Require Import Relations Omega.
Require Import mathcomp.ssreflect.ssreflect.

Require Import Kstar_def demo hilbert_ref gen_def gen_ref.

Main Results

Soundness and Completeness of the Hilbert System

Lemma soundness s : prv s -> forall (M:cmodel) (w:M), eval s w.

Theorem informative_completeness s :
    prv (fImp s fF)
  + (exists2 M:fmodel, #|M| <= 2^(2 * f_size s) & exists w:M, eval s w).

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| <= 2^(2 * f_size s) & exists w:M, eval s w).

Correctness of the Gentzen System
Equivalence to Segerberg Style Hilbert System

Theorem prv_Sprv s : prv s <-> S.prv s.