Require Import Undecidability.Synthetic.Definitions Undecidability.Synthetic.Undecidability.
Require Import Undecidability.FOL.ZF.
From Undecidability.FOL.Util Require Import FullTarski FullDeduction_facts Aczel_CE Aczel_TD ZF_model.
From Undecidability.FOL.Reductions Require Import PCPb_to_ZFeq PCPb_to_ZF PCPb_to_ZFD.
From Undecidability.PCP Require Import PCP PCP_undec Util.PCP_facts Reductions.PCPb_iff_dPCPb.
(* Semantic entailment in full ZF restricted to extensional models *)
Theorem PCPb_entailment_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> PCPb ⪯ entailment_ZF.
Proof.
intros H. exists solvable. intros B. apply PCP_ZF. apply H.
Qed.
Theorem undecidable_entailment_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> undecidable entailment_ZF.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZF.
Qed.
Corollary undecidable_model_entailment_ZF :
CE -> TD -> undecidable entailment_ZF.
Proof.
intros H1 H2. now apply undecidable_entailment_ZF, normaliser_model.
Qed.
(* Semantic entailment in ZF' restricted to extensional models *)
Theorem PCPb_entailment_ZF' :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, In psi ZF' -> rho ⊨ psi) -> PCPb ⪯ entailment_ZF'.
Proof.
intros H. exists solvable. intros B. apply PCP_ZF'. apply H.
Qed.
Theorem undecidable_entailment_ZF' :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, In psi ZF' -> rho ⊨ psi) -> undecidable entailment_ZF'.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZF'.
Qed.
Corollary undecidable_model_entailment_ZF' :
CE -> undecidable entailment_ZF'.
Proof.
intros H. now apply undecidable_entailment_ZF', extensionality_model.
Qed.
(* Semantic entailment in ZFeq' allowing for intensional models *)
Theorem PCPb_entailment_ZFeq' :
PCPb ⪯ entailment_ZFeq'.
Proof.
exists solvable. intros B. split; intros H.
- eapply PCP_ZFD, soundness in H. intros D M rho H'. apply H, H'.
- now apply PCP_ZFeq'; try apply intensional_model.
Qed.
Theorem undecidable_entailment_ZFeq' :
undecidable entailment_ZFeq'.
Proof.
apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZFeq'.
Qed.
(* Intuitionistic deduction in full ZFeq *)
Theorem PCPb_deduction_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> PCPb ⪯ deduction_ZF.
Proof.
intros (V & M & H1 & H2 & H3).
exists solvable. intros B. split.
- intros H % (@PCP_ZFD intu). exists ZFeq'. split; eauto using ZFeq.
- intros H'. specialize (tsoundness H'). clear H'. intros H'.
apply PCPb_iff_dPCPb. eapply PCP_ZF2; eauto using ZF.
apply (H' V M (fun _ => ∅)). intros psi [].
+ apply extensional_eq; eauto using ZF.
+ apply H3. constructor 2.
+ apply H3. constructor 3.
Qed.
Theorem undecidable_deduction_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> undecidable deduction_ZF.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_deduction_ZF.
Qed.
Corollary undecidable_model_deduction_ZF :
CE -> TD -> undecidable deduction_ZF.
Proof.
intros H1 H2. now apply undecidable_deduction_ZF, normaliser_model.
Qed.
(* Intuitionistic deduction in ZFeq' *)
Theorem PCPb_deduction_ZF' :
PCPb ⪯ deduction_ZF'.
Proof.
exists solvable. intros B. split; try apply PCP_ZFD.
intros H' % soundness. apply PCP_ZFeq'; try apply intensional_model.
intros D M rho H. apply H', H.
Qed.
Corollary undecidable_deduction_entailment_ZF' :
undecidable deduction_ZF'.
Proof.
apply (undecidability_from_reducibility PCPb_undec), PCPb_deduction_ZF'.
Qed.
Require Import Undecidability.FOL.ZF.
