inductive lsubc (RP:lenv→predicate term): relation lenv ≝
| lsubc_atom: lsubc RP (⋆) (⋆)
| lsubc_pair: ∀I,L1,L2,V. lsubc RP L1 L2 → lsubc RP (L1. ⓑ{I} V) (L2. ⓑ{I} V)
-| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ [RP] ϵ 〚A〛 → L2 ⊢ W ⁝ A →
+| lsubc_abbr: ∀L1,L2,V,W,A. ⦃L1, V⦄ ϵ[RP] 〚A〛 → L2 ⊢ W ⁝ A →
lsubc RP L1 L2 → lsubc RP (L1. ⓓV) (L2. ⓛW)
.
(* Basic inversion lemmas ***************************************************)
-fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L1 = ⋆ → L2 = ⋆.
+fact lsubc_inv_atom1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L1 = ⋆ → L2 = ⋆.
#RP #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
qed.
(* Basic_1: was: csubc_gen_sort_r *)
-lemma lsubc_inv_atom1: ∀RP,L2. ⋆ [RP] ⊑ L2 → L2 = ⋆.
+lemma lsubc_inv_atom1: ∀RP,L2. ⋆ ⊑[RP] L2 → L2 = ⋆.
/2 width=4/ qed-.
-fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
- (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ⁝ A &
- K1 [RP] ⊑ K2 &
+fact lsubc_inv_pair1_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K1,V. L1 = K1. ⓑ{I} V →
+ (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
+ K1 ⊑[RP] K2 &
L2 = K2. ⓛW & I = Abbr.
#RP #L1 #L2 * -L1 -L2
[ #I #K1 #V #H destruct
qed.
(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V [RP] ⊑ L2 →
- (∃∃K2. K1 [RP] ⊑ K2 & L2 = K2. ⓑ{I} V) ∨
- ∃∃K2,W,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ⁝ A &
- K1 [RP] ⊑ K2 &
+lemma lsubc_inv_pair1: ∀RP,I,K1,L2,V. K1. ⓑ{I} V ⊑[RP] L2 →
+ (∃∃K2. K1 ⊑[RP] K2 & L2 = K2. ⓑ{I} V) ∨
+ ∃∃K2,W,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
+ K1 ⊑[RP] K2 &
L2 = K2. ⓛW & I = Abbr.
/2 width=3/ qed-.
-fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → L2 = ⋆ → L1 = ⋆.
+fact lsubc_inv_atom2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → L2 = ⋆ → L1 = ⋆.
#RP #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
qed.
(* Basic_1: was: csubc_gen_sort_l *)
-lemma lsubc_inv_atom2: ∀RP,L1. L1 [RP] ⊑ ⋆ → L1 = ⋆.
+lemma lsubc_inv_atom2: ∀RP,L1. L1 ⊑[RP] ⋆ → L1 = ⋆.
/2 width=4/ qed-.
-fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 [RP] ⊑ L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
- (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ⁝ A &
- K1 [RP] ⊑ K2 &
+fact lsubc_inv_pair2_aux: ∀RP,L1,L2. L1 ⊑[RP] L2 → ∀I,K2,W. L2 = K2. ⓑ{I} W →
+ (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
+ K1 ⊑[RP] K2 &
L1 = K1. ⓓV & I = Abst.
#RP #L1 #L2 * -L1 -L2
[ #I #K2 #W #H destruct
qed.
(* Basic_1: was: csubc_gen_head_l *)
-lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 [RP] ⊑ K2. ⓑ{I} W →
- (∃∃K1. K1 [RP] ⊑ K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃K1, V⦄ [RP] ϵ 〚A〛 & K2 ⊢ W ⁝ A &
- K1 [RP] ⊑ K2 &
+lemma lsubc_inv_pair2: ∀RP,I,L1,K2,W. L1 ⊑[RP] K2. ⓑ{I} W →
+ (∃∃K1. K1 ⊑[RP] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃K1, V⦄ ϵ[RP] 〚A〛 & K2 ⊢ W ⁝ A &
+ K1 ⊑[RP] K2 &
L1 = K1. ⓓV & I = Abst.
/2 width=3/ qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: csubc_refl *)
-lemma lsubc_refl: ∀RP,L. L [RP] ⊑ L.
+lemma lsubc_refl: ∀RP,L. L ⊑[RP] L.
#RP #L elim L -L // /2 width=1/
qed.