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ArrowGentzen.v
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(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(****************************************************************************)
(* Signes Project *)
(* 2002-2003 *)
(* Houda ANOUN *)
(* Pierre Casteran *)
(* LaBRI/INRIA *)
(****************************************************************************)
Require Export Sequent.
Set Implicit Arguments.
Unset Strict Implicit.
Section arrowGen.
Variable Atoms : Set.
(* Auxiliary lemma that will be used to prove that gentzen presentation
is equivalent to the axiomatic one *)
Definition replace_arrow :
forall (T1 T2 Gamma Gamma' : Term Atoms) (X : arrow_extension),
replace Gamma Gamma' T1 T2 ->
arrow X (deltaTranslation T2) (deltaTranslation T1) ->
arrow X (deltaTranslation Gamma') (deltaTranslation Gamma).
simple induction 1.
intros; auto with ctl.
intros.
simpl in |- *.
apply Dot_mono_left.
auto.
intros.
simpl in |- *.
apply Dot_mono_right.
auto.
Defined.
Definition replace_arrow' :
forall (T1 T2 Gamma Gamma' : Term Atoms) (X : arrow_extension)
(C : Form Atoms),
replace Gamma Gamma' T1 T2 ->
arrow X (deltaTranslation T2) (deltaTranslation T1) ->
arrow X (deltaTranslation Gamma) C -> arrow X (deltaTranslation Gamma') C.
intros T1 T2 Gamma.
intros.
apply comp with (deltaTranslation Gamma).
eapply replace_arrow; eauto.
assumption.
Defined.
(* from axiomatic presentation to sequent calculus *)
Definition arrowToGentzenExt (X : arrow_extension)
(E : gentzen_extension) :=
forall A B : Form Atoms, X Atoms A B -> gentzenSequent E (OneForm A) B.
Definition NLToNL_Sequent : arrowToGentzenExt NL NL_Sequent.
unfold arrowToGentzenExt in |- *.
simple induction 1.
Defined.
Definition NLPToNLP_Sequent : arrowToGentzenExt NLP NLP_Sequent.
unfold arrowToGentzenExt in |- *.
simple induction 1.
intros.
apply NLPextensionSimplDot.
apply genExtendsRef.
apply Ax.
Defined.
Definition LToL_Sequent : arrowToGentzenExt L L_Sequent.
unfold arrowToGentzenExt in |- *.
simple induction 1.
intros.
apply LextensionSimplDot'.
apply genExtendsRef.
apply Ax.
intros.
apply LextensionSimplDot.
apply genExtendsRef.
apply Ax.
Defined.
Definition LPToLP_Sequent : arrowToGentzenExt LP LP_Sequent.
unfold arrowToGentzenExt in |- *.
intros A B H.
unfold LP in H.
unfold add_extension in H.
elim H.
intro.
apply gentzenExtends with NLP_Sequent.
apply LPextendsNLP.
apply genExtendsRef.
apply NLPToNLP_Sequent.
assumption.
intro.
apply gentzenExtends with L_Sequent.
apply LPextendsL.
apply genExtendsRef.
apply LToL_Sequent.
assumption.
Defined.
Definition arrowToGentzen :
forall (A B : Form Atoms) (E : gentzen_extension) (X : arrow_extension),
arrowToGentzenExt X E ->
arrow X A B -> gentzenSequent E (OneForm A) B.
intros A B E X H0 H.
elim H.
apply Ax.
intros.
eapply CutRuleSimpl; eauto.
intros.
apply RightSlashDot.
assumption.
intros.
apply DotRightSlash'.
assumption.
intros.
apply RightBackslashDot.
assumption.
intros.
apply DotRightBackslash'.
assumption.
intros.
unfold arrowToGentzenExt in H0.
auto.
Defined.
(* particular cases for NLP, L; LP and NL systems *)
Definition arrowToGentzenNL :
forall A B : Form Atoms,
arrow NL A B -> gentzenSequent NL_Sequent (OneForm A) B.
intros.
apply arrowToGentzen with NL.
apply NLToNL_Sequent.
assumption.
Defined.
Definition arrowToGentzenNLP :
forall A B : Form Atoms,
arrow NLP A B -> gentzenSequent NLP_Sequent (OneForm A) B.
intros.
apply arrowToGentzen with NLP.
apply NLPToNLP_Sequent.
assumption.
