inductive tps: nat → nat → lenv → relation term ≝
| tps_atom : ∀L,I,d,e. tps d e L (𝕒{I}) (𝕒{I})
| tps_subst: ∀L,K,V,W,i,d,e. d ≤ i → i < d + e →
- â\86\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\86\91[0, i + 1] V â\89¡ W â\86\92 tps d e L (#i) W
+ â\87\93[0, i] L â\89¡ K. ð\9d\95\93{Abbr} V â\86\92 â\87\91[0, i + 1] V â\89¡ W â\86\92 tps d e L (#i) W
| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
tps d e L V1 V2 → tps (d + 1) e (L. 𝕓{I} V2) T1 T2 →
tps d e L (𝕓{I} V1. T1) (𝕓{I} V2. T2)
qed.
(* Basic_1: was: subst1_ex *)
-lemma tps_full: â\88\80K,V,T1,L,d. â\86\93[0, d] L â\89¡ (K. ð\9d\95\93{Abbr} V) â\86\92
- â\88\83â\88\83T2,T. L â\8a¢ T1 [d, 1] â\89« T2 & â\86\91[d, 1] T â\89¡ T2.
+lemma tps_full: â\88\80K,V,T1,L,d. â\87\93[0, d] L â\89¡ (K. ð\9d\95\93{Abbr} V) â\86\92
+ â\88\83â\88\83T2,T. L â\8a¢ T1 [d, 1] â\89« T2 & â\87\91[d, 1] T â\89¡ T2.
#K #V #T1 elim T1 -T1
[ * #i #L #d #HLK /2 width=4/
elim (lt_or_eq_or_gt i d) #Hid /3 width=4/
fact tps_inv_atom1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀I. T1 = 𝕒{I} →
T2 = 𝕒{I} ∨
∃∃K,V,i. d ≤ i & i < d + e &
- â\86\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
- â\86\91[O, i + 1] V â\89¡ T2 &
+ â\87\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
+ â\87\91[O, i + 1] V â\89¡ T2 &
I = LRef i.
#L #T1 #T2 #d #e * -L -T1 -T2 -d -e
[ #L #I #d #e #J #H destruct /2 width=1/
lemma tps_inv_atom1: ∀L,T2,I,d,e. L ⊢ 𝕒{I} [d, e] ≫ T2 →
T2 = 𝕒{I} ∨
∃∃K,V,i. d ≤ i & i < d + e &
- â\86\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
- â\86\91[O, i + 1] V â\89¡ T2 &
+ â\87\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
+ â\87\91[O, i + 1] V â\89¡ T2 &
I = LRef i.
/2 width=3/ qed-.
lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
T2 = #i ∨
∃∃K,V. d ≤ i & i < d + e &
- â\86\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
- â\86\91[O, i + 1] V â\89¡ T2.
+ â\87\93[O, i] L â\89¡ K. ð\9d\95\93{Abbr} V &
+ â\87\91[O, i + 1] V â\89¡ T2.
#L #T2 #i #d #e #H
elim (tps_inv_atom1 … H) -H /2 width=1/
* #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=4/