--- /dev/null
+Require lift_props.
+Require drop_props.
+Require pc3_props.
+Require pc3_gen.
+Require ty0_defs.
+Require ty0_props.
+Require ty0_sred.
+
+(*#* #caption "corollaries of subject reduction" #clauses *)
+
+(*#* #stop file *)
+
+ Section ty0_gen. (********************************************************)
+
+ Tactic Definition IH e :=
+ Match Context With
+ [ H0: (t:?; d:?) ?1 = (lift ?2 d t) -> ?; H1: ?1 = (lift ?2 ?3 ?4) |- ? ] ->
+ LApply (H0 ?4 ?3); [ Clear H0 H1; Intros H0 | XAuto ];
+ LApply (H0 e); [ Clear H0; Intros H0 | XEAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap t2 #alpha c in C1, e in C2, t1 in T, x in T1, d in i *)
+
+ Theorem ty0_gen_lift : (g:?; c:?; t1,x:?; h,d:?)
+ (ty0 g c (lift h d t1) x) ->
+ (e:?) (wf0 g e) -> (drop h d c e) ->
+ (EX t2 | (pc3 c (lift h d t2) x) & (ty0 g e t1 t2)).
+
+(*#* #stop file *)
+
+ Intros until 1; InsertEq H '(lift h d t1);
+ UnIntro H d; UnIntro H t1; XElim H; Intros;
+ Rename x0 into t3; Rename x1 into d0.
+(* case 1 : ty0_conv *)
+ IH e; XEAuto.
+(* case 2 : ty0_sort *)
+ LiftGenBase; Rewrite H0; Clear H0 t.
+ EApply ex2_intro; [ Rewrite lift_sort; XAuto | XAuto ].
+(* case 3 : ty0_abbr *)
+ Apply (lt_le_e n d0); Intros.
+(* case 3.1 : n < d0 *)
+ LiftGenBase; DropS; Rewrite H3; Clear H3 t3.
+ Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ DropDis; IH x1 | XAuto ].
+ EApply ex2_intro;
+ [ Rewrite lift_d; [ EApply pc3_lift; XEAuto | XEAuto ]
+ | EApply ty0_abbr; XEAuto ].
+(* case 3.2 : n >= d0 *)
+ Apply (lt_le_e n (plus d0 h)); Intros.
+(* case 3.2.1 : n < d0 + h *)
+ LiftGenBase.
+(* case 3.2.2 : n >= d0 + h *)
+ Rewrite (le_plus_minus_sym h n) in H3; [ Idtac | XEAuto ].
+ LiftGenBase; DropDis; Rewrite H3; Clear H3 t3.
+ EApply ex2_intro; [ Idtac | EApply ty0_abbr; XEAuto ].
+ Rewrite lift_free; [ Idtac | XEAuto | XAuto ].
+ Rewrite <- plus_n_Sm; Rewrite <- le_plus_minus; XEAuto.
+(* case 4 : ty0_abst *)
+ Apply (lt_le_e n d0); Intros.
+(* case 4.1 : n < d0 *)
+ LiftGenBase; Rewrite H3; Clear H3 t3.
+ Rewrite (le_plus_minus (S n) d0); [ Idtac | XAuto ].
+ Rewrite (lt_plus_minus n d0) in H5; [ DropDis; Rewrite H0; IH x1 | XAuto ].
+ EApply ex2_intro; [ Rewrite lift_d | EApply ty0_abst ]; XEAuto.
+(* case 4.2 : n >= d0 *)
+ Apply (lt_le_e n (plus d0 h)); Intros.
+(* case 4.2.1 : n < d0 + h *)
+ LiftGenBase.
+(* case 4.2.2 : n >= d0 + h *)
+ Rewrite (le_plus_minus_sym h n) in H3; [ Idtac | XEAuto ].
+ LiftGenBase; DropDis; Rewrite H3; Clear H3 t3.
+ EApply ex2_intro; [ Idtac | EApply ty0_abst; XEAuto ].
+ Rewrite lift_free; [ Idtac | XEAuto | XAuto ].
+ Rewrite <- plus_n_Sm; Rewrite <- le_plus_minus; XEAuto.
+(* case 5 : ty0_bind *)
+ LiftGenBase; Rewrite H5; Rewrite H8; Rewrite H8 in H2; Clear H5 t3.
