--- /dev/null
+(*#* #stop file *)
+
+Require Export ty0_defs.
+
+ Inductive csub0 [g:G] : C -> C -> Prop :=
+(* structural rules *)
+ | csub0_sort : (n:?) (csub0 g (CSort n) (CSort n))
+ | csub0_tail : (c1,c2:?) (csub0 g c1 c2) -> (k,u:?)
+ (csub0 g (CTail c1 k u) (CTail c2 k u))
+(* axioms *)
+ | csub0_void : (c1,c2:?) (csub0 g c1 c2) -> (b:?) ~b=Void -> (u1,u2:?)
+ (csub0 g (CTail c1 (Bind Void) u1) (CTail c2 (Bind b) u2))
+ | csub0_abst : (c1,c2:?) (csub0 g c1 c2) -> (u,t:?) (ty0 g c2 u t) ->
+ (csub0 g (CTail c1 (Bind Abst) t) (CTail c2 (Bind Abbr) u)).
+
+ Hint csub0 : ltlc := Constructors csub0.
+
+ Section csub0_props. (****************************************************)
+
+ Theorem csub0_refl : (g:?; c:?) (csub0 g c c).
+ XElim c; XAuto.
+ Qed.
+
+ End csub0_props.
+
+ Hints Resolve csub0_refl : ltlc.
+
+ Section csub0_drop. (*****************************************************)
+
+ Theorem csub0_drop_abbr : (g:?; n:?; c1,c2:?) (csub0 g c1 c2) -> (d1,u:?)
+ (drop n (0) c1 (CTail d1 (Bind Abbr) u)) ->
+ (EX d2 | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abbr) u))
+ ).
+ XElim n.
+(* case 1 : n = 0 *)
+ Intros; DropGenBase; Rewrite H0 in H; Inversion H; XEAuto.
+(* case 2 : n > 0 *)
+ Intros until 2; XElim H0.
+(* case 2.1 : csub0_sort *)
+ Intros; Inversion H0.
+(* case 2.2 : csub0_tail *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u0); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H1 d1 u0); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; XEAuto.
+(* case 2.3 : csub0_void *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+(* case 2.4 : csub0_abst *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 u0); [ Clear H; Intros H | XAuto ].
+ XElim H; XEAuto.
+ Qed.
+
+ Theorem csub0_drop_abst : (g:?; n:?; c1,c2:?) (csub0 g c1 c2) -> (d1,t:?)
+ (drop n (0) c1 (CTail d1 (Bind Abst) t)) ->
+ (EX d2 | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abst) t))
+
+ ) \/
+ (EX d2 u | (csub0 g d1 d2) &
+ (drop n (0) c2 (CTail d2 (Bind Abbr) u)) &
+ (ty0 g d2 u t)
+ ).
+ XElim n.
+(* case 1 : n = 0 *)
+ Intros; DropGenBase; Rewrite H0 in H; Inversion H; XEAuto.
+(* case 2 : n > 0 *)
+ Intros until 2; XElim H0.
+(* case 2.1 : csub0_sort *)
+ Intros; Inversion H0.
+(* case 2.2 : csub0_tail *)
+ XElim k; Intros; DropGenBase.
+(* case 2.2.1 : Bind *)
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+(* case 2.2.2 : Flat *)
+ LApply (H1 d1 t); [ Clear H1; Intros H1 | XAuto ].
+ XElim H1; Intros; XElim H1; XEAuto.
+(* case 2.3 : csub0_void *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+(* case 2.4 : csub0_abst *)
+ Intros; DropGenBase.
+ LApply (H c0 c3); [ Clear H; Intros H | XAuto ].
+ LApply (H d1 t0); [ Clear H; Intros H | XAuto ].
+ XElim H; Intros; XElim H; XEAuto.
+ Qed.
+
+ End csub0_drop.
