matita/matita/contribs/lambda-delta/Basic-2/reduction/tpr_tps.ma

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  14
  15 include "Basic-2/substitution/tps_tps.ma".
  16 include "Basic-2/substitution/ltps_tps.ma".
  17 include "Basic-2/reduction/ltpr_drop.ma".
  18
  19 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
  20
  21 (* Note: the constant 1 comes from tps_subst *)
  22 (* Basic-1: was: pr0_subst1 *)
  23 lemma tpr_tps_ltpr: ∀T1,T2. T1 ⇒ T2 →
  24                     ∀L1,d,U1. L1 ⊢ T1 [d, 1] ≫ U1 →
  25                     ∀L2. L1 ⇒ L2 →
  26                     ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, 1] ≫ U2.
  27 #T1 #T2 #H elim H -H T1 T2
  28 [ #I #L1 #d #X #H
  29   elim (tps_inv_atom1 … H) -H
  30   [ #H destruct -X /2/
  31   | * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct -I;
  32     elim (ltpr_drop_conf … HLK1 … HL12) -HLK1 HL12 #X #HLK2 #H
  33     elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct -X;
  34     elim (lift_total V2 0 (i+1)) #U2 #HVU2
  35     lapply (tpr_lift … HV12 … HVU1 … HVU2) -HV12 HVU1 #HU12
  36     @ex2_1_intro [2: @HU12 | skip | /2/ ] (**) (* /3 width=6/ is too slow *)
  37   ]
  38 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #X #H #L2 #HL12
  39   elim (tps_inv_flat1 … H) -H #W1 #U1 #HVW1 #HTU1 #H destruct -X;
  40   elim (IHV12 … HVW1 … HL12) -IHV12 HVW1;
  41   elim (IHT12 … HTU1 … HL12) -IHT12 HTU1 HL12 /3 width=5/
  42 | #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #X #H #L2 #HL12
  43   elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X;
  44   elim (tps_inv_bind1 … HY) -HY #WW #TT1 #_ #HTT1 #H destruct -Y;
  45   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  46   elim (IHT12 … HTT1 (L2. 𝕓{Abst} WW) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  47   lapply (tps_leq_repl_dx … HTT2 (L2. 𝕓{Abbr} VV2) ?) -HTT2 /3 width=5/
  48 | #I #V1 #V2 #T1 #T2 #U2 #HV12 #_ #HTU2 #IHV12 #IHT12 #L1 #d #X #H #L2 #HL12
  49   elim (tps_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X;
  50   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  51   elim (IHT12 … HTT1 (L2. 𝕓{I} VV2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  52   elim (tps_conf_neq … HTT2 … HTU2 ?) -HTT2 HTU2 T2 /2/ #T2 #HTT2 #HUT2
  53   lapply (tps_leq_repl_dx … HTT2 (L2. 𝕓{I} V2) ?) -HTT2 /2/ #HTT2
  54   elim (ltps_tps_conf … HTT2 (L2. 𝕓{I} VV2) (d + 1) 1 ?) -HTT2 /2/ #W2 #HTTW2 #HTW2
  55   lapply (tps_leq_repl_dx … HTTW2 (⋆. 𝕓{I} VV2) ?) -HTTW2 /2/ #HTTW2
  56   lapply (tps_trans_ge … HUT2 … HTW2 ?) -HUT2 HTW2 // #HUW2
  57   /3 width=5/
  58 | #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #X #H #L2 #HL12
  59   elim (tps_inv_flat1 … H) -H #VV1 #Y #HVV1 #HY #HX destruct -X;
  60   elim (tps_inv_bind1 … HY) -HY #WW1 #TT1 #HWW1 #HTT1 #H destruct -Y;
  61   elim (IHV12 … HVV1 … HL12) -IHV12 HVV1 #VV2 #HVV12 #HVV2
  62   elim (IHW12 … HWW1 … HL12) -IHW12 HWW1 #WW2 #HWW12 #HWW2
  63   elim (IHT12 … HTT1 (L2. 𝕓{Abbr} WW2) ?) -IHT12 HTT1 /2/ -HL12 #TT2 #HTT12 #HTT2
  64   elim (lift_total VV2 0 1) #VV #H2VV
  65   lapply (tps_lift_ge … HVV2 (L2. 𝕓{Abbr} WW2) … HV2 H2VV ?) -HVV2 HV2 /2/ #HVV
  66   @ex2_1_intro [2: @tpr_theta |1: skip |3: @tps_bind [2: @tps_flat ] ] /width=11/ (**) (* /4 width=11/ is too slow *)
  67 | #V1 #TT1 #T1 #T2 #HT1 #_ #IHT12 #L1 #d #X #H #L2 #HL12
  68   elim (tps_inv_bind1 … H) -H #V2 #TT2 #HV12 #HTT12 #H destruct -X;
  69   elim (tps_inv_lift1_ge … HTT12 L1 … HT1 ?) -HTT12 HT1 /2/ #T2 #HT12 #HTT2
  70   elim (IHT12 … HT12 … HL12) -IHT12 HT12 HL12 <minus_plus_m_m /3/
  71 | #V1 #T1 #T2 #_ #IHT12 #L1 #d #X #H #L2 #HL12
  72   elim (tps_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct -X;
  73   elim (IHT12 … HTT1 … HL12) -IHT12 HTT1 HL12 /3/
  74 ]
  75 qed.
  76
  77 lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
  78                     ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
  79                     ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
  80 /3 width=7/ qed.