--- /dev/null
+Require lift_gen.
+Require subst1_gen.
+Require csubst1_defs.
+Require pr0_lift.
+Require pr0_subst1.
+Require cpr0_defs.
+Require cpr0_props.
+Require pc3_props.
+Require pc3_gen.
+Require ty0_defs.
+Require ty0_lift.
+Require ty0_props.
+Require ty0_subst0.
+Require ty0_gen_context.
+Require csub0_defs.
+
+(*#* #caption "subject reduction" #clauses *)
+
+(*#* #stop file *)
+
+ Section ty0_sred_cpr0_pr0. (**********************************************)
+
+ Tactic Definition IH H c2 t2 :=
+ LApply (H c2); [ Intros H_x | XEAuto ];
+ LApply H_x; [ Clear H_x; Intros H_x | XAuto ];
+ LApply (H_x t2); [ Clear H_x; Intros | XEAuto ].
+
+ Tactic Definition IH0 :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ IH H1 ?5 ?3.
+
+ Tactic Definition IH0c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 ?2 ?3 ?4) |- ? ] ->
+ IH H1 ?5 ?3; Clear H1.
+
+ Tactic Definition IH0B :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?6) ?7) ?3.
+
+ Tactic Definition IH0Bc :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?6) ?7) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (ty0 ?1 (CTail ?2 (Bind ?6) ?7) ?3 ?4) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?6) ?7) ?3; Clear H1.
+
+ Tactic Definition IH1 :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 ?6.
+
+ Tactic Definition IH1c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 ?6; Clear H1.
+
+ Tactic Definition IH1Bc :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?8) ?6; Clear H1.
+
+ Tactic Definition IH1BLc :=
+ Match Context With
+ [ 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);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?8) '(lift ?10 ?11 ?6); Clear H1.
+
+ Tactic Definition IH1T :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6) |- ? ] ->
+ IH H1 ?5 '(TTail ?7 ?8 ?6).
+
+ Tactic Definition IH1T2c :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 ?2 c2)->(t2:T)(pr0 (TTail ?7 ?8 ?3) t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
+ IH H1 ?5 '(TTail ?7 ?9 ?6); Clear H1.
+
+ Tactic Definition IH3B :=
+ Match Context With
+ [ H1: (c2:C) (wf0 ?1 c2)->(cpr0 (CTail ?2 (Bind ?7) ?8) c2)->(t2:T)(pr0 ?3 t2)->(ty0 ?1 c2 t2 ?4);
+ H2: (cpr0 ?2 ?5); H3: (pr0 ?3 ?6); H4: (pr0 ?8 ?9) |- ? ] ->
+ IH H1 '(CTail ?5 (Bind ?7) ?9) ?6.
+
+(*#* #start file *)
+
+(*#* #caption "base case" *)
+(*#* #cap #cap c1, c2 #alpha t1 in T, t2 in T1, t in T2 *)
+
+ Theorem ty0_sred_cpr0_pr0 : (g:?; c1:?; t1,t:?) (ty0 g c1 t1 t) ->
+ (c2:?) (wf0 g c2) -> (cpr0 c1 c2) ->
+ (t2:?) (pr0 t1 t2) -> (ty0 g c2 t2 t).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : ty0_conv *)
+ IH1c; IH0c; EApply ty0_conv; XEAuto.
+(* case 2 : ty0_sort *)
+ Inversion H2; XAuto.
+(* case 3 : ty0_abbr *)
+ Inversion H5; Cpr0Drop; IH1c; XEAuto.
+(* case 4 : ty0_abst *)
+ Intros; Inversion H5; Cpr0Drop; IH0; IH1.
+ EApply ty0_conv;
+ [ EApply ty0_lift; [ Idtac | XAuto | XEAuto ]
+ | EApply ty0_abst
+ | EApply pc3_lift ]; XEAuto.
+(* case 5 : ty0_bind *)
+ Intros; Inversion H7; Clear H7.
+(* case 5.1 : pr0_refl *)
+ IH0c; IH0Bc; IH0Bc.
+ EApply ty0_bind; XEAuto.
+(* case 5.2 : pr0_cont *)
+ IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
+ EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
+(* case 5.3 : pr0_delta *)
+ Rewrite <- H8 in H1; Rewrite <- H8 in H2;
+ Rewrite <- H8 in H3; Rewrite <- H8 in H4; Clear H8 b.
+ IH0; IH0B; Ty0Correct; IH3B; Ty0Correct.
+ EApply ty0_conv; [ EApply ty0_bind | EApply ty0_bind | Idtac ]; XEAuto.
+(* case 5.4 : pr0_zeta *)
+ Rewrite <- H11 in H1; Rewrite <- H11 in H2; Clear H8 H9 H10 H11 b0 t2 t7 u0.
+ IH0; IH1BLc; Move H3 after H8; IH0Bc; Ty0Correct; Move H8 after H4; Clear H H0 H1 H3 H6 c c1 t t1;
+ NewInduction b.
+(* case 5.4.1 : Abbr *)
+ Ty0GenContext; Subst1Gen; LiftGen; Rewrite H in H1; Clear H x0.
+ EApply ty0_conv;
+ [ EApply ty0_bind; XEAuto | XEAuto
+ | EApply pc3_pr3_x;
+ EApply (pr3_t (TTail (Bind Abbr) u (lift (1) (0) x1))); XEAuto ].
