matita/matita/contribs/lambdadelta/basic_2A/computation/lsubc_drop.ma

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  14
  15 include "basic_2A/static/aaa_lift.ma".
  16 include "basic_2A/computation/lsubc.ma".
  17
  18 (* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC 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,m. ⬇[s, 0, m] L2 ≡ K2 →
  25                            ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
  26 #RP #G #L1 #L2 #H elim H -L1 -L2
  27 [ #X #s #m #H elim (drop_inv_atom1 … H) -H /4 width=3 by drop_atom, ex2_intro/
  28 | #I #L1 #L2 #V #_ #IHL12 #X #s #m #H
  29   elim (drop_inv_O1_pair1 … H) -H * #Hm #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 #m #H
  35   elim (drop_inv_O1_pair1 … H) -H * #Hm #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. gcp RR RS RP →
  45                         ∀G,L1,K1,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
  46                         ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇[Ⓕ, l, m] L2 ≡ K2.
  47 #RR #RS #RP #Hgcp #G #L1 #K1 #l #m #H elim H -L1 -K1 -l -m
  48 [ #l #m #Hm #X #H elim (lsubc_inv_atom1 … H) -H
  49   >Hm /2 width=3 by ex2_intro/
  50 | #L1 #I #V1 #X #H
  51   elim (lsubc_inv_pair1 … H) -H *
  52   [ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
  53   | #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
  54     /3 width=4 by lsubc_beta, drop_pair, ex2_intro/
  55   ]
  56 | #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12
  57   elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
  58 | #I #L1 #K1 #V1 #V2 #l #m #HLK1 #HV21 #IHLK1 #X #H
  59   elim (lsubc_inv_pair1 … H) -H *
  60   [ #K2 #HK12 #H destruct
  61     elim (IHLK1 … HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
  62   | #K2 #V #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
  63     elim (lift_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
  64     elim (IHLK1 … HK12) #K #HL1K #HK2
  65     lapply (gcr_lift … Hgcp … HV2 … HLK1 … HV3) -HV2
  66     lapply (gcr_lift … Hgcp … H1W2 … HLK1 … HW23) -H1W2
  67     /4 width=11 by lsubc_beta, aaa_lift, drop_skip, ex2_intro/
  68   ]
  69 ]
  70 qed-.