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 (* SINGLE STEP PARALLEL REDUCTION ON TERMS **********************************)
16 inductive pr: lenv → term → term → Prop ≝
17 | pr_sort : ∀L,k. pr L (⋆k) (⋆k)
18 | pr_lref : ∀L,i. pr L (#i) (#i)
19 | pr_bind : ∀L,I,V1,V2,T1,T2. pr L V1 V2 → pr (L. 𝕓{I} V1) T1 T2 →
20 pr L (𝕓{I} V1. T1) (𝕓{I} V2. T2)
21 | pr_flat : ∀L,I,V1,V2,T1,T2. pr L V1 V2 → pr L T1 T2 →
22 pr L (𝕗{I} V1. T1) (𝕗{I} V2. T2)
23 | pr_beta : ∀L,V1,V2,W,T1,T2.
24 pr L V1 V2 → pr (L. 𝕓{Abst} W) T1 T2 → (*𝕓*)
25 pr L (𝕚{Appl} V1. 𝕚{Abst} W. T1) (𝕚{Abbr} V2. T2)
26 | pr_delta: ∀L,K,V1,V2,V,i.
27 ↑[0,i] K. 𝕓{Abbr} V1 ≡ L → pr K V1 V2 → ↑[0,i+1] V2 ≡ V →
29 | pr_theta: ∀L,V,V1,V2,W1,W2,T1,T2.
30 pr L V1 V2 → ↑[0,1] V2 ≡ V → pr L W1 W2 → pr (L. 𝕓{Abbr} W1) T1 T2 → (*𝕓*)
31 pr L (𝕚{Appl} V1. 𝕚{Abbr} W1. T1) (𝕚{Abbr} W2. 𝕚{Appl} V. T2)
32 | pr_zeta : ∀L,V,T,T1,T2. ↑[0,1] T1 ≡ T → pr L T1 T2 →
33 pr L (𝕚{Abbr} V. T) T2
34 | pr_tau : ∀L,V,T1,T2. pr L T1 T2 → pr L (𝕚{Cast} V. T1) T2
38 "single step parallel reduction (term)"
39 'PR L T1 T2 = (pr L T1 T2).
41 (* Basic properties *********************************************************)
43 lemma pr_refl: ∀T,L. L ⊢ T ⇒ T.
48 (* The basic inversion lemmas ***********************************************)
50 lemma pr_inv_lref2_aux: ∀L,T1,T2. L ⊢ T1 ⇒ T2 → ∀i. T2 = #i →
52 | ∃∃K,V1,j. j < i & K ⊢ V1 ⇒ #(i-j-1) &
53 ↑[O,j] K. 𝕓{Abbr} V1 ≡ L & T1 = #j
54 | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & L ⊢ T0 ⇒ #i &
56 | ∃∃V,T. L ⊢ T ⇒ #i & T1 = 𝕗{Cast} V. T.
57 #L #T1 #T2 #H elim H -H L T1 T2
58 [ #L #k #i #H destruct
60 | #L #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
61 | #L #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
62 | #L #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #i #H destruct
63 | #L #K #V1 #V2 #V #j #HLK #HV12 #HV2 #_ #i #H destruct -V;
64 elim (lift_inv_lref2 … HV2) -HV2 * #H1 #H2
65 [ elim (lt_zero_false … H1)
66 | destruct -V2 /3 width=7/
68 | #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #i #H destruct
69 | #L #V #T #T1 #T2 #HT1 #HT12 #_ #i #H destruct /3 width=6/
70 | #L #V #T1 #T2 #HT12 #_ #i #H destruct /3/
74 lemma pr_inv_lref2: ∀L,T1,i. L ⊢ T1 ⇒ #i →
76 | ∃∃K,V1,j. j < i & K ⊢ V1 ⇒ #(i-j-1) &
77 ↑[O,j] K. 𝕓{Abbr} V1 ≡ L & T1 = #j
78 | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & L ⊢ T0 ⇒ #i &
80 | ∃∃V,T. L ⊢ T ⇒ #i & T1 = 𝕗{Cast} V. T.