+(* case 5.4.2 : Abst *)
+ EqFalse.
+(* case 5.4.3 : Void *)
+ Ty0GenContext; Rewrite H0; Rewrite H0 in H2; Clear H0 t3.
+ LiftGen; Rewrite <- H in H1; Clear H x0.
+ EApply ty0_conv; [ EApply ty0_bind; XEAuto | XEAuto | XAuto ].
+(* case 6 : ty0_appl *)
+ Intros; Inversion H5; Clear H5.
+(* case 6.1 : pr0_refl *)
+ IH0c; IH0c; EApply ty0_appl; XEAuto.
+(* case 6.2 : pr0_cont *)
+ Clear H6 H7 H8 H9 c1 k t t1 t2 t3 u1.
+ IH0; Ty0Correct; Ty0GenBase; IH1c; IH0; IH1c.
+ EApply ty0_conv;
+ [ EApply ty0_appl; [ XEAuto | EApply ty0_bind; XEAuto ]
+ | EApply ty0_appl; XEAuto
+ | XEAuto ].
+(* case 6.3 : pr0_beta *)
+ Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H9 c1 t t1 t2 v v1.
+ IH1T; IH0c; Ty0Correct; Ty0GenBase; IH0; IH1c.
+ Move H5 after H13; Ty0GenBase; Pc3Gen; Repeat CSub0Ty0.
+ EApply ty0_conv;
+ [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]
+ | EApply ty0_bind
+ | Apply (pc3_t (TTail (Bind Abbr) v2 t0))
+ ]; XEAuto.
+(* case 6.4 : pr0_delta *)
+ Rewrite <- H7 in H1; Rewrite <- H7 in H2; Clear H6 H7 H11 c1 t t1 t2 v v1.
+ IH1T2c; Clear H1; Ty0Correct; NonLinear; Ty0GenBase; IH1; IH0c.
+ Move H5 after H1; Ty0GenBase; Pc3Gen; Rewrite lift_bind in H0.
+ Move H1 after H0; Ty0Lift b u2; Rewrite lift_bind in H17.
+ Ty0GenBase.
+ EApply ty0_conv;
+ [ Apply ty0_appl; [ Idtac | EApply ty0_bind ]; XEAuto
+ | EApply ty0_bind;
+ [ Idtac
+ | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_conv ]
+ | EApply ty0_appl; [ EApply ty0_lift | EApply ty0_bind ]
+ ]; XEAuto
+ | Idtac ].
+ Rewrite <- lift_bind; Apply pc3_pc1;
+ Apply (pc1_u (TTail (Flat Appl) v2 (TTail (Bind b) u2 (lift (1) (0) (TTail (Bind Abst) u t0))))); XAuto.
+(* case 7 : ty0_cast *)
+ Intros; Inversion H5; Clear H5.
+(* case 7.1 : pr0_refl *)
+ IH0c; IH0c; EApply ty0_cast; XEAuto.
+(* case 7.2 : pr0_cont *)
+ Clear H6 H7 H8 H9 c1 k u1 t t1 t4 t5.
+ IH0; IH1c; IH1c.
+ EApply ty0_conv;
+ [ XEAuto
+ | EApply ty0_cast; [ EApply ty0_conv; XEAuto | XEAuto ]
+ | XAuto ].
+(* case 7.3 : pr0_eps *)
+ XAuto.
+ Qed.
+
+ End ty0_sred_cpr0_pr0.
+
+ Section ty0_sred_pr3. (**********************************************)
+
+ Theorem ty0_sred_pr1 : (c:?; t1,t2:?) (pr1 t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr1_r *)
+ XAuto.
+(* case 2 : pr1_u *)
+ EApply H1; EApply ty0_sred_cpr0_pr0; XEAuto.
+ Qed.
+
+ Theorem ty0_sred_pr2 : (c:?; t1,t2:?) (pr2 c t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr2_pr0 *)
+ EApply ty0_sred_cpr0_pr0; XEAuto.
+(* case 2 : pr2_u *)
+ XEAuto.
+ Qed.
+
+(*#* #start file *)
+
+(*#* #caption "general case" *)
+(*#* #cap #cap c #alpha t1 in T, t2 in T1, t in T2 *)
+
+ Theorem ty0_sred_pr3 : (c:?; t1,t2:?) (pr3 c t1 t2) ->
+ (g:?; t:?) (ty0 g c t1 t) ->
+ (ty0 g c t2 t).
+
+(*#* #stop file *)
+
+ Intros until 1; XElim H; Intros.
+(* case 1 : pr3_r *)
+ XAuto.
+(* case 2 : pr3_u *)
+ EApply H1; EApply ty0_sred_pr2; XEAuto.
+ Qed.
+
+ End ty0_sred_pr3.
+
+ Tactic Definition Ty0SRed :=
+ Match Context With
+ | [ H1: (pr3 ?1 ?2 ?3); H2: (ty0 ?4 ?1 ?2 ?5) |- ? ] ->
+ LApply (ty0_sred_pr3 ?1 ?2 ?3); [ Intros H_x | XAuto ];
+ LApply (H_x ?4 ?5); [ Clear H2 H_x; Intros | XAuto ].
+
+(*#* #start file *)
+
+(*#* #single *)
projects / helm.git / blobdiff