2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda-delta/substitution/pts_split.ma".
14 (* PARTIAL TELESCOPIC SUBSTITUTION ******************************************)
16 (* Main properties **********************************************************)
18 lemma pts_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
20 #L #T1 #T #d #e #H elim H -L T1 T d e
23 | #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
24 lapply (drop_fwd_drop2 … HLK) #H
25 elim (pts_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
26 @pts_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
27 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
28 elim (pts_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
29 @pts_bind /2/ @IHT12 @(pts_leq_repl … HT2) /2/
30 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
31 elim (pts_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
35 axiom pts_conf: ∀L,T0,d,e,T1. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
36 ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
39 Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
40 (u1,u:?; i:?) (subst0 i u u1 u2) ->
41 (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
43 Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
44 (u1,u:?; i:?) (subst0 i u u2 u1) ->
45 (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).