lemma tpss_ind: ∀d,e,L,T1. ∀R:predicate term. R T1 →
(∀T,T2. L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → R T → R T2) →
∀T2. L ⊢ T1 ▶* [d, e] T2 → R T2.
-#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
+#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma tpss_ind_dx: ∀d,e,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → R T → R T1) →
+ ∀T1. L ⊢ T1 ▶* [d, e] T2 → R T1.
+#d #e #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
qed-.
(* Basic properties *********************************************************)
-lemma tpss_strap: ∀L,T1,T,T2,d,e.
- L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+lemma tpss_strap1: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶ [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+lemma tpss_strap2: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 ▶ [d, e] T → L ⊢ T ▶* [d, e] T2 → L ⊢ T1 ▶* [d, e] T2.
/2 width=3/ qed.
-lemma tpss_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
- ∀L2. L1 ≼ [d, e] L2 → L2 ⊢ T1 ▶* [d, e] T2.
+lemma tpss_lsubs_trans: ∀L1,T1,T2,d,e. L1 ⊢ T1 ▶* [d, e] T2 →
+ ∀L2. L2 ≼ [d, e] L1 → L2 ⊢ T1 ▶* [d, e] T2.
/3 width=3/ qed.
lemma tpss_refl: ∀d,e,L,T. L ⊢ T ▶* [d, e] T.
/2 width=1/ qed.
lemma tpss_bind: ∀L,V1,V2,d,e. L ⊢ V1 ▶* [d, e] V2 →
- ∀I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
- L ⊢ ⓑ{I} V1. T1 ▶* [d, e] ⓑ{I} V2. T2.
+ ∀a,I,T1,T2. L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 →
+ L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] ⓑ{a,I} V2. T2.
#L #V1 #V2 #d #e #HV12 elim HV12 -V2
-[ #V2 #HV12 #I #T1 #T2 #HT12 elim HT12 -T2
+[ #V2 #HV12 #a #I #T1 #T2 #HT12 elim HT12 -T2
[ /3 width=5/
| #T #T2 #_ #HT2 #IHT @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
-| #V #V2 #_ #HV12 #IHV #I #T1 #T2 #HT12
- lapply (tpss_lsubs_conf … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
- lapply (IHV … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
+| #V #V2 #_ #HV12 #IHV #a #I #T1 #T2 #HT12
+ lapply (tpss_lsubs_trans … HT12 (L. ⓑ{I} V) ?) -HT12 /2 width=1/ #HT12
+ lapply (IHV a … HT12) -IHV -HT12 #HT12 @step /2 width=5/ (**) (* /3 width=5/ is too slow *)
]
qed.
lapply (tpss_weak_top … HT12) //
qed.
+lemma tpss_append: ∀K,T1,T2,d,e. K ⊢ T1 ▶* [d, e] T2 →
+ ∀L. L @@ K ⊢ T1 ▶* [d, e] T2.
+#K #T1 #T2 #d #e #H @(tpss_ind … H) -T2 // /3 width=3/
+qed.
+
(* Basic inversion lemmas ***************************************************)
(* Note: this can be derived from tpss_inv_atom1 *)
]
qed-.
-lemma tpss_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ ⓑ{I} V1. T1 ▶* [d, e] U2 →
+lemma tpss_inv_bind1: ∀d,e,L,a,I,V1,T1,U2. L ⊢ ⓑ{a,I} V1. T1 ▶* [d, e] U2 →
∃∃V2,T2. L ⊢ V1 ▶* [d, e] V2 &
L. ⓑ{I} V2 ⊢ T1 ▶* [d + 1, e] T2 &
- U2 = ⓑ{I} V2. T2.
-#d #e #L #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
+ U2 = ⓑ{a,I} V2. T2.
+#d #e #L #a #I #V1 #T1 #U2 #H @(tpss_ind … H) -U2
[ /2 width=5/
| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
elim (tps_inv_bind1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+ lapply (tpss_lsubs_trans … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #[T1] ≤ #[T2].
+lemma tpss_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 → #{T1} ≤ #{T2}.
#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
#T #T2 #_ #HT2 #IHT1
lapply (tps_fwd_tw … HT2) -HT2 #HT2