list_enumerator L p -> x el L m -> p x.
Proof.
firstorder.
Qed.
Definition containsL `{falsity_flag} A (T : form -> Prop) :=
forall psi, psi el A -> T psi.
Lemma enum_containsL `{falsity_flag} T L :
cumulative L -> list_enumerator L T -> list_enumerator (L_con L) (fun A => containsL A T).
Proof with try (eapply cum_ge'; eauto; lia).
intros HL He. split.
- induction x as [| phi A]; intros HT.
+ exists 0. firstorder.
+ destruct IHA as [m1], (enum_el phi He) as [m2]. 1,2,3: firstorder.
exists (1 + m1 + m2). cbn. in_app 2. in_collect (phi, A)...
- intros [m]. induction m in x, H0 |-*; cbn in *.
+ destruct H0 as [<- | []]. firstorder.
+ inv_collect. apply IHm in H2. apply (enum_p He) in H0. unfold containsL in *. firstorder congruence.
Qed.
Fixpoint L_tded {p : peirce} {b : falsity_flag} (L : nat -> list form) (n : nat) : list form :=
match n with
| 0 => nil
| S n => L_tded L n ++ concat ([ L_ded A n | A ∈ L_con L n ])
end.
Lemma enum_tprv {p : peirce} {b : falsity_flag} T L :
list_enumerator L T -> list_enumerator (L_tded (cumul L)) (tprv T).
Proof with try (eapply cum_ge'; eauto; lia).
intros He.
assert (HL : list_enumerator (cumul L) T) by now apply list_enumerator_to_cumul.
split.
- intros (A & [m1] % (enum_el A (@enum_containsL _ _ (cumul L) (to_cumul_cumulative L) HL)) & [m2] % (enum_el x (enum_prv A))).
exists (1 + m1 + m2). cbn. in_app 2. eapply in_concat_iff. eexists. split. 2: in_collect A... idtac...
- intros [m]. induction m in x, H |-*; cbn in *. 1: contradiction. inv_collect. exists x1. split.
+ eapply (enum_p (enum_containsL (to_cumul_cumulative L) HL)); eauto.
+ eapply (enum_p (enum_prv x1)); eassumption.
Qed.
End Enumerability.
Section TheoryManipulation.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Context {p : peirce} {ff : falsity_flag}.
Context {HF : eq_dec syms} {HP : eq_dec preds}.
Lemma prv_T_impl T phi psi :
(T ⋄ phi) ⊢T psi -> T ⊢T (phi → psi).
Proof using HF HP ff p Σ_funcs Σ_preds.
intros (A & HA1 & HA2). exists (remove (dec_form _ _ _ _) phi A); split.
- intros f [[] % HA1 Hf2] % in_remove; subst; intuition.
- eapply II, Weak. 1: apply HA2. transitivity (phi :: A). 1: eauto. intros a [-> | Ha].
+ now left.
+ destruct (dec_form _ _ _ _ a phi) as [->|Hr]. 1:now left.
right. now eapply in_in_remove.
Qed.
Lemma prv_T_remove T phi psi :
T ⊢T phi -> T ⋄ phi ⊢T psi -> T ⊢T psi.
Proof using HF HP ff p Σ_funcs Σ_preds.
intros (A & HA1 & HA2) (B & HB1 & HB2) % prv_T_impl.
use_theory (A ++ B).
- now intros psi1 [H1|H2]%in_app_iff; [apply HA1 | apply HB1].
- eapply IE; eapply Weak. 1,3: eassumption.
all: intros x Hx; apply in_app_iff; eauto.
Qed.
Lemma prv_T_comp T phi psi xi :
T ⋄ phi ⊢T xi -> T ⋄ psi ⊢T phi -> T ⋄ psi ⊢T xi.
Proof using HF HP.
intros (A & HA1 & HA2) % prv_T_impl (B & HB1 & HB2) % prv_T_impl.
use_theory (psi :: A ++ B).
- now intros psi1 [->|[H1|H2]%in_app_iff]; [now right | |]; left; [apply HA1 | apply HB1].
- eapply IE. 1: eapply Weak. 1: apply HA2. 2: eapply IE.
2-3: eapply Weak. 2: eapply HB2. 3: apply (@Ctx _ _ _ _ (psi::nil)); now left.
