matita/matita/contribs/lambda_delta/Basic_2/reduction/tpr_tpss.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
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  14
  15 include "Basic_2/unfold/ltpss_ltpss.ma".
  16 include "Basic_2/reduction/ltpr_ldrop.ma".
  17
  18 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
  19
  20 (* Basic_1: was: pr0_subst1 *)
  21 lemma tpr_tps_ltpr: ∀T1,T2. T1 ⇒ T2 →
  22                     ∀L1,d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 →
  23                     ∀L2. L1 ⇒ L2 →
  24                     ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫* U2.
  25 #T1 #T2 #H elim H -H T1 T2
  26 [ #I #L1 #d #e #X #H
  27   elim (tps_inv_atom1 … H) -H
  28   [ #H destruct -X /2/
  29   | * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct -I;
  30     elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 HL12 #X #HLK2 #H
  31     elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct -X;
  32     elim (lift_total V2 0 (i+1)) #U2 #HVU2
  33     lapply (tpr_lift … HV12 … HVU1 … HVU2) -HV12 HVU1 #HU12
  34     @ex2_1_intro [2: @HU12 | skip | /3/ ] (**) (* /4 width=6/ is too slow *)
  35   ]
  36 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
  37   elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct -X;
  38   elim (IHV12 … HVW1 … HL12) -IHV12 HVW1;
  39   elim (IHT12 … HTU1 … HL12) -IHT12 HTU1 HL12 /3 width=5/
  40 | #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
  41   elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X;
  42   elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct -Y;
  43   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  44   elim (IHT12 … HTT1 (L2. 𝕓{Abst} WW) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  45   lapply (tpss_lsubs_conf … HTT2 (L2. 𝕓{Abbr} VV2) ?) -HTT2 /3 width=5/
  46 | #I #V1 #V2 #T1 #T2 #U2 #HV12 #_ #HTU2 #IHV12 #IHT12 #L1 #d #e #X #H #L2 #HL12
  47   elim (tps_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X;
  48   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  49   elim (IHT12 … HTT1 (L2. 𝕓{I} VV2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  50   elim (tpss_strip_neq … HTT2 … HTU2 ?) -HTT2 HTU2 T2 /2/ #T2 #HTT2 #HUT2
  51   lapply (tps_lsubs_conf … HTT2 (L2. 𝕓{I} V2) ?) -HTT2 /2/ #HTT2
  52   elim (ltpss_tps_conf … HTT2 (L2. 𝕓{I} VV2) (d + 1) e ?) -HTT2 /2/ #W2 #HTTW2 #HTW2
  53   lapply (tpss_lsubs_conf … HTTW2 (⋆. 𝕓{I} VV2) ?) -HTTW2 /2/ #HTTW2
  54   lapply (tpss_tps … HTTW2) -HTTW2 #HTTW2
  55   lapply (tpss_lsubs_conf … HTW2 (L2. 𝕓{I} VV2) ?) -HTW2 /2/ #HTW2
  56   lapply (tpss_trans_eq … HUT2 … HTW2) -HUT2 HTW2 /3 width=5/
  57 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #e #X #H #L2 #HL12
  58   elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X;
  59   elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct -Y;
  60   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  61   elim (IHW12 … HWW1 … HL12) -IHW12 HWW1 #WW2 #HWW12 #HWW2
  62   elim (IHT12 … HTT1 (L2. 𝕓{Abbr} WW2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  63   elim (lift_total VV2 0 1) #VV #H2VV
  64   lapply (tpss_lift_ge … HVV2 (L2. 𝕓{Abbr} WW2) … HV2 … H2VV) -HVV2 HV2 /2/ #HVV
  65   @ex2_1_intro [2: @tpr_theta |1: skip |3: @tpss_bind [2: @tpss_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
  66 | #V1 #TT1 #T1 #T2 #HT1 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
  67   elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct -X;
  68   elim (tps_inv_lift1_ge … HTT12 L1 … HT1 ?) -HTT12 HT1 /2/ #T2 #HT12 #HTT2
  69   elim (IHT12 … HT12 … HL12) -IHT12 HT12 HL12 <minus_plus_m_m /3/
  70 | #V1 #T1 #T2 #_ #IHT12 #L1 #d #e #X #H #L2 #HL12
  71   elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X;
  72   elim (IHT12 … HTT1 … HL12) -IHT12 HTT1 HL12 /3/
  73 ]
  74 qed.
  75
  76 lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
  77                     ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
  78                     ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
  79 #I #V1 #V2 #T1 #T2 #U1 #HV12 #HT12 #HTU1
  80 elim (tpr_tps_ltpr … HT12 … HTU1 (⋆. 𝕓{I} V2) ?) -HT12 HTU1 /3/
  81 qed.
  82
  83 lemma tpr_tpss_ltpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
  84                      ∀d,e,U1. L1 ⊢ T1 [d, e] ≫* U1 →
  85                      ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫* U2.
  86 #L1 #L2 #HL12 #T1 #T2 #HT12 #d #e #U1 #HTU1 @(tpss_ind … HTU1) -U1
  87 [ /2/
  88 | -HT12 #U #U1 #_ #HU1 * #T #HUT #HT2
  89   elim (tpr_tps_ltpr … HUT … HU1 … HL12) -HUT HU1 HL12 #U2 #HU12 #HTU2
  90   lapply (tpss_trans_eq … HT2 … HTU2) -T /2/
  91 ]
  92 qed.