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/drop_defs.ma".
14 (* PARALLEL SUBSTITUTION ****************************************************)
16 inductive ps: lenv → term → nat → nat → term → Prop ≝
17 | ps_sort : ∀L,k,d,e. ps L (⋆k) d e (⋆k)
18 | ps_lref : ∀L,i,d,e. ps L (#i) d e (#i)
19 | ps_subst: ∀L,K,V,U1,U2,i,d,e.
21 ↓[0, i] L ≡ K. 𝕓{Abbr} V → ps K V 0 (d + e - i - 1) U1 →
22 ↑[0, i + 1] U1 ≡ U2 → ps L (#i) d e U2
23 | ps_bind : ∀L,I,V1,V2,T1,T2,d,e.
24 ps L V1 d e V2 → ps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
25 ps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
26 | ps_flat : ∀L,I,V1,V2,T1,T2,d,e.
27 ps L V1 d e V2 → ps L T1 d e T2 →
28 ps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
31 interpretation "parallel substritution" 'PSubst L T1 d e T2 = (ps L T1 d e T2).
33 (* Basic properties *********************************************************)
35 lemma subst_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
40 (* Basic inversion lemmas ***************************************************)
42 lemma ps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
43 ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
44 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
45 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
47 #d #e #L #U1 #U2 #H elim H -H d e L U1 U2
48 [ #L #k #d #e #I #V1 #T1 #H destruct
49 | #L #i #d #e #I #V1 #T1 #H destruct
50 | #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
51 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
52 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
56 lemma subst_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
57 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
58 L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
62 lemma subst_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
63 ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
64 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
66 #d #e #L #U1 #U2 #H elim H -H d e L U1 U2
67 [ #L #k #d #e #I #V1 #T1 #H destruct
68 | #L #i #d #e #I #V1 #T1 #H destruct
69 | #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
70 | #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #I #V #T #H destruct
71 | #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #_ #_ #I #V #T #H destruct /2 width=5/
75 lemma subst_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
76 ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &