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Frame_Validation.v
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Frame_Validation.v
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Require Import Modal_Library Classical.
Theorem validation_frame_reflexivo_ida:
forall M Ψ,
reflexivity_frame (F M) ->
(M |= .[] Ψ .-> Ψ).
Proof.
intros;
unfold validate_model in *;
simpl in *; intuition.
Qed.
Theorem validation_frame_reflexivo_volta:
forall M Ψ,
(M |= .[] Ψ .-> Ψ) ->
reflexivity_frame (F M).
Proof.
Admitted.
Theorem validation_frame_transitivo_ida:
forall M φ,
transitivity_frame (F M) ->
(M |= .[]φ .-> .[].[]φ).
Proof.
unfold validate_model, transitivity_frame.
simpl; intros.
apply H0.
eapply H; split.
- apply H1.
- assumption.
Qed.
Theorem validation_frame_transitivo_volta:
forall M φ,
(M |= .[]φ .-> .[].[]φ) ->
transitivity_frame (F M).
Proof.
Admitted.
Theorem validation_frame_simetria_ida:
forall M φ,
simmetry_frame (F M) ->
(M |= φ .-> .[] .<> φ).
Proof.
unfold validate_model, simmetry_frame.
simpl in *; intros; exists w.
apply and_comm; split.
- apply H0.
- eauto.
Qed.
Theorem validation_frame_simetria_volta:
forall M φ,
(M |= φ .-> .[] .<> φ) ->
simmetry_frame (F M).
Proof.
Admitted.
Theorem validation_frame_eucliadiana_ida:
forall M φ,
euclidian_frame (F M) ->
(M |= .<> φ .-> .[] .<> φ).
Proof.
unfold euclidian_frame, validate_model.
simpl in *; intros.
edestruct H0.
exists x; split.
- eapply H; split.
* apply H1.
* intuition.
- intuition.
Qed.
Theorem validation_frame_eucliadiana_volta:
forall M φ,
(M |= .<> φ .-> .[] .<> φ) ->
euclidian_frame (F M).
Proof.
Admitted.
Theorem validation_frame_serial_ida:
forall M φ,
serial_frame (F M) ->
(M |= .[] φ .-> .<> φ).
Proof.
unfold validate_model, serial_frame.
simpl in *; intros.
edestruct H.
exists x; split.
- apply H1.
- eauto.
Qed.
Theorem validation_frame_serial_volta:
forall M φ,
(M |= .[] φ .-> .<> φ) ->
serial_frame (F M).
Proof.
Admitted.
Theorem validation_frame_funcional_ida:
forall M φ,
functional_frame (F M) ->
(M |= .<> φ .-> .[] φ).
Proof.
intros; unfold validate_model;
unfold functional_frame in *.
intros w H0 w1 H1.
edestruct H0.
edestruct H with (w:=w) (w':=w1) (w'':=x).
- split.
+ apply H1.
+ intuition.
- intuition.
Qed.
Theorem validation_frame_funcional_volta:
forall M φ,
(M |= .<> φ .-> .[] φ) ->
functional_frame (F M).
Proof.
Admitted.
Theorem validation_frame_densa_ida:
forall M φ,
dense_frame (F M) ->
(M |= .[] .[] φ .-> .[] φ).
Proof.
Admitted.
Theorem validation_frame_densa_volta:
forall M φ,
(M |= .[] .[] φ .-> .[] φ) ->
dense_frame (F M).
Proof.
Admitted.
Theorem validation_frame_convergente_ida:
forall M φ,
convergente_frame (F M) ->
(M |= .<> .[] φ .-> .[] .<> φ).
Proof.
unfold convergente_frame, validate_model.
simpl in *; intros.
edestruct H0.
destruct H with (w:=w) (x:=x) (y:=w').
destruct H0; auto.
exists x0.
split; intuition.
Qed.
Theorem validation_frame_convergente_volta:
forall (M: Model) (φ: modalFormula),
(M |= .<> .[] φ .-> .[] .<> φ) ->
convergente_frame (F M).
Proof.
Admitted.