--- /dev/null
+(*#* #stop file *)
+
+Require Export subst0_defs.
+
+ Inductive pr0 : T -> T -> Prop :=
+(* structural rules *)
+ | pr0_refl : (t:?) (pr0 t t)
+ | pr0_thin : (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (k:?) (pr0 (TTail k u1 t1) (TTail k u2 t2))
+(* axiom rules *)
+ | pr0_beta : (k,v1,v2:?) (pr0 v1 v2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (pr0 (TTail (Flat Appl) v1 (TTail (Bind Abst) k t1))
+ (TTail (Bind Abbr) v2 t2))
+ | pr0_gamma : (b:?) ~b=Abst -> (v1,v2:?) (pr0 v1 v2) ->
+ (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (pr0 (TTail (Flat Appl) v1 (TTail (Bind b) u1 t1))
+ (TTail (Bind b) u2 (TTail (Flat Appl) (lift (1) (0) v2) t2)))
+ | pr0_delta : (u1,u2:?) (pr0 u1 u2) -> (t1,t2:?) (pr0 t1 t2) ->
+ (w:?) (subst0 (0) u2 t2 w) ->
+ (pr0 (TTail (Bind Abbr) u1 t1) (TTail (Bind Abbr) u2 w))
+ | pr0_zeta : (b:?) ~b=Abst -> (t1,t2:?) (pr0 t1 t2) ->
+ (u:?) (pr0 (TTail (Bind b) u (lift (1) (0) t1)) t2)
+ | pr0_eps : (t1,t2:?) (pr0 t1 t2) ->
+ (u:?) (pr0 (TTail (Flat Cast) u t1) t2).
+
+ Hint pr0 : ltlc := Constructors pr0.
+
+ Section pr0_gen. (********************************************************)
+
+ Theorem pr0_gen_sort : (x:?; n:?) (pr0 (TSort n) x) -> x = (TSort n).
+ Intros; Inversion H; XAuto.
+ Qed.
+
+ Theorem pr0_gen_bref : (x:?; n:?) (pr0 (TBRef n) x) -> x = (TBRef n).
+ Intros; Inversion H; XAuto.
+ Qed.
+
+ Theorem pr0_gen_abst : (u1,t1,x:?) (pr0 (TTail (Bind Abst) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ).
+
+ Intros; Inversion H; Clear H.
+(* case 1 : pr0_refl *)
+ XEAuto.
+(* case 2 : pr0_cont *)
+ XEAuto.
+(* case 3 : pr0_zeta *)
+ XElim H4; XAuto.
+ Qed.
+
+ Theorem pr0_gen_appl : (u1,t1,x:?) (pr0 (TTail (Flat Appl) u1 t1) x) -> (OR
+ (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ) |
+ (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
+ x = (TTail (Bind Abbr) u2 t2) &
+ (pr0 u1 u2) & (pr0 z1 t2)
+ ) |
+ (EX b y1 z1 u2 v2 t2 |
+ ~b=Abst &
+ t1 = (TTail (Bind b) y1 z1) &
+ x = (TTail (Bind b) v2 (TTail (Flat Appl) (lift (1) (0) u2) t2)) &
+ (pr0 u1 u2) & (pr0 y1 v2) & (pr0 z1 t2))
+ ).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ Theorem pr0_gen_cast : (u1,t1,x:?) (pr0 (TTail (Flat Cast) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
+ (pr0 u1 u2) & (pr0 t1 t2)
+ ) \/
+ (pr0 t1 x).
+ Intros; Inversion H; XEAuto.
+ Qed.
+
+ End pr0_gen.
+
+ Hints Resolve pr0_gen_sort pr0_gen_bref : ltlc.
+
+ Tactic Definition Pr0GenBase :=
+ Match Context With
+ | [ H: (pr0 (TTail (Bind Abst) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_abst ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros
+ | [ H: (pr0 (TTail (Flat Appl) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_appl ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; Intros H; XElim H; Intros
+ | [ H: (pr0 (TTail (Flat Cast) ?1 ?2) ?3) |- ? ] ->
+ LApply (pr0_gen_cast ?1 ?2 ?3); [ Clear H; Intros H | XAuto ];
+ XElim H; [ Intros H; XElim H; Intros | Intros ].