matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma

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  14
  15 include "basic_2/static/aaa_lift.ma".
  16 include "basic_2/computation/lsubc.ma".
  17
  18 (* LOCAL ENVIRONMENT REFINEMENT FOR ABSTRACT CANDIDATES OF REDUCIBILITY *****)
  19
  20 (* Properties concerning basic local environment slicing ********************)
  21
  22 (* Basic_1: was: csubc_drop_conf_O *)
  23 (* Note: the constant 0 can not be generalized *)
  24 lemma lsubc_drop_O1_trans: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀K2,s,e. ⇩[s, 0, e] L2 ≡ K2 →
  25                            ∃∃K1. ⇩[s, 0, e] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
  26 #RP #G #L1 #L2 #H elim H -L1 -L2
  27 [ #X #s #e #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
  28 | #I #L1 #L2 #V #_ #IHL12 #X #s #e #H
  29   elim (drop_inv_O1_pair1 … H) -H * #He #H destruct
  30   [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
  31     /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
  32   | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
  33   ]
  34 | #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #e #H
  35   elim (drop_inv_O1_pair1 … H) -H * #He #H destruct
  36   [ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 … H) -H
  37     /3 width=8 by lsubc_beta, drop_pair, ex2_intro/
  38   | elim (IHL12 … H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
  39   ]
  40 ]
  41 qed-.
  42
  43 (* Basic_1: was: csubc_drop_conf_rev *)
  44 lemma drop_lsubc_trans: ∀RR,RS,RP.
  45                         acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
  46                         ∀G,L1,K1,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
  47                         ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⇩[Ⓕ, d, e] L2 ≡ K2.
  48 #RR #RS #RP #Hacp #Hacr #G #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
  49 [ #d #e #He #X #H elim (lsubc_inv_atom1 … H) -H
  50   >He /2 width=3 by ex2_intro/
  51 | #L1 #I #V1 #X #H
  52   elim (lsubc_inv_pair1 … H) -H *
  53   [ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
  54   | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
  55     /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
  56   ]
  57 | #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12
  58   elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
  59 | #I #L1 #K1 #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H
  60   elim (lsubc_inv_pair1 … H) -H *
  61   [ #K2 #HK12 #H destruct
  62     elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
  63   | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
  64     elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
  65     elim (IHLK1 … HK12) #K #HL1K #HK2
  66     lapply (aacr_acr … Hacp Hacr A) -Hacp -Hacr #HA
  67     lapply (s0 … HA … HV2 … HLK1 HV3) -HV2
  68     lapply (s0 … HA … H1W2 … HLK1 HW23) -H1W2
  69     /4 width=11 by lsubc_beta, aaa_lift, drop_skip, ex2_intro/
  70   ]
  71 ]
  72 qed-.