*)
inductive lsred: ptr → relation term ≝
| lsred_beta : ∀B,A. lsred (◊) (@B.𝛌.A) ([⬐B]A)
-| lsred_abst : ∀p,A1,A2. lsred p A1 A2 → lsred (rc::p) (𝛌.A1) (𝛌.A2)
+| lsred_abst : ∀p,A1,A2. lsred p A1 A2 → lsred (sn::p) (𝛌.A1) (𝛌.A2)
| lsred_appl_sn: ∀p,B1,B2,A. lsred p B1 B2 → lsred (sn::p) (@B1.A) (@B2.A)
| lsred_appl_dx: ∀p,B,A1,A2. lsred p A1 A2 → lsred (dx::p) (@B.A1) (@B.A2)
.
]
qed-.
-lemma lsred_inv_rc: ∀p,M,N. M ⇀[p] N → ∀q. rc::q = p →
- ∃∃A1,A2. A1 ⇀[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N.
-#p #M #N * -p -M -N
-[ #B #A #q #H destruct
-| #p #A1 #A2 #HA12 #q #H destruct /2 width=5/
-| #p #B1 #B2 #A #_ #q #H destruct
-| #p #B #A1 #A2 #_ #q #H destruct
-]
-qed-.
-
lemma lsred_inv_sn: ∀p,M,N. M ⇀[p] N → ∀q. sn::q = p →
+ (∃∃A1,A2. A1 ⇀[q] A2 & 𝛌.A1 = M & 𝛌.A2 = N) ∨
∃∃B1,B2,A. B1 ⇀[q] B2 & @B1.A = M & @B2.A = N.
#p #M #N * -p -M -N
[ #B #A #q #H destruct
-| #p #A1 #A2 #_ #q #H destruct
-| #p #B1 #B2 #A #HB12 #q #H destruct /2 width=6/
+| #p #A1 #A2 #HA12 #q #H destruct /3 width=5/
+| #p #B1 #B2 #A #HB12 #q #H destruct /3 width=6/
| #p #B #A1 #A2 #_ #q #H destruct
]
qed-.
theorem lsred_mono: ∀p. singlevalued … (lsred p).
#p #M #N1 #H elim H -p -M -N1
[ #B #A #N2 #H elim (lsred_inv_nil … H ?) -H // #D #C #H #HN2 destruct //
-| #p #A1 #A2 #_ #IHA12 #N2 #H elim (lsred_inv_rc … H ??) -H [3: // |2: skip ] #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/ (**) (* simplify line *)
-| #p #B1 #B2 #A #_ #IHB12 #N2 #H elim (lsred_inv_sn … H ??) -H [3: // |2: skip ] #D1 #D2 #C #HD12 #H #HN2 destruct /3 width=1/ (**) (* simplify line *)
+| #p #A1 #A2 #_ #IHA12 #N2 #H elim (lsred_inv_sn … H ??) -H [4: // |2: skip ] * (**) (* simplify line *)
+ [ #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/
+ | #D1 #D2 #C #_ #H destruct
+ ]
+| #p #B1 #B2 #A #_ #IHB12 #N2 #H elim (lsred_inv_sn … H ??) -H [4: // |2: skip ] * (**) (* simplify line *)
+ [ #C1 #C2 #_ #H destruct
+ | #D1 #D2 #C #HD12 #H #HN2 destruct /3 width=1/
+ ]
| #p #B #A1 #A2 #_ #IHA12 #N2 #H elim (lsred_inv_dx … H ??) -H [3: // |2: skip ] #D #C1 #C2 #HC12 #H #HN2 destruct /3 width=1/ (**) (* simplify line *)
]
qed-.