(* THINNING *****************************************************************)
inductive thin: lenv → nat → nat → lenv → Prop ≝
- | thin_refl: ∀L. thin L 0 0 L
- | thin_thin: ∀L1,L2,I,V,e.
- thin L1 0 e L2 → thin (L1. ♭I V) 0 (e + 1) L2
- | thin_skip: ∀L1,L2,I,V1,V2,d,e.
- thin L1 d e L2 → L1 ⊢ ↓[d,e] V1 ≡ V2 →
- thin (L1. ♭I V1) (d + 1) e (L2. ♭I V2)
+| thin_refl: ∀L. thin L 0 0 L
+| thin_thin: ∀L1,L2,I,V,e. thin L1 0 e L2 → thin (L1. 𝕓{I} V) 0 (e + 1) L2
+| thin_skip: ∀L1,L2,I,V1,V2,d,e.
+ thin L1 d e L2 → L1 ⊢ ↓[d,e] V1 ≡ V2 →
+ thin (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
.
interpretation "thinning" 'RSubst L1 d e L2 = (thin L1 d e L2).
∀e2,L2. ↓[0,e2] L ≡ L2 → d1 + e1 ≤ e2 → ↓[0,e2-e1] L1 ≡ L2.
axiom thin_conf_lt: ∀d1,e1,L,L1. ↓[d1,e1] L ≡ L1 →
- ∀e2,K2,I,V2. ↓[0,e2] L ≡ K2. ♭I V2 →
+ ∀e2,K2,I,V2. ↓[0,e2] L ≡ K2. 𝕓{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃K1,V1. ↓[0,e2] L1 ≡ K1. ♭I V1 ∧ ↓[d,e1] K2 ≡ K1 ∧ ↑[d,e1] V1 ≡ V2.
+ ∃∃K1,V1. ↓[0,e2] L1 ≡ K1. 𝕓{I} V1 & ↓[d,e1] K2 ≡ K1 & ↑[d,e1] V1 ≡ V2.