-(*#* #stop file *)
-
Require Export drop_defs.
Require Export pr0_defs.
- Inductive pr2 [c:C; t1,t2:T] : Prop :=
+(*#* #caption "current axioms for the relation $\\PrS{}{}{}$",
+ "context-free case", "context-dependent $\\delta$-expansion"
+*)
+(*#* #cap #cap c, d, t, t1, t2 #alpha u in V *)
+
+ Inductive pr2 [c:C; t1:T] : T -> Prop :=
(* structural rules *)
- | pr2_pr0 : (pr0 t1 t2) -> (pr2 c t1 t2)
+ | pr2_free : (t2:?) (pr0 t1 t2) -> (pr2 c t1 t2)
(* axiom rules *)
- | pr2_delta : (d:?; u:?; i:?)
- (drop i (0) c (CTail d (Bind Abbr) u)) ->
- (subst0 i u t1 t2) -> (pr2 c t1 t2).
+ | pr2_delta: (d:?; u:?; i:?)
+ (drop i (0) c (CTail d (Bind Abbr) u)) ->
+ (t2:?) (pr0 t1 t2) -> (t:?) (subst0 i u t2 t) ->
+ (pr2 c t1 t).
+
+(*#* #stop file *)
Hint pr2 : ltlc := Constructors pr2.
Section pr2_gen_base. (***************************************************)
- Theorem pr2_gen_sort : (c:?; x:?; n:?) (pr2 c (TSort n) x) ->
- x = (TSort n).
- Intros; Inversion H;
- Try Subst0GenBase; XEAuto.
+ Theorem pr2_gen_sort: (c:?; x:?; n:?) (pr2 c (TSort n) x) ->
+ x = (TSort n).
+ Intros; Inversion H; Pr0GenBase;
+ [ XAuto | Rewrite H1 in H2; Subst0GenBase ].
Qed.
- Theorem pr2_gen_bref : (c:?; x:?; n:?) (pr2 c (TBRef n) x) ->
- x = (TBRef n) \/
- (EX d u | (drop n (0) c (CTail d (Bind Abbr) u)) &
- x = (lift (S n) (0) u)
- ).
- Intros; Inversion H;
- Try Subst0GenBase; Try Rewrite <- H1 in H0; XEAuto.
+ Theorem pr2_gen_lref: (c:?; x:?; n:?) (pr2 c (TLRef n) x) ->
+ x = (TLRef n) \/
+ (EX d u | (drop n (0) c (CTail d (Bind Abbr) u)) &
+ x = (lift (S n) (0) u)
+ ).
+ Intros; Inversion H; Pr0GenBase;
+ [ XAuto | Rewrite H1 in H2; Subst0GenBase; Rewrite <- H2 in H0; XEAuto ].
Qed.
- Theorem pr2_gen_abst : (c:?; u1,t1,x:?)
- (pr2 c (TTail (Bind Abst) u1 t1) x) ->
- (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
- (pr2 c u1 u2) & (b:?; u:?)
- (pr2 (CTail c (Bind b) u) t1 t2)
- ).
- Intros; Inversion H;
- Try Pr0GenBase; Try Subst0GenBase; XDEAuto 6.
+ Theorem pr2_gen_abst: (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Bind Abst) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Bind Abst) u2 t2) &
+ (pr2 c u1 u2) & (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) t1 t2)
+ ).
+ Intros; Inversion H; Pr0GenBase;
+ [ XEAuto | Rewrite H1 in H2; Subst0GenBase; XDEAuto 6 ].
Qed.
- Theorem pr2_gen_appl : (c:?; u1,t1,x:?)
- (pr2 c (TTail (Flat Appl) u1 t1) x) -> (OR
- (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
- (pr2 c u1 u2) & (pr2 c 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;
- Try Pr0GenBase; Try Subst0GenBase; XEAuto.
+ Theorem pr2_gen_appl: (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Flat Appl) u1 t1) x) -> (OR
+ (EX u2 t2 | x = (TTail (Flat Appl) u2 t2) &
+ (pr2 c u1 u2) & (pr2 c t1 t2)
+ ) |
+ (EX y1 z1 u2 t2 | t1 = (TTail (Bind Abst) y1 z1) &
+ x = (TTail (Bind Abbr) u2 t2) &
+ (pr2 c u1 u2) & (b:?; u:?)
+ (pr2 (CTail c (Bind b) u) z1 t2)
+ ) |
+ (EX b y1 z1 z2 u2 v2 t2 |
+ ~b=Abst &
+ t1 = (TTail (Bind b) y1 z1) &
+ x = (TTail (Bind b) v2 z2) &
+ (pr2 c u1 u2) & (pr2 c y1 v2) & (pr0 z1 t2))
+ ).
+ Intros; Inversion H; Pr0GenBase;
+ Try Rewrite H1 in H2; Try Rewrite H4 in H2; Try Rewrite H5 in H2;
+ Try Subst0GenBase; XDEAuto 7.
Qed.
- Theorem pr2_gen_cast : (c:?; u1,t1,x:?)
