+++ /dev/null
-Require lift_props.
-Require drop_props.
-Require pc3_props.
-Require pc3_gen.
-Require ty0_defs.
-Require ty0_gen.
-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; Pc3Unfold; Repeat Ty0SRed; XEAuto.
- Qed.
-
-
- End ty0_sred_props.
-
-(*#* #start file *)
-
-(*#* #single *)
projects / helm.git / blobdiff