Defined.
Definition arrowToGentzenL :
forall A B : Form Atoms,
arrow L A B -> gentzenSequent L_Sequent (OneForm A) B.
intros.
apply arrowToGentzen with L.
apply LToL_Sequent.
assumption.
Defined.
Definition arrowToGentzenLP :
forall A B : Form Atoms,
arrow LP A B -> gentzenSequent LP_Sequent (OneForm A) B.
intros.
apply arrowToGentzen with LP.
apply LPToLP_Sequent.
assumption.
Defined.
(* from sequent calculus to axiomatic presentation *)
(* condition on structural rules *)
Definition gentzenToArrowExt (E : gentzen_extension)
(X : arrow_extension) :=
forall T1 T2 : Term Atoms,
E Atoms T1 T2 -> X Atoms (deltaTranslation T2) (deltaTranslation T1).
Definition NL_SequentToNL : gentzenToArrowExt NL_Sequent NL.
unfold gentzenToArrowExt in |- *.
intros.
case H.
Defined.
Definition NLP_SequentToNLP : gentzenToArrowExt NLP_Sequent NLP.
unfold gentzenToArrowExt in |- *.
simple induction 1.
simpl in |- *.
constructor 1.
Defined.
Definition L_SequentToL : gentzenToArrowExt L_Sequent L.
unfold gentzenToArrowExt in |- *.
simple induction 1.
simpl in |- *.
constructor 2.
simpl in |- *.
constructor 1.
Defined.
Definition LP_SequentToLP : gentzenToArrowExt LP_Sequent LP.
unfold gentzenToArrowExt in |- *.
simple induction 1.
unfold LP in |- *.
unfold add_extension in |- *.
intro.
left.
apply NLP_SequentToNLP.
assumption.
intro.
unfold LP in |- *.
unfold add_extension in |- *.
right.
apply L_SequentToL.
assumption.
Defined.
Definition gentzenToArrow :
forall (T : Term Atoms) (A : Form Atoms) (X : arrow_extension)
(E : gentzen_extension),
gentzenToArrowExt E X ->
gentzenSequent E T A -> arrow X (deltaTranslation T) A.
intros T A X E H0 H.
elim H.
intros; simpl in |- *; auto with ctl.
intros; simpl in |- *; auto with ctl.
intros; simpl in |- *; auto with ctl.
intros; simpl in |- *; apply Dot_mono; assumption.
intros Delta Gamma.
intros.
eapply replace_arrow'; eauto.
simpl in |- *.
apply beta'.
apply Slash_antimono_right.
assumption.
intros Delta Gamma.
intros.
eapply replace_arrow'; eauto.
simpl in |- *.
apply gamma'.
apply Backslash_antimono_left.
assumption.
intro Gamma; intros.
eapply replace_arrow'; eauto.
simpl in |- *.
apply one.
intros Delta Gamma; intros.
eapply replace_arrow'; eauto.
intros.
eapply replace_arrow'; eauto.
unfold gentzenToArrowExt in H0.
constructor 7.
auto.
Defined.
Definition NLGentzenToArrow :
forall (T : Term Atoms) (A : Form Atoms),
gentzenSequent NL_Sequent T A -> arrow NL (deltaTranslation T) A.
intros.
eapply gentzenToArrow.
apply NL_SequentToNL.
assumption.
Defined.
Definition NLPGentzenToArrow :
forall (T : Term Atoms) (A : Form Atoms),
gentzenSequent NLP_Sequent T A -> arrow NLP (deltaTranslation T) A.
intros.
eapply gentzenToArrow.
apply NLP_SequentToNLP.
assumption.
Defined.
Definition LGentzenToArrow :
forall (T : Term Atoms) (A : Form Atoms),
gentzenSequent L_Sequent T A -> arrow L (deltaTranslation T) A.
intros.
eapply gentzenToArrow.
apply L_SequentToL.
assumption.
Defined.
Definition LPGentzenToArrow :
forall (T : Term Atoms) (A : Form Atoms),
gentzenSequent LP_Sequent T A -> arrow LP (deltaTranslation T) A.
intros.
eapply gentzenToArrow.
apply LP_SequentToLP.
assumption.
Defined.
End arrowGen.