matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_ltpss.ma

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  14
  15 include "basic_2/unfold/tpss_tpss.ma".
  16 include "basic_2/unfold/ltpss_tpss.ma".
  17
  18 (* PARTIAL UNFOLD ON LOCAL ENVIRONMENTS *************************************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma ltpss_tpss_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ▶* U2 →
  23                        ∀L1,d1,e1. L0 [d1, e1] ▶* L1 →
  24                        ∃∃T. L1 ⊢ T2 [d2, e2] ▶* T &
  25                             L1 ⊢ U2 [d1, e1] ▶* T.
  26 #L0 #T2 #U2 #d2 #e2 #H #L1 #d1 #e1 #HL01 @(tpss_ind … H) -U2 /2 width=3/
  27 #U #U2 #_ #HU2 * #X2 #HTX2 #HUX2
  28 elim (ltpss_tps_conf … HU2 … HL01) -L0 #X1 #HUX1 #HU2X1
  29 elim (tpss_strip_eq … HUX2 … HUX1) -U #X #HX2 #HX1
  30 lapply (tpss_trans_eq … HU2X1 … HX1) -X1 /3 width=3/
  31 qed.
  32
  33 lemma ltpss_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
  34                              L1 [d1, e1] ▶* L0 → L0 ⊢ T2 [d2, e2] ▶* U2 →
  35                              ∃∃T. L1 ⊢ T2 [d2, e2] ▶* T & L0 ⊢ T [d1, e1] ▶* U2.
  36 #L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
  37 [ /2 width=3/
  38 | #U #U2 #_ #HU2 * #T #HT2 #HTU
  39   elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
  40   elim (ltpss_tps_trans … HTU … HL10) -HTU -HL10 #X #HTX #HXU
  41   lapply (tpss_trans_eq … HXU HU2) -U /3 width=3/
  42 ]
  43 qed.
  44
  45 (* Main properties **********************************************************)
  46
  47 fact ltpss_conf_aux: ∀K,K1,L1,d1,e1. K1 [d1, e1] ▶* L1 →
  48                      ∀K2,L2,d2,e2. K2 [d2, e2] ▶* L2 → K1 = K → K2 = K →
  49                      ∃∃L. L1 [d2, e2] ▶* L & L2 [d1, e1] ▶* L.
  50 #K @(lw_wf_ind … K) -K #K #IH #K1 #L1 #d1 #e1 * -K1 -L1 -d1 -e1
  51 [ -IH /2 width=3/
  52 | -IH #K1 #I1 #V1 #K2 #L2 #d2 #e2 * -K2 -L2 -d2 -e2
  53   [ /2 width=3/
  54   | #K2 #I2 #V2 #H1 #H2 destruct /2 width=3/
  55   | #K2 #L2 #I2 #W2 #V2 #e2 #HKL2 #HWV2 #H1 #H2 destruct /3 width=3/
  56   | #K2 #L2 #I2 #W2 #V2 #d2 #e2 #HKL2 #HWV2 #H1 #H2 destruct /3 width=3/
  57   ]
  58 | #K1 #L1 #I1 #W1 #V1 #e1 #HKL1 #HWV1 #K2 #L2 #d2 #e2 * -K2 -L2 -d2 -e2
  59   [ -IH #d2 #e2 #H1 #H2 destruct
  60   | -IH #K2 #I2 #V2 #H1 #H2 destruct /3 width=5/
  61   | #K2 #L2 #I2 #W2 #V2 #e2 #HKL2 #HWV2 #H1 #H2 destruct
  62     elim (IH … HKL1 … HKL2 ? ?) -IH [2,4: // |3: skip |5: /2 width=1/ ] -K1 #L #HL1 #HL2
  63     elim (ltpss_tpss_conf … HWV1 … HL1) #U1 #HWU1 #HVU1
  64     elim (ltpss_tpss_conf … HWV2 … HL2) #U2 #HWU2 #HVU2
  65     elim (tpss_conf_eq … HWU1 … HWU2) -W1 #W #HU1W #HU2W
  66     lapply (tpss_trans_eq … HVU1 HU1W) -U1
  67     lapply (tpss_trans_eq … HVU2 HU2W) -U2 /3 width=5/
  68   | #K2 #L2 #I2 #W2 #V2 #d2 #e2 #HKL2 #HWV2 #H1 #H2 destruct
  69     elim (IH … HKL1 … HKL2 ? ?) -IH [2,4: // |3: skip |5: /2 width=1/ ] -K1 #L #HL1 #HL2
  70     elim (ltpss_tpss_conf … HWV1 … HL1) #U1 #HWU1 #HVU1
  71     elim (ltpss_tpss_conf … HWV2 … HL2) #U2 #HWU2 #HVU2
  72     elim (tpss_conf_eq … HWU1 … HWU2) -W1 #W #HU1W #HU2W
  73     lapply (tpss_trans_eq … HVU1 HU1W) -U1
  74     lapply (tpss_trans_eq … HVU2 HU2W) -U2 /3 width=5/
  75   ]
  76 | #K1 #L1 #I1 #W1 #V1 #d1 #e1 #HKL1 #HWV1 #K2 #L2 #d2 #e2 * -K2 -L2 -d2 -e2
  77   [ -IH #d2 #e2 #H1 #H2 destruct
  78   | -IH #K2 #I2 #V2 #H1 #H2 destruct /3 width=5/
  79   | #K2 #L2 #I2 #W2 #V2 #e2 #HKL2 #HWV2 #H1 #H2 destruct
  80     elim (IH … HKL1 … HKL2 ? ?) -IH [2,4: // |3: skip |5: /2 width=1/ ] -K1 #L #HL1 #HL2
  81     elim (ltpss_tpss_conf … HWV1 … HL1) #U1 #HWU1 #HVU1
  82     elim (ltpss_tpss_conf … HWV2 … HL2) #U2 #HWU2 #HVU2
  83     elim (tpss_conf_eq … HWU1 … HWU2) -W1 #W #HU1W #HU2W
  84     lapply (tpss_trans_eq … HVU1 HU1W) -U1
  85     lapply (tpss_trans_eq … HVU2 HU2W) -U2 /3 width=5/
  86   | #K2 #L2 #I2 #W2 #V2 #d2 #e2 #HKL2 #HWV2 #H1 #H2 destruct
  87     elim (IH … HKL1 … HKL2 ? ?) -IH [2,4: // |3: skip |5: /2 width=1/ ] -K1 #L #HL1 #HL2
  88     elim (ltpss_tpss_conf … HWV1 … HL1) #U1 #HWU1 #HVU1
  89     elim (ltpss_tpss_conf … HWV2 … HL2) #U2 #HWU2 #HVU2
  90     elim (tpss_conf_eq … HWU1 … HWU2) -W1 #W #HU1W #HU2W
  91     lapply (tpss_trans_eq … HVU1 HU1W) -U1
  92     lapply (tpss_trans_eq … HVU2 HU2W) -U2 /3 width=5/
  93   ]
  94 ]
  95 qed.
  96
  97 lemma ltpss_conf: ∀L0,L1,d1,e1. L0 [d1, e1] ▶* L1 →
  98                   ∀L2,d2,e2. L0 [d2, e2] ▶* L2 →
  99                   ∃∃L. L1 [d2, e2] ▶* L & L2 [d1, e1] ▶* L.
 100 /2 width=7/ qed.