Local Open Scope logic_base.
Local Open Scope syntax.
Local Open Scope kripke_model.
Section TruthLemma.
Context {L: Language}
{minL: MinimunLanguage L}
{pL: PropositionalLanguage L}
{Gamma: ProofTheory L}
{bSC: BasicSequentCalculus L Gamma}
{minSC: MinimunSequentCalculus L Gamma}
{ipSC: IntuitionisticPropositionalSequentCalculus L Gamma}
{MD: Model}
{kMD: KripkeModel MD}
{M: Kmodel}
{R: Relation (Kworlds M)}
{SM: Semantics L MD}
{kminSM: KripkeMinimunSemantics L MD M SM}
{kpSM: KripkePropositionalSemantics L MD M SM}.
Context (P: context -> Prop)
(rel: bijection (Kworlds M) (sig P)).
Hypothesis H_R: forall m n Phi Psi, rel m Phi -> rel n Psi -> (m <= n <-> Included _ (proj1_sig Phi) (proj1_sig Psi)).

(* Specifications: *)
Lemma truth_lemma_falsep (AL_CONSI: at_least consistent P):
forall m Phi, rel m Phi -> (KRIPKE: M, m |= falsep <-> proj1_sig Phi falsep).
Lemma truth_lemma_andp
(AL_DC: at_least derivable_closed P)
(x y: expr)
(IHx: forall m Phi, rel m Phi -> (KRIPKE: M, m |= x <-> proj1_sig Phi x))
(IHy: forall m Phi, rel m Phi -> (KRIPKE: M, m |= y <-> proj1_sig Phi y)):
forall m Phi, rel m Phi -> (KRIPKE: M, m |= x && y <-> proj1_sig Phi (x && y)).
Lemma truth_lemma_orp
(AL_DC: at_least derivable_closed P)
(AL_OW: at_least orp_witnessed P)
(x y: expr)
(IHx: forall m Phi, rel m Phi -> (KRIPKE: M, m |= x <-> proj1_sig Phi x))
(IHy: forall m Phi, rel m Phi -> (KRIPKE: M, m |= y <-> proj1_sig Phi y)):
forall m Phi, rel m Phi -> (KRIPKE: M, m |= x || y <-> proj1_sig Phi (x || y)).