From Undecidability.FOL.Util Require Import FullTarski FullDeduction_facts Aczel_CE Aczel_TD ZF_model.
From Undecidability.FOL.Reductions Require Import PCPb_to_ZFeq PCPb_to_ZF PCPb_to_ZFD.
From Undecidability.PCP Require Import PCP PCP_undec Util.PCP_facts Reductions.PCPb_iff_dPCPb.
(* Semantic entailment in full ZF restricted to extensional models *)
Theorem PCPb_entailment_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> PCPb ⪯ entailment_ZF.
Proof.
intros H. exists solvable. intros B. apply PCP_ZF. apply H.
Qed.
Theorem undecidable_entailment_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> undecidable entailment_ZF.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZF.
Qed.
Corollary undecidable_model_entailment_ZF :
CE -> TD -> undecidable entailment_ZF.
Proof.
intros H1 H2. now apply undecidable_entailment_ZF, normaliser_model.
Qed.
(* Semantic entailment in ZF' restricted to extensional models *)
Theorem PCPb_entailment_ZF' :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, In psi ZF' -> rho ⊨ psi) -> PCPb ⪯ entailment_ZF'.
Proof.
intros H. exists solvable. intros B. apply PCP_ZF'. apply H.
Qed.
Theorem undecidable_entailment_ZF' :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, In psi ZF' -> rho ⊨ psi) -> undecidable entailment_ZF'.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZF'.
Qed.
Corollary undecidable_model_entailment_ZF' :
CE -> undecidable entailment_ZF'.
Proof.
intros H. now apply undecidable_entailment_ZF', extensionality_model.
Qed.
(* Semantic entailment in ZFeq' allowing for intensional models *)
Theorem PCPb_entailment_ZFeq' :
PCPb ⪯ entailment_ZFeq'.
Proof.
exists solvable. intros B. split; intros H.
- eapply PCP_ZFD, soundness in H. intros D M rho H'. apply H, H'.
- now apply PCP_ZFeq'; try apply intensional_model.
Qed.
Theorem undecidable_entailment_ZFeq' :
undecidable entailment_ZFeq'.
Proof.
apply (undecidability_from_reducibility PCPb_undec), PCPb_entailment_ZFeq'.
Qed.
(* Intuitionistic deduction in full ZFeq *)
Theorem PCPb_deduction_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> PCPb ⪯ deduction_ZF.
Proof.
intros (V & M & H1 & H2 & H3).
exists solvable. intros B. split.
- intros H % (@PCP_ZFD intu). exists ZFeq'. split; eauto using ZFeq.
- intros H'. specialize (tsoundness H'). clear H'. intros H'.
apply PCPb_iff_dPCPb. eapply PCP_ZF2; eauto using ZF.
apply (H' V M (fun _ => ∅)). intros psi [].
+ apply extensional_eq; eauto using ZF.
+ apply H3. constructor 2.
+ apply H3. constructor 3.
Qed.
Theorem undecidable_deduction_ZF :
(exists V (M : interp V), extensional M /\ standard M /\ forall rho psi, ZF psi -> rho ⊨ psi) -> undecidable deduction_ZF.
Proof.
intros H. now apply (undecidability_from_reducibility PCPb_undec), PCPb_deduction_ZF.
Qed.
Corollary undecidable_model_deduction_ZF :
CE -> TD -> undecidable deduction_ZF.
Proof.
intros H1 H2. now apply undecidable_deduction_ZF, normaliser_model.
Qed.
(* Intuitionistic deduction in ZFeq' *)
Theorem PCPb_deduction_ZF' :
PCPb ⪯ deduction_ZF'.
Proof.
exists solvable. intros B. split; try apply PCP_ZFD.
intros H' % soundness. apply PCP_ZFeq'; try apply intensional_model.
intros D M rho H. apply H', H.
Qed.
Corollary undecidable_deduction_entailment_ZF' :
undecidable deduction_ZF'.
Proof.
apply (undecidability_from_reducibility PCPb_undec), PCPb_deduction_ZF'.
Qed.