- (pr2 c (TTail (Flat Cast) u1 t1) x) ->
- (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
- (pr2 c u1 u2) & (pr2 c t1 t2)
- ) \/
- (pr0 t1 x).
- Intros; Inversion H;
- Try Pr0GenBase; Try Subst0GenBase; XEAuto.
+ Theorem pr2_gen_cast: (c:?; u1,t1,x:?)
+ (pr2 c (TTail (Flat Cast) u1 t1) x) ->
+ (EX u2 t2 | x = (TTail (Flat Cast) u2 t2) &
+ (pr2 c u1 u2) & (pr2 c t1 t2)
+ ) \/
+ (pr2 c t1 x).
+ Intros; Inversion H; Pr0GenBase;
+ Try Rewrite H1 in H2; Try Subst0GenBase; XEAuto.
Qed.
End pr2_gen_base.
Tactic Definition Pr2GenBase :=
Match Context With
- | [ H: (pr2 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
+ | [ H: (pr2 ?1 (TSort ?2) ?3) |- ? ] ->
+ LApply (pr2_gen_sort ?1 ?3 ?2); [ Clear H; Intros | XAuto ]
+ | [ H: (pr2 ?1 (TLRef ?2) ?3) |- ? ] ->
+ LApply (pr2_gen_lref ?1 ?3 ?2); [ Clear H; Intros H | XAuto ];
+ XDecompose H
+ | [ H: (pr2 ?1 (TTail (Bind Abst) ?2 ?3) ?4) |- ? ] ->
LApply (pr2_gen_abst ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
- XElim H; Intros.
+ XDecompose H
+ | [ H: (pr2 ?1 (TTail (Flat Appl) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr2_gen_appl ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XDecompose H
+ | [ H: (pr2 ?1 (TTail (Flat Cast) ?2 ?3) ?4) |- ? ] ->
+ LApply (pr2_gen_cast ?1 ?2 ?3 ?4); [ Clear H; Intros H | XAuto ];
+ XDecompose H.
Section pr2_props. (******************************************************)
- Theorem pr2_thin_dx : (c:?; t1,t2:?) (pr2 c t1 t2) ->
- (u:?; f:?) (pr2 c (TTail (Flat f) u t1)
- (TTail (Flat f) u t2)).
- Intros; Inversion H; XEAuto.
+ Theorem pr2_thin_dx: (c:?; t1,t2:?) (pr2 c t1 t2) ->
+ (u:?; f:?) (pr2 c (TTail (Flat f) u t1)
+ (TTail (Flat f) u t2)).
+ Intros; XElim H; XEAuto.
Qed.
- Theorem pr2_tail_1 : (c:?; u1,u2:?) (pr2 c u1 u2) ->
- (k:?; t:?) (pr2 c (TTail k u1 t) (TTail k u2 t)).
- Intros; Inversion H; XEAuto.
+ Theorem pr2_tail_1: (c:?; u1,u2:?) (pr2 c u1 u2) ->
+ (k:?; t:?) (pr2 c (TTail k u1 t) (TTail k u2 t)).
+ Intros; XElim H; XEAuto.
Qed.
- Theorem pr2_tail_2 : (c:?; u,t1,t2:?; k:?) (pr2 (CTail c k u) t1 t2) ->
- (pr2 c (TTail k u t1) (TTail k u t2)).
- XElim k; Intros;
- (Inversion H; [ XAuto | Clear H ];
- (NewInduction i; DropGenBase; [ Inversion H; XEAuto | XEAuto ])).
+ Theorem pr2_tail_2: (c:?; u,t1,t2:?; k:?) (pr2 (CTail c k u) t1 t2) ->
+ (pr2 c (TTail k u t1) (TTail k u t2)).
+ XElim k; Intros; (
+ XElim H; [ XAuto | XElim i; Intros; DropGenBase; CGenBase; XEAuto ]).
Qed.
-
+
Hints Resolve pr2_tail_2 : ltlc.
- Theorem pr2_shift : (i:?; c,e:?) (drop i (0) c e) ->
- (t1,t2:?) (pr2 c t1 t2) ->
- (pr2 e (app c i t1) (app c i t2)).
+ Theorem pr2_shift: (i:?; c,e:?) (drop i (0) c e) ->
+ (t1,t2:?) (pr2 c t1 t2) ->
+ (pr2 e (app c i t1) (app c i t2)).
XElim i.
-(* case 1 : i = 0 *)
- Intros.
- DropGenBase; Rewrite H in H0.
+(* case 1: i = 0 *)
+ Intros; DropGenBase; Rewrite H in H0.
Repeat Rewrite app_O; XAuto.
-(* case 2 : i > 0 *)
+(* case 2: i > 0 *)
XElim c.
-(* case 2.1 : CSort *)
+(* case 2.1: CSort *)
Intros; DropGenBase; Rewrite H0; XAuto.
-(* case 2.2 : CTail *)
+(* case 2.2: CTail *)
XElim k; Intros; Simpl; DropGenBase; XAuto.
Qed.
-
+
End pr2_props.
Hints Resolve pr2_thin_dx pr2_tail_1 pr2_tail_2 pr2_shift : ltlc.
projects / helm.git / blobdiff