∃∃M2. M1 ↦[p] M2 & {⊤}⇑M2 = F2.
#p @(in_whd_ind … p) -p
[ #M1 #F2 #H
- elim (pl_st_inv_nil … H ?) -H // #V #T #HM1 #H
+ elim (pl_st_inv_nil … H …) -H // #V #T #HM1 #H
elim (boolean_inv_appl … (sym_eq … HM1)) -HM1 #B #N #_ #HB #HN #HM1
elim (boolean_inv_abst … HN) -HN #A #_ #HA #HN destruct /2 width=3/
| #p #_ #IHp #M1 #F2 #H
- elim (pl_st_inv_dx … H ??) -H [3: // |2:skip ] #b #V #T1 #T2 #HT12 #HM1 #H (**) (* simplify line *)
+ elim (pl_st_inv_dx … H …) -H [3: // |2:skip ] #b #V #T1 #T2 #HT12 #HM1 #H (**) (* simplify line *)
elim (boolean_inv_appl … (sym_eq … HM1)) -HM1 #B #A #Hb #HB #HA #HM1 destruct
elim (IHp … HT12) -IHp -HT12 #C #HAC #H destruct
@(ex2_intro … (@B.C)) // /2 width=1/ (**) (* auto needs some help here *)
theorem pl_st_mono: ∀p. singlevalued … (pl_st p).
#p #F #G1 #H elim H -p -F -G1
-[ #V #T #G2 #H elim (pl_st_inv_nil … H ?) -H //
+[ #V #T #G2 #H elim (pl_st_inv_nil … H …) -H //
#W #U #H #HG2 destruct //
-| #b #p #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_rc … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
+| #b #p #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_rc … H …) -H [3: // |2: skip ] (**) (* simplify line *)
#c #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
-| #b #p #V1 #V2 #T #_ #IHV12 #G2 #H elim (pl_st_inv_sn … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
+| #b #p #V1 #V2 #T #_ #IHV12 #G2 #H elim (pl_st_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
#c #W1 #W2 #U #HW12 #H #HG2 destruct /3 width=1/
-| #b #p #V #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_dx … H ??) -H [3: // |2: skip ] (**) (* simplify line *)
+| #b #p #V #T1 #T2 #_ #IHT12 #G2 #H elim (pl_st_inv_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
#c #W #U1 #U2 #HU12 #H #HG2 destruct /3 width=1/
]
qed-.
theorem pl_st_fwd_sle: ∀p1,F1,F. F1 Ⓡ↦[p1] F →
∀p2,F2. F Ⓡ↦[p2] F2 → p1 ≤ p2.
#p1 #F1 #F #H elim H -p1 -F1 -F //
-[ #b #p #T1 #T #_ #IHT1 #p2 #F2 #H elim (pl_st_inv_abst … H ???) -H [3: // |2,4: skip ] (**) (* simplify line *)
+[ #b #p #T1 #T #_ #IHT1 #p2 #F2 #H elim (pl_st_inv_abst … H …) -H [3: // |2,4: skip ] (**) (* simplify line *)
#q #T2 #HT2 #H1 #H2 destruct /3 width=2/
-| #b #p #V1 #V #T #_ #IHV1 #p2 #F2 #H elim (pl_st_inv_appl … H ????) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
+| #b #p #V1 #V #T #_ #IHV1 #p2 #F2 #H elim (pl_st_inv_appl … H …) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
[ #U #H destruct
| #q #V2 #H1 #HV2 #H2 destruct /3 width=2/
- | #q #U #_ #H elim (pl_st_inv_empty … H ??) [ // | skip ] (**) (* simplify line *)
+ | #q #U #_ #H elim (pl_st_inv_empty … H …) [ // | skip ] (**) (* simplify line *)
]
-| #b #p #V #T1 #T #HT1 #IHT1 #p2 #F2 #H elim (pl_st_inv_appl … H ????) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
+| #b #p #V #T1 #T #HT1 #IHT1 #p2 #F2 #H elim (pl_st_inv_appl … H …) -H [7: // |2,3,4: skip ] * (**) (* simplify line *)
[ #U #_ #H1 #H2 #_ -b -V -F2 -IHT1
elim (pl_st_fwd_abst … HT1 … H2) // -H1 * #q #H
elim (pl_st_inv_rc … HT1 … H) -HT1 -H #b #U1 #U2 #_ #_ #H -b -q -T1 -U1 destruct