matita/matita/contribs/lambdadelta/basic_2/etc_2A1/lsubr/ssta_lpss.etc

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   3 (*      ||M||                                                             *)
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   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/substitution/lpss_ldrop.ma".
  16 include "basic_2/static/ssta_lift.ma".
  17
  18 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
  19
  20 (* Properties about sn parallel substitution ********************************)
  21
  22 (* Note: apparently this was missing in basic_1 *)
  23 lemma ssta_cpss_lpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
  24                            ∀T2. L1 ⊢ T1 ▶* T2 → ∀L2. L1 ⊢ ▶* L2 →
  25                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
  26 #h #g #L1 #T1 #U1 #l #H elim H -L1 -T1 -U1 -l
  27 [ #L1 #k1 #l1 #Hkl1 #X #H
  28   >(cpss_inv_sort1 … H) -H /3 width=3/
  29 | #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #_ #HWU1 #IHVW1 #X #H #L2 #HL12
  30   elim (cpss_inv_lref1 … H) -H
  31   [ #H destruct
  32     elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  33     elim (lpss_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
  34     lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
  35     elim (IHVW1 … HV12 … HK12) -IHVW1 -HV12 -HK12 #W2 #HVW2 #HW12
  36     elim (lift_total W2 0 (i+1)) #U2 #HWU2
  37     lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
  38   | * #Y #Z #V2 #H #HV12 #HV2
  39     lapply (ldrop_mono … H … HLK1) -H #H destruct
  40     elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #Z #H #HLK2
  41     elim (lpss_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
  42     lapply (ldrop_fwd_ldrop2 … HLK2) -V0 #HLK2
  43     lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
  44     elim (IHVW1 … HV12 … HK12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
  45     elim (lift_total W2 0 (i+1)) #U2 #HWU2
  46     lapply (ssta_lift … HVW2 … HLK2 … HV2 … HWU2) -HVW2 -HLK2 -HV2
  47     lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 -HWU2 /3 width=3/
  48   ]
  49 | #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #_ #HWU1 #IHWV1 #X #H #L2 #HL12
  50   elim (cpss_inv_lref1 … H) -H [ | -IHWV1 -HWU1 -HL12 ]
  51   [ #H destruct
  52     elim (lpss_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  53     elim (lpss_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
  54     lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 #HLK1
  55     elim (IHWV1 … HW12 … HK12) -IHWV1 -HK12 #V2 #HWV2 #_
  56     elim (lift_total W2 0 (i+1)) #U2 #HWU2
  57     lapply (cpss_lift … HW12 … HLK1 … HWU1 … HWU2) -HW12 -HLK1 -HWU1 /3 width=6/
  58   | * #K2 #V2 #W2 #HLK2 #_ #_
  59     lapply (ldrop_mono … HLK2 … HLK1) -HLK1 -HLK2 #H destruct
  60   ]
  61 | #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
  62   elim (cpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  63   elim (IHTU1 … HT12 (L2.ⓑ{I}V2)) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
  64 | #L1 #V1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
  65   elim (cpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  66   elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=5/
  67 | #L1 #W1 #T1 #U1 #l #_ #IHTU1 #X #H #L2 #HL12
  68   elim (cpss_inv_flat1 … H) -H #W2 #T2 #HW12 #HT12 #H destruct
  69   elim (IHTU1 … HT12 … HL12) -IHTU1 -HT12 -HL12 /3 width=3/
  70 ]
  71 qed-.
  72
  73 lemma ssta_cpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
  74                       ∀T2. L ⊢ T1 ▶* T2 →
  75                       ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L ⊢ U1 ▶* U2.
  76 /2 width=3 by ssta_cpss_lpss_conf/ qed-.
  77
  78 lemma ssta_lpss_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g] ⦃l, U1⦄ →
  79                       ∀L2. L1 ⊢ ▶* L2 →
  80                       ∃∃U2. ⦃h, L2⦄ ⊢ T •[g] ⦃l, U2⦄ & L1 ⊢ U1 ▶* U2.
  81 /2 width=3 by ssta_cpss_lpss_conf/ qed-.