+ Move H0 after H2; IH e; IH '(CTail e (Bind b) x0); Ty0Correct.
+ EApply ex2_intro; [ Rewrite lift_bind; XEAuto | XEAuto ].
+(* case 6 : ty0_appl *)
+ LiftGenBase; Rewrite H3; Rewrite H6; Clear H3 c t3 x y.
+ IH e; IH e; Pc3Gen; Pc3T; Pc3Gen; Pc3T.
+ Move H3 after H12; Ty0Correct; Ty0SRed; Ty0GenBase; Wf0Ty0.
+ EApply ex2_intro;
+ [ Rewrite lift_flat; Apply pc3_thin_dx;
+ Rewrite lift_bind; Apply pc3_tail_21; [ EApply pc3_pr3_x | Idtac ]
+ | EApply ty0_appl;
+ [ EApply ty0_conv
+ | EApply ty0_conv; [ EApply ty0_bind | Idtac | Idtac ] ]
+ ]; XEAuto.
+(* case 7 : ty0_cast *)
+ LiftGenBase; Rewrite H3; Rewrite H6; Rewrite H6 in H0.
+ IH e; IH e; Pc3Gen; XEAuto.
+ Qed.
+
+ End ty0_gen.
+
+ Tactic Definition Ty0Gen :=
+ Match Context With
+ | [ H0: (ty0 ?1 ?2 (lift ?3 ?4 ?5) ?6);
+ H1: (drop ?3 ?4 ?2 ?7) |- ? ] ->
+ LApply (ty0_gen_lift ?1 ?2 ?5 ?6 ?3 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XEAuto ];
+ LApply H0; [ Clear H0 H1; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | [ H0: (ty0 ?1 ?2 (lift ?3 ?4 ?5) ?6);
+ _ : (wf0 ?1 ?7) |- ? ] ->
+ LApply (ty0_gen_lift ?1 ?2 ?5 ?6 ?3 ?4); [ Clear H0; Intros H0 | XAuto ];
+ LApply (H0 ?7); [ Clear H0; Intros H0 | XAuto ];
+ LApply H0; [ Clear H0; Intros H0 | XAuto ];
+ XElim H0; Intros
+ | _ -> Ty0GenContext.
+
+ Section ty0_sred_props. (*************************************************)
+
+(*#* #start file *)
+
+(*#* #caption "drop preserves well-formedness" *)
+(*#* #cap #alpha c in C1, e in C2, d in i *)
+
+ Theorem wf0_drop : (c,e:?; d,h:?) (drop h d c e) ->
+ (g:?) (wf0 g c) -> (wf0 g e).
+
+(*#* #stop proof *)
+
+ XElim c.
+(* case 1 : CSort *)
+ Intros; DropGenBase; Rewrite H; XAuto.
+(* case 2 : CTail k *)
+ Intros c IHc; XElim k; (
+ XElim d;
+ [ XEAuto
+ | Intros d IHd; Intros;
+ DropGenBase; Rewrite H; Rewrite H1 in H0; Clear IHd H H1 e t;
+ Inversion H0; Clear H3 H4 b0 u ]).
+(* case 2.1 : Bind, d > 0 *)
+ Ty0Gen; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "type reduction" *)
+(*#* #cap #cap c, t1, t2 #alpha u in T *)
+
+ Theorem ty0_tred : (g:?; c:?; u,t1:?) (ty0 g c u t1) ->
+ (t2:?) (pr3 c t1 t2) -> (ty0 g c u t2).
+
+(*#* #stop proof *)
+
+ Intros; Ty0Correct; Ty0SRed; EApply ty0_conv; XEAuto.
+ Qed.
+
+(*#* #start proof *)
+
+(*#* #caption "subject conversion" *)
+(*#* #cap #cap c, u1, u2, t1, t2 *)
+
+ Theorem ty0_sconv : (g:?; c:?; u1,t1:?) (ty0 g c u1 t1) ->
+ (u2,t2:?) (ty0 g c u2 t2) ->
+ (pc3 c u1 u2) -> (pc3 c t1 t2).
+
+(*#* #stop file *)
+
+ Intros; Pc3Confluence; Repeat Ty0SRed; XEAuto.
+ Qed.
+
+
+ End ty0_sred_props.
+
+(*#* #start file *)
+
+(*#* #single *)
projects / helm.git / blobdiff