Require Import mathcomp.ssreflect.ssreflect.
Require Import CTL_def dags demo agreement.
Require Import hilbert hilbert_ref hilbert_LS hilbert_Eme90.
Require Import gen_def gen_dec gen_hsound gen_ref.
Require Import CTL_def dags demo agreement.
Require Import hilbert hilbert_ref hilbert_LS hilbert_Eme90.
Require Import gen_def gen_dec gen_hsound gen_ref.
Theorem IC_soundness s : IC.prv s -> forall (M:cmodel) (w:M), eval s w.
Theorem IC_LS_equivalence s : IC.prv s <-> LS.prv s.
Theorem IC_Eme90_equivalence s : IC.prv s <-> Eme90.prv s.
Soundness and Completeness for IC
Import IC.
Theorem soundness s : prv s -> forall (M:cmodel) (w:M), eval s w.
Theorem informative_completeness s :
( prv (~~: s) )
+ (exists2 M : fmodel, #|M| <= f_size s * 2^(4 * f_size s + 2) & exists (w:M), eval s w).
Corollary fin_completeness s : (forall (M:fmodel) (w:M), eval s w) -> prv s.
Theorem soundness s : prv s -> forall (M:cmodel) (w:M), eval s w.
Theorem informative_completeness s :
( prv (~~: s) )
+ (exists2 M : fmodel, #|M| <= f_size s * 2^(4 * f_size s + 2) & exists (w:M), eval s w).
Corollary fin_completeness s : (forall (M:fmodel) (w:M), eval s w) -> prv s.
Decidability and Small-Model-Property
Corollary prv_dec s : decidable (prv s).
Corollary sat_dec s : decidable (exists (M:cmodel) (w:M), eval s w).
Corollary valid_dec s : decidable (forall (M:cmodel) (w:M), eval s w).
Corollary small_models s :
(exists (M:cmodel) (w:M), eval s w) ->
(exists2 M : fmodel, #|M| <= f_size s * 2^(4 * f_size s + 2) & exists (w:M), eval s w).
Gentzen System
Theorem plain_soundness C : gen (C,aVoid) -> prv (~~: [af C]).
Theorem gen_complete C :
gen (C,aVoid) + (exists (M:fmodel) (w:M), forall s, s \in C -> eval (interp s) w).
Theorem gen_dec (A : clause * annot) : decidable (gen A).
Agreement of Path semantics with Inductive Semantics
Lemma evalP2 (M:fmodel) s (w : M) : (@satisfies M s w) <-> (@eval M s w).
Lemma fin_path_soundness s : prv s -> forall (M : fmodel) (w:M), @satisfies M s w.
Lemma sts_agreement : XM -> DC -> forall (M:sts) (w :M) s, eval s w <-> satisfies s w.
Soundness wrt. to Path semantics implies XM and DX
Lemma XM_required :
(forall s, prv s -> forall (M : sts) (w : M), satisfies s w) -> XM.
Lemma DC_required :
(forall s, prv s -> forall (M : sts) (w : M), satisfies s w) -> DC.
Agreement of coinductive AR and disjunctive AR implies LPO