15 Require ty0_gen_context.
19 (*#* #caption "subject reduction" #clauses *)
23 Section ty0_sred_cpr0_pr0. (**********************************************)
25 Tactic Definition IH H c2 t2 :=
26 LApply (H c2); [ Intros H_x | XEAuto ];
27 LApply H_x; [ Clear H_x; Intros H_x | XAuto ];
28 LApply (H_x t2); [ Clear H_x; Intros | XEAuto ].
30 Tactic Definition IH0 :=
32 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
33 H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
36 Tactic Definition IH0c :=
38 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
39 H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
40 IH H1 ?5 ?3; Clear H1.
42 Tactic Definition IH0B :=
44 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
45 H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
46 IH H1 '(CTail ?5 (Bind ?6) ?7) ?3.
48 Tactic Definition IH0Bc :=
50 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
51 H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
52 IH H1 '(CTail ?5 (Bind ?6) ?7) ?3; Clear H1.
54 Tactic Definition IH1 :=
56 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
57 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
60 Tactic Definition IH1c :=
62 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
63 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
64 IH H1 ?5 ?6; Clear H1.
66 Tactic Definition IH1Bc :=
68 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
69 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
70 IH H1 '(CTail ?5 (Bind ?7) ?8) ?6; Clear H1.
72 Tactic Definition IH1BLc :=
74 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 (lift ?10 ?11 ?3) t2)->(ty0 ?1 c2 t2 ?4);
75 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
76 IH H1 '(CTail ?5 (Bind ?7) ?8) '(lift ?10 ?11 ?6); Clear H1.
78 Tactic Definition IH1T :=
80 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
81 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
82 IH H1 ?5 '(TTail ?7 ?8 ?6).
84 Tactic Definition IH1T2c :=
86 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
87 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
88 IH H1 ?5 '(TTail ?7 ?9 ?6); Clear H1.
90 Tactic Definition IH3B :=
92 [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
93 H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
94 IH H1 '(CTail ?5 (Bind ?7) ?9) ?6.
98 (*#* #caption "base case" *)
99 (*#* #cap #cap c1, c2 #alpha t1 in T, t2 in T1, t in T2 *)
101 Theorem ty0_sred_cpr0_pr0: (g:?; c1:?; t1,t:?) (ty0 g c1 t1 t) ->
102 (c2:?) (wf0 g c2) -> (cpr0 c1 c2) ->
103 (t2:?) (pr0 t1 t2) -> (ty0 g c2 t2 t).
107 Intros until 1; XElim H; Intros.
108 (* case 1: ty0_conv *)
109 IH1c; IH0c; EApply ty0_conv; XEAuto.
110 (* case 2: ty0_sort *)
112 (* case 3: ty0_abbr *)
113 Inversion H5; Cpr0Drop; IH1c; XEAuto.
114 (* case 4: ty0_abst *)
115 Intros; Inversion H5; Cpr0Drop; IH0; IH1.
117 [ EApply ty0_lift; [ Idtac | XAuto | XEAuto ]
119 | EApply pc3_lift ]; XEAuto.
120 (* case 5: ty0_bind *)
121 Intros; Inversion H7; Clear H7.
122 (* case 5.1: pr0_refl *)
124 EApply ty0_bind; XEAuto.
125 (* case 5.2: pr0_cont *)
126 IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
127 EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
128 (* case 5.3: pr0_delta *)
129 Rewrite <- H8 in H1; Rewrite <- H8 in H2;
130 Rewrite <- H8 in H3; Rewrite <- H8 in H4; Clear H8 b.
131 IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
132 EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
133 (* case 5.4: pr0_zeta *)
134 Rewrite <- H11 in H1; Rewrite <- H11 in H2; Clear H8 H9 H10 H11 b0 t2 t7 u0.
135 IH0; IH1BLc; Move H3 after H8; IH0Bc; Ty0Correct; Move H8 after H4; Clear H H0 H1 H3 H6 c c1 t t1;
137 (* case 5.4.1: Abbr *)
