NaturalDeduction: add nd_swap, nd_prod_split, and some tactics

[coq-hetmet.git] / src / NaturalDeduction.v

diff --git a/src/NaturalDeduction.v b/src/NaturalDeduction.v
index 3a06d66..4ecc166 100644 (file)
--- a/src/NaturalDeduction.v
+++ b/src/NaturalDeduction.v
@@ -479,6 +479,25 @@ Section Natural_Deduction.
     ; cndr_sndr  := sndr
     }.
 
     ; cndr_sndr  := sndr
     }.
 

+    (* a proof is Analytic if it does not use cut *)
+    (*
+    Definition Analytic_Rule : NDPredicate :=
+      fun h c f => forall c, not (snd_cut _ _ c = f).
+    Definition AnalyticND := NDPredicateClosure Analytic_Rule.
+
+    (* a proof system has cut elimination if, for every proof, there is an analytic proof with the same conclusion *)
+    Class CutElimination :=
+    { ce_eliminate : forall h c, h/⋯⋯/c -> h/⋯⋯/c
+    ; ce_analytic  : forall h c f, AnalyticND (ce_eliminate h c f)
+    }.
+
+    (* cut elimination is strong if the analytic proof is furthermore equivalent to the original proof *)
+    Class StrongCutElimination :=
+    { sce_ce       <: CutElimination
+    ; ce_strong     : forall h c f, f === ce_eliminate h c f
+    }.
+    *)
+

   End ContextND.
 
   Close Scope nd_scope.
   End ContextND.
 
   Close Scope nd_scope.

@@ -570,6 +589,44 @@ Section ND_Relation_Facts.
     intros; apply (ndr_builtfrom_structural f); auto.
     Defined.
 
     intros; apply (ndr_builtfrom_structural f); auto.
     Defined.
 

+  Ltac nd_prod_preserves_comp_ltac P EQV :=
+    match goal with
+      [ |- context [ (?A ** ?B) ;; (?C ** ?D) ] ] => 
+        set (@ndr_prod_preserves_comp _ _ EQV _ _ A _ _ B _ C _ D) as P
+    end.
+
+  Lemma nd_swap A B C D (f:ND _ A B) (g:ND _ C D) :
+    (f ** nd_id C) ;; (nd_id B ** g) ===
+    (nd_id A ** g) ;; (f ** nd_id D).
+    setoid_rewrite <- ndr_prod_preserves_comp.
+    setoid_rewrite ndr_comp_left_identity.
+    setoid_rewrite ndr_comp_right_identity.
+    reflexivity.
+    Qed.
+
+  (* this tactical searches the environment; setoid_rewrite doesn't seem to be able to do that properly sometimes *)
+  Ltac nd_swap_ltac P EQV :=
+    match goal with
+      [ |- context [ (?F ** nd_id _) ;; (nd_id _ ** ?G) ] ] => 
+        set (@nd_swap _ _ EQV _ _ _ _ F G) as P
+    end.
+
+  Lemma nd_prod_split_left A B C D (f:ND _ A B) (g:ND _ B C) :
+    nd_id D ** (f ;; g) ===
+    (nd_id D ** f) ;; (nd_id D ** g).
+    setoid_rewrite <- ndr_prod_preserves_comp.
+    setoid_rewrite ndr_comp_left_identity.
+    reflexivity.
+    Qed.
+
+  Lemma nd_prod_split_right A B C D (f:ND _ A B) (g:ND _ B C) :
+    (f ;; g) ** nd_id D ===
+    (f ** nd_id D) ;; (g ** nd_id D).
+    setoid_rewrite <- ndr_prod_preserves_comp.
+    setoid_rewrite ndr_comp_left_identity.
+    reflexivity.
+    Qed.
+

 End ND_Relation_Facts.
 
 (* a generalization of the procedure used to build (nd_id n) from nd_id0 and nd_id1 *)
 End ND_Relation_Facts.
 
 (* a generalization of the procedure used to build (nd_id n) from nd_id0 and nd_id1 *)