matita/matita/contribs/lambdadelta/basic_2/etc/cpr/tpr_lift.etc

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  14
  15 include "basic_2/substitution/tps_lift.ma".
  16 include "basic_2/reducibility/tpr.ma".
  17
  18 (* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
  19
  20 (* Relocation properties ****************************************************)
  21
  22 (* Basic_1: was: pr0_lift *)
  23 lemma tpr_lift: t_liftable tpr.
  24 #T1 #T2 #H elim H -T1 -T2
  25 [ * #i #U1 #d #e #HU1 #U2 #HU2
  26   lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
  27   [ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
  28   | lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
  29   | lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
  30   ]
  31 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
  32   elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
  33   elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
  34 | #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
  35   elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
  36   elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
  37   elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
  38 | #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
  39   elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
  40   elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
  41   elim (lift_total T (d + 1) e) #U #HTU
  42   @tpr_delta
  43   [4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
  44 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
  45   elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
  46   elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
  47   elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
  48   elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
  49   elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
  50 | #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
  51   elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
  52   elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
  53 | #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
  54   elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
  55 ]
  56 qed.
  57
  58 (* Basic_1: was: pr0_gen_lift *)
  59 lemma tpr_inv_lift1: t_deliftable_sn tpr.
  60 #T1 #T2 #H elim H -T1 -T2
  61 [ * #i #X #d #e #HX
  62   [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
  63   | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
  64   | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
  65   ]
  66 | #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
  67   elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
  68   elim (IHV12 … HV01) -V1
  69   elim (IHT12 … HT01) -T1 /3 width=5/
  70 | #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
  71   elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
  72   elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
  73   elim (IHV12 … HV01) -V1
  74   elim (IHT12 … HT01) -T1 /3 width=5/
  75 | #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
  76   elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
  77   elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
  78   elim (IHT1 … HUT1) -T1 #U #HUT #HU1
  79   elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
  80   @ex2_intro  [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
  81 | #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
  82   elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
  83   elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
  84   elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
  85   elim (IHW12 … HW01) -W1 #W3 #HW32 #HW03
  86   elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
  87   elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
  88   @ex2_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
  89 | #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
  90   elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
  91   elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
  92   elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
  93 | #V #T1 #T2 #_ #IHT12 #X #d #e #HX
  94   elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
  95   elim (IHT12 … HT01) -T1 /3 width=3/
  96 ]
  97 qed-.
  98
  99 (* Advanced inversion lemmas ************************************************)
 100
 101 (* Basic_1: was pr0_gen_abst *)
 102 lemma tpr_inv_abst1: ∀a,V1,T1,U2. ⓛ{a}V1. T1 ➡ U2 →
 103                      ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓛ{a}V2. T2.
 104 #a #V1 #T1 #U2 #H elim (tpr_inv_bind1 … H) -H *
 105 [ #V2 #T #T2 #HV12 #HT1 #HT2
 106   lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /2 width=5/
 107 | #T2 #_ #_ #_ #H destruct
 108 ]
 109 qed-.
 110
 111 (* Advanced forward lemmas **************************************************)
 112
 113 lemma tpr_fwd_abst1: ∀a,V1,T1,U2. ⓛ{a}V1.T1 ➡ U2 → ∀b,I,W.
 114                      ∃∃V2,T2. ⓑ{b,I}W.T1 ➡ ⓑ{b,I}W.T2 &
 115                               U2 = ⓛ{a}V2.T2.
 116 #a #V1 #T1 #U2 #H #b #I #W elim (tpr_inv_bind1 … H) -H *
 117 [ #V2 #T #T2 #HV12 #HT1 #HT2
 118   lapply (tps_inv_refl_SO2 … HT2 ???) -HT2 // /3 width=4/
 119 | #T2 #_ #_ #_ #H destruct
 120 ]
 121 qed-.