138 Ty0GenContext; Subst1Gen; LiftGen; Rewrite H in H1; Clear H x0.
140 [ EApply ty0_bind; XEAuto | XEAuto
142 EApply (pr3_t (TTail (Bind Abbr) u (lift (1) (0) x1))); XEAuto ].
143 (* case 5.4.2: Abst *)
145 (* case 5.4.3: Void *)
146 Ty0GenContext; Rewrite H0; Rewrite H0 in H2; Clear H0 t3.
147 LiftGen; Rewrite <- H in H1; Clear H x0.
148 EApply ty0_conv; [ EApply ty0_bind; XEAuto | XEAuto | XAuto ].
149 (* case 6: ty0_appl *)
150 Intros; Inversion H5; Clear H5.
151 (* case 6.1: pr0_refl *)
152 IH0c; IH0c; EApply ty0_appl; XEAuto.
153 (* case 6.2: pr0_cont *)
154 Clear H6 H7 H8 H9 c1 k t t1 t2 t3 u1.
155 IH0; Ty0Correct; Ty0GenBase; IH1c; IH0; IH1c.
157 [ EApply ty0_appl; [ XEAuto | EApply ty0_bind; XEAuto ]
158 | EApply ty0_appl; XEAuto
160 (* case 6.3: pr0_beta *)
161 Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H9 c1 t t1 t2 v v1.
162 IH1T; IH0c; Ty0Correct; Ty0GenBase; IH0; IH1c.
163 Move H5 after H13; Ty0GenBase; Pc3Gen; Repeat CSub0Ty0.
165 [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]
167 | Apply (pc3_t (TTail (Bind Abbr) v2 t0))
169 (* case 6.4: pr0_delta *)
170 Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H11 c1 t t1 t2 v v1.
171 IH1T2c; Clear H1; Ty0Correct; NonLinear; Ty0GenBase; IH1; IH0c.
172 Move H5 after H1; Ty0GenBase; Pc3Gen; Rewrite lift_bind in H0.
173 Move H1 after H0; Ty0Lift b u2; Rewrite lift_bind in H17.
176 [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]; XEAuto
179 | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_conv ]
180 | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_bind ]
183 Rewrite <- lift_bind; Apply pc3_pc1;
184 Apply (pc1_pr0_u2 (TTail (Flat Appl) v2 (TTail (Bind b) u2 (lift (1) (0) (TTail (Bind Abst) u t0))))); XAuto.
185 (* case 7: ty0_cast *)
186 Intros; Inversion H5; Clear H5.
187 (* case 7.1: pr0_refl *)
188 IH0c; IH0c; EApply ty0_cast; XEAuto.
189 (* case 7.2: pr0_cont *)
190 Clear H6 H7 H8 H9 c1 k u1 t t1 t4 t5.
194 | EApply ty0_cast; [ EApply ty0_conv; XEAuto | XEAuto ]
196 (* case 7.3: pr0_epsilon *)
200 End ty0_sred_cpr0_pr0.
202 Section ty0_sred_pr3. (**********************************************)
204 Theorem ty0_sred_pr1: (c:?; t1,t2:?) (pr1 t1 t2) ->
205 (g:?; t:?) (ty0 g c t1 t) ->
207 Intros until 1; XElim H; Intros.
211 EApply H1; EApply ty0_sred_cpr0_pr0; XEAuto.
214 Theorem ty0_sred_pr2: (c:?; t1,t2:?) (pr2 c t1 t2) ->
215 (g:?; t:?) (ty0 g c t1 t) ->
217 Intros until 1; XElim H; Intros.
218 (* case 1: pr2_free *)
219 EApply ty0_sred_cpr0_pr0; XEAuto.
221 EApply ty0_subst0; Try EApply ty0_sred_cpr0_pr0; XEAuto.
226 (*#* #caption "general case without the reduction in the context" *)
227 (*#* #cap #cap c #alpha t1 in T, t2 in T1, t in T2 *)
229 Theorem ty0_sred_pr3: (c:?; t1,t2:?) (pr3 c t1 t2) ->
230 (g:?; t:?) (ty0 g c t1 t) ->
235 Intros until 1; XElim H; Intros.
236 (* case 1: pr3_refl *)
238 (* case 2: pr3_sing *)
239 EApply H1; EApply ty0_sred_pr2; XEAuto.
244 Tactic Definition Ty0SRed :=
246 | [ H1: (pr3 ?1 ?2 ?3); H2: (ty0 ?4 ?1 ?2 ?5) |- ? ] ->
247 LApply (ty0_sred_pr3 ?1 ?2 ?3); [ Intros H_x | XAuto ];
248 LApply (H_x ?4 ?5); [ Clear H2 H_x; Intros | XAuto ].