| lsubd_atom: lsubd h g G (⋆) (⋆)
| lsubd_pair: ∀I,L1,L2,V. lsubd h g G L1 L2 →
lsubd h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubd_beta: ∀L1,L2,W,V,l. ⦃G, L1⦄ ⊢ V ▪[h, g] l+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] l →
+| lsubd_beta: ∀L1,L2,W,V,d. ⦃G, L1⦄ ⊢ V ▪[h, g] d+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d →
lsubd h g G L1 L2 → lsubd h g G (L1.ⓓⓝW.V) (L2.ⓛW)
.
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
]
qed-.
fact lsubd_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
∀I,K1,X. L1 = K1.ⓑ{I}X →
(∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+ ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
G ⊢ K1 ⫃▪[h, g] K2 &
I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
]
qed-.
lemma lsubd_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃▪[h, g] L2 →
(∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,l. ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+ ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
G ⊢ K1 ⫃▪[h, g] K2 &
I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
/2 width=3 by lsubd_inv_pair1_aux/ qed-.
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
]
qed-.
fact lsubd_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
∀I,K2,W. L2 = K2.ⓑ{I}W →
(∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,l. ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+ ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
]
qed-.
lemma lsubd_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃▪[h, g] K2.ⓑ{I}W →
(∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,l. ⦃G, K1⦄ ⊢ V ▪[h, g] l+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] l &
+ ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
/2 width=3 by lsubd_inv_pair2_aux/ qed-.
(* Note: the constant 0 cannot be generalized *)
lemma lsubd_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀K1,s,e. ⬇[s, 0, e] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, e] L2 ≡ K2.
+ ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
| elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK1
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
(* Note: the constant 0 cannot be generalized *)
lemma lsubd_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀K2,s,e. ⬇[s, 0, e] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, e] L1 ≡ K1.
+ ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
| elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 … H) -H * #He #HLK2
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/