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PA.v
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(* * Peano Arithmetic *)
(* ** Problems in PA *)
From Undecidability.FOL.Arithmetics Require Export Signature FA Robinson PA.
Import Vector.VectorNotations.
Require Import List.
Notation extensional M :=
(forall x y, @i_atom _ _ _ M Eq ([x ; y]) <-> x = y).
(* Semantic entailment restricted to FA *)
Definition entailment_FA phi :=
valid_ctx FAeq phi.
(* Deductive entailment restricted to intuitionistic rules and FA. *)
Definition deduction_FA phi :=
FAeq ⊢I phi.
(* Semantic entailment restricted to Q *)
Definition entailment_Q phi :=
valid_ctx Qeq phi.
(* Deductive entailment restricted to intuitionistic rules and FA. *)
Definition deduction_Q phi :=
Qeq ⊢I phi.
(* Semantic entailment for PA *)
Definition entailment_PA phi :=
forall D (I : interp D) rho, (forall psi rho, PAeq psi -> rho ⊨ psi) -> rho ⊨ phi.
(* Semantic entailment restricted to extensional models of PA *)
Definition ext_entailment_PA phi :=
forall D (I : interp D) rho, extensional I -> (forall psi rho, PA psi -> rho ⊨ psi) -> rho ⊨ phi.
(* Deductive entailment restricted to intuitionistic rules. *)
Definition deduction_PA phi :=
PAeq ⊢TI phi.