matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr.ma

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  14
  15 include "Basic_2/reducibility/tpr.ma".
  16
  17 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
  18
  19 inductive ltpr: relation lenv ≝
  20 | ltpr_stom: ltpr (⋆) (⋆)
  21 | ltpr_pair: ∀K1,K2,I,V1,V2.
  22              ltpr K1 K2 → V1 ➡ V2 → ltpr (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) (*ⓑ*)
  23 .
  24
  25 interpretation
  26   "context-free parallel reduction (environment)"
  27   'PRed L1 L2 = (ltpr L1 L2).
  28
  29 (* Basic properties *********************************************************)
  30
  31 lemma ltpr_refl: ∀L:lenv. L ➡ L.
  32 #L elim L -L // /2 width=1/
  33 qed.
  34
  35 (* Basic inversion lemmas ***************************************************)
  36
  37 fact ltpr_inv_atom1_aux: ∀L1,L2. L1 ➡ L2 → L1 = ⋆ → L2 = ⋆.
  38 #L1 #L2 * -L1 -L2
  39 [ //
  40 | #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
  41 ]
  42 qed.
  43
  44 (* Basic_1: was: wcpr0_gen_sort *)
  45 lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
  46 /2 width=3/ qed-.
  47
  48 fact ltpr_inv_pair1_aux: ∀L1,L2. L1 ➡ L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
  49                          ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
  50 #L1 #L2 * -L1 -L2
  51 [ #K1 #I #V1 #H destruct
  52 | #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
  53 ]
  54 qed.
  55
  56 (* Basic_1: was: wcpr0_gen_head *)
  57 lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
  58                       ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
  59 /2 width=3/ qed-.
  60
  61 fact ltpr_inv_atom2_aux: ∀L1,L2. L1 ➡ L2 → L2 = ⋆ → L1 = ⋆.
  62 #L1 #L2 * -L1 -L2
  63 [ //
  64 | #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
  65 ]
  66 qed.
  67
  68 lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
  69 /2 width=3/ qed-.
  70
  71 fact ltpr_inv_pair2_aux: ∀L1,L2. L1 ➡ L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
  72                          ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
  73 #L1 #L2 * -L1 -L2
  74 [ #K2 #I #V2 #H destruct
  75 | #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
  76 ]
  77 qed.
  78
  79 lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
  80                       ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
  81 /2 width=3/ qed-.
  82
  83 (* Basic forward lemmas *****************************************************)
  84
  85 lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
  86 #L1 #L2 #H elim H -L1 -L2 normalize //
  87 qed-.
  88
  89 (* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)