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

Lemma truth_lemma_impp
(AL_DC: at_least derivable_closed cP)
(LIN_CD: forall x, Lindenbaum_constructable (cannot_derive x) cP)
(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)).