6 Section csub0_pc3. (*****************************************************)
8 Theorem csub0_pr2: (g:?; c1:?; t1,t2:?) (pr2 c1 t1 t2) ->
9 (c2:?) (csub0 g c1 c2) -> (pr2 c2 t1 t2).
10 Intros until 1; XElim H; Intros.
11 (* case 1: pr2_free *)
13 (* case 2: pr2_delta *)
17 Hints Resolve csub0_pr2.
21 Theorem csub0_pc3: (g:?; c1:?; t1,t2:?) (pc3 c1 t1 t2) ->
22 (c2:?) (csub0 g c1 c2) -> (pc3 c2 t1 t2).
23 Intros until 1; XElimUsing pc3_ind_left H; XEAuto.
28 Hints Resolve csub0_pc3 : ltlc.
30 Section csub0_ty0. (*****************************************************)
32 Theorem csub0_ty0: (g:?; c1:?; t1,t2:?) (ty0 g c1 t1 t2) ->
33 (c2:?) (wf0 g c2) -> (csub0 g c1 c2) ->
35 Intros until 1; XElim H; Intros.
36 (* case 1: ty0_conv *)
37 EApply ty0_conv; XEAuto.
38 (* case 2: ty0_sort *)
40 (* case 3: ty0_abbr *)
41 CSub0Drop; EApply ty0_abbr; XEAuto.
42 (* case 4: ty0_abst *)
43 CSub0Drop; [ EApply ty0_abst | EApply ty0_abbr ]; XEAuto.
44 (* case 5: ty0_bind *)
45 EApply ty0_bind; XEAuto.
46 (* case 6: ty0_appl *)
47 EApply ty0_appl; XEAuto.
48 (* case 7: ty0_cast *)
49 EApply ty0_cast; XAuto.
52 Theorem csub0_ty0_ld: (g:?; c:?; u,v:?) (ty0 g c u v) -> (t1,t2:?)
53 (ty0 g (CTail c (Bind Abst) v) t1 t2) ->
54 (ty0 g (CTail c (Bind Abbr) u) t1 t2).
55 Intros; EApply csub0_ty0; XEAuto.
60 Hints Resolve csub0_ty0 csub0_ty0_ld : ltlc.
62 Tactic Definition CSub0Ty0 :=
64 [ _: (ty0 ?1 ?2 ?4 ?); _: (ty0 ?1 ?2 ?3 ?7); _: (pc3 ?2 ?4 ?7);
65 H: (ty0 ?1 (CTail ?2 (Bind Abst) ?4) ?5 ?6) |- ? ] ->
66 LApply (csub0_ty0_ld ?1 ?2 ?3 ?4); [ Intros H_x | EApply ty0_conv; XEAuto ];
67 LApply (H_x ?5 ?6); [ Clear H_x H; Intros | XAuto ].