-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "Basic-2/substitution/tps_lift.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Split properties *********************************************************)
-
-lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
-#L #T1 #T2 #d #e #H elim H -L T1 T2 d e
-[ /2/
-| /2/
-| #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide Hjde;
- >(plus_minus_m_m_comm j d) in ⊢ (% → ?) // -Hdj /3/
- | -Hdi Hdj; #Hid
- generalize in match Hide -Hide (**) (* rewriting in the premises, rewrites in the goal too *)
- >(plus_minus_m_m_comm … Hjde) in ⊢ (% → ?) -Hjde /4/
- ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
- -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
- @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
- [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi Hide /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
- ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
-qed.