+
+ Tactic Definition CSub0Drop :=
+ Match Context With
+ | [ H1: (csub0 ?1 ?2 ?3);
+ H2: (drop ?4 (0) ?2 (CTail ?5 (Bind Abbr) ?6)) |- ? ] ->
+ LApply (csub0_drop_abbr ?1 ?4 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros
+ | [ H1: (csub0 ?1 ?2 ?3);
+ H2: (drop ?4 (0) ?2 (CTail ?5 (Bind Abst) ?6)) |- ? ] ->
+ LApply (csub0_drop_abst ?1 ?4 ?2 ?3); [ Clear H1; Intros H1 | XAuto ];
+ LApply (H1 ?5 ?6); [ Clear H1 H2; Intros H1 | XAuto ];
+ XElim H1; Intros H1; XElim H1; Intros.
+
+ Section csub0_pc3. (*****************************************************)
+
+ Theorem csub0_pr2 : (g:?; c1:?; t1,t2:?) (pr2 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pr2 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr2_pr0 *)
+ XAuto.
+(* case 2 : pr2_delta *)
+ CSub0Drop; XEAuto.
+ Qed.
+
+ Theorem csub0_pc2 : (g:?; c1:?; t1,t2:?) (pc2 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pc2 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pc2_r *)
+ Apply pc2_r; EApply csub0_pr2; XEAuto.
+(* case 2 : pc2_x *)
+ Apply pc2_x; EApply csub0_pr2; XEAuto.
+ Qed.
+
+ Theorem csub0_pc3 : (g:?; c1:?; t1,t2:?) (pc3 c1 t1 t2) ->
+ (c2:?) (csub0 g c1 c2) -> (pc3 c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pc3_r *)
+ XAuto.
+(* case 2 : pc3_u *)
+ EApply pc3_u; [ EApply csub0_pc2; XEAuto | XAuto ].
+ Qed.
+
+ End csub0_pc3.
+
+ Hints Resolve csub0_pc3 : ltlc.
+
+ Section csub0_ty0. (*****************************************************)
+
+ Theorem csub0_ty0 : (g:?; c1:?; t1,t2:?) (ty0 g c1 t1 t2) ->
+ (c2:?) (wf0 g c2) -> (csub0 g c1 c2) ->
+ (ty0 g c2 t1 t2).
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ EApply ty0_conv; XEAuto.
+(* case 2 : ty0_sort *)
+ XEAuto.
+(* case 3 : ty0_abbr *)
+ CSub0Drop; EApply ty0_abbr; XEAuto.
+(* case 4 : ty0_abst *)
+ CSub0Drop; [ EApply ty0_abst | EApply ty0_abbr ]; XEAuto.
+(* case 5 : ty0_bind *)
+ EApply ty0_bind; XEAuto.
+(* case 6 : ty0_appl *)
+ EApply ty0_appl; XEAuto.
+(* case 7 : ty0_cast *)
+ EApply ty0_cast; XAuto.
+ Qed.
+
+ Theorem csub0_ty0_ld : (g:?; c:?; u,v:?) (ty0 g c u v) -> (t1,t2:?)
+ (ty0 g (CTail c (Bind Abst) v) t1 t2) ->
+ (ty0 g (CTail c (Bind Abbr) u) t1 t2).
+ Intros; EApply csub0_ty0; XEAuto.
+ Qed.
+
+ End csub0_ty0.
+
+ Hints Resolve csub0_ty0 csub0_ty0_ld : ltlc.
+
+ Tactic Definition CSub0Ty0 :=
+ Match Context With
+ [ _: (ty0 ?1 ?2 ?4 ?); _: (ty0 ?1 ?2 ?3 ?7); _: (pc3 ?2 ?4 ?7);
+ H: (ty0 ?1 (CTail ?2 (Bind Abst) ?4) ?5 ?6) |- ? ] ->
+ LApply (csub0_ty0_ld ?1 ?2 ?3 ?4); [ Intros H_x | EApply ty0_conv; XEAuto ];
+ LApply (H_x ?5 ?6); [ Clear H_x H; Intros | XAuto ].
projects / helm.git / blobdiff