+ intros a Ha; right; apply in_app_iff; now left.
+ intros b Hb; right; apply in_app_iff; now right.
+ intros x [->|[]]; now left.
Qed.
Lemma elem_prv T phi :
phi ∈ T -> T ⊢T phi.
Proof.
intros ?. use_theory [phi]. 2:apply Ctx; now left.
intros psi [->|[]]. apply H.
Qed.
Lemma Weak_T T1 T2 phi :
T1 ⊢T phi -> T1 ⊑ T2 -> T2 ⊢T phi.
Proof.
intros (A & HA1 & HA2) HT2. exists A. split. 2:easy. intros psi H; now apply HT2, HA1.
Qed.
End TheoryManipulation.
Section Closedness.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Context {p : peirce} {ff : falsity_flag}.
Definition capture_subs n x := $(x + n).
Lemma capture_extract n A phi :
A ⊢ subst_form (capture_subs n) (capture All n phi) -> A ⊢ subst_form (capture_subs 0) phi.
Proof.
induction n; comp; intuition. apply IHn. apply (AllE (var n)) in H. rewrite subst_comp in H.
erewrite subst_ext. 1: apply H. intros [| x]; unfold capture_subs; cbn; f_equal; lia.
Qed.
Context {Funcs_eq_dec : eq_dec Σ_funcs}.
Context {Preds_eq_dec : eq_dec Σ_preds}.
Lemma close_extract A phi :
A ⊢ close All phi -> A ⊢ phi.
Proof.
intros H. assert (Hclosed : closed (close All phi)) by apply close_closed.
unfold close in *. destruct (find_bounded phi) as [n Hn]; cbn in *.
erewrite <- subst_id with (sigma := fun x => $x) in H.
erewrite (@bounded_subst _ _ _ _ _ _ _ (capture_subs n)) in H.
2: exact Hclosed. 3:easy. 2: lia.
apply capture_extract in H. rewrite subst_id in H; intuition. unfold capture_subs. f_equal. lia.
Qed.
End Closedness.
Section DNFacts.
Context {Σ_funcs : funcs_signature}.
Context {Σ_preds : preds_signature}.
Lemma DN A phi: A ⊢C (¬ (¬ phi)) -> A ⊢C phi.
Proof.
intros H. eapply IE. 1: apply Pc. eapply II.
apply Exp. eapply IE. 2: apply Ctx; now left.
eapply Weak. 1: apply H. intros a Ha; now right.
Qed.
Lemma XMc A (phi chi : form) : (phi::A) ⊢C chi -> ((¬ phi)::A) ⊢C chi -> A ⊢C chi.
Proof.
intros HA HB. eapply IE with ((phi → chi) → (¬phi → chi) → chi).
- eapply II. eapply IE. 1: eapply IE. 1: apply Ctx; now left.
1-2: eapply Weak; [eapply II|]. 1: apply HA. 2: apply HB. 1-2: intros a Ha; now right.
- eapply DN. apply II. eapply IE. 1: apply Ctx; now left. eapply II. eapply II.
eapply IE. 1: apply Ctx; now left.
eapply II. eapply IE. 1: apply Ctx; do 3 right; now left. do 2 eapply II.
eapply IE. 1: apply Ctx; right; now left.
apply Ctx; now eauto.
Qed.
Lemma XM A (phi : form) : A ⊢C (phi ∨ ¬ phi).
Proof.
apply (@XMc _ phi).
- apply DI1, Ctx. now left.
- apply DI2, Ctx. now left.
Qed.
End DNFacts.
Section RefutationComp.
Context {Σ_funcs : funcs_signature} {Σ_preds : preds_signature}.
Context {HF : eq_dec Σ_funcs} {HP : eq_dec Σ_preds}.
Lemma refutation_prv T phi :
T ⊢TC phi <-> (T ⋄ ¬ phi) ⊢TC ⊥.
Proof using HF HP.
split.
- intros (A & HA1 & HA2). use_theory (¬ phi :: A).
+ intros psi [<-|Ha]. 1: now right. left. now apply HA1.
+ eapply IE. 1: apply Ctx; now left. eapply Weak. 1: apply HA2. intros a Ha; now right.
- intros (A & HA1 & HA2) % prv_T_impl. use_theory A. now apply DN.
Qed.
End RefutationComp.