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15 include "basic_2/notation/relations/lazypredsn_5.ma".
16 include "basic_2/relocation/llpx_sn.ma".
17 include "basic_2/reduction/cpr.ma".
19 (* LAZY SN PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ************************)
21 definition llpr: genv → relation4 ynat term lenv lenv ≝ λG. llpx_sn (cpr G).
23 interpretation "lazy parallel reduction (local environment, sn variant)"
24 'LazyPRedSn G L1 L2 T d = (llpr G d T L1 L2).
26 (* Basic inversion lemmas ***************************************************)
28 lemma llpr_inv_flat: ∀I,G,L1,L2,V,T,d. ⦃G, L1⦄ ⊢ ➡[ⓕ{I}V.T, d] L2 →
29 ⦃G, L1⦄ ⊢ ➡[V, d] L2 ∧ ⦃G, L1⦄ ⊢ ➡[T, d] L2.
30 /2 width=2 by llpx_sn_inv_flat/ qed-.
32 (* Basic forward lemmas *****************************************************)
34 lemma llpr_fwd_length: ∀G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡[T, d] L2 → |L1| = |L2|.
35 /2 width=4 by llpx_sn_fwd_length/ qed-.
37 (* Basic properties *********************************************************)
39 lemma llpr_lref: ∀I,G,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
40 ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
41 ⦃G, K1⦄ ⊢ ➡[V1, 0] K2 → ⦃G, K1⦄ ⊢ V1 ➡ V2 → ⦃G, L1⦄ ⊢ ➡[#i, d] L2.
42 /2 width=9 by llpx_sn_lref/ qed.
45 lemma llpr_refl: ∀G,T,d. reflexive … (llpr G d T).
46 /2 width=1 by llpx_sn_refl/ qed.
48 (* Basic_1: removed theorems 5: wcpr0_gen_sort wcpr0_gen_head
49 wcpr0_getl wcpr0_getl_back