matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma

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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/substitution/lleq_ext.ma".
  16 include "basic_2/reduction/lpx_ldrop.ma".
  17
  18 (* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
  19
  20 (* Properties on lazy equivalence for local environments ********************)
  21 (*
  22 lamma pippo: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 → ∀L1. |L1| = |L2| →
  23              ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & |K1| = |K2| &
  24                    (∀T,d. L1 ⋕[T, d] L2 ↔ K1 ⋕[T, d] K2).
  25 #h #g #G #L2 #K2 #H elim H -L2 -K2
  26 [ #L1 #H >(length_inv_zero_dx … H) -L1 /3 width=5 by ex3_intro, conj/
  27 | #I2 #L2 #K2 #V2 #W2 #_ #HVW2 #IHLK2 #Y #H
  28   elim (length_inv_pos_dx … H) -H #I #L1 #V1 #HL12 #H destruct
  29   elim (IHLK2 … HL12) -IHLK2 #K1 #HLK1 #HK12 #IH
  30   elim (eq_term_dec V1 V2) #H destruct
  31   [ @(ex3_intro … (K1.ⓑ{I}W2)) normalize /2 width=1 by /
  32 *)
  33 axiom lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
  34                       ∀L1,T,d. L1 ⋕[T, d] L2 →
  35                       ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & K1 ⋕[T, d] K2.
  36 (*
  37 #h #g #G #L2 #K2 #H elim H -L2 -K2
  38 [ #L1 #T #d #H lapply (lleq_fwd_length … H) -H
  39   #H >(length_inv_zero_dx … H) -L1 /2 width=3 by ex2_intro/
  40 | #I2 #L2 #K2 #V2 #W2 #HLK2 #HVW2 #IHLK2 #Y #T #d #HT
  41   lapply (lleq_fwd_length … HT) #H
  42   elim (length_inv_pos_dx … H) -H #I1 #L1 #V1 #HL12 #H destruct
  43   elim (eq_term_dec V1 V2) #H destruct
  44   [ @ex2_intro …
  45 *)
  46
  47 lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
  48                           ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ⋕[T1, 0] L1 →
  49                           ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ⋕[T2, 0] L2.
  50 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  51 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
  52   #K0 #V0 #H1KL1 #_ #H destruct
  53   elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
  54   #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
  55   /2 width=4 by fqu_lref_O, ex3_intro/
  56 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
  57   [ elim (lleq_inv_bind … H)
  58   | elim (lleq_inv_flat … H)
  59   ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
  60 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
  61   /3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
  62 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
  63   /2 width=4 by fqu_flat_dx, ex3_intro/
  64 | #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
  65   elim (ldrop_O1_le (e+1) K1)
  66   [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
  67     #H2KL elim (lpx_ldrop_trans_O1 … H1KL1 … HL1) -L1
  68     #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
  69     /3 width=4 by fqu_drop, ex3_intro/
  70   | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g
  71     lapply (lleq_fwd_length … H2KL1) //
  72   ]
  73 ]
  74 qed-.
  75
  76 lemma lpx_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
  77                            ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ⋕[T1, 0] L1 →
  78                            ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ⋕[T2, 0] L2.
  79 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
  80 elim (fquq_inv_gen … H) -H
  81 [ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
  82   /3 width=4 by fqu_fquq, ex3_intro/
  83 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
  84 ]
  85 qed-.
  86
  87 lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
  88                            ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ⋕[T1, 0] L1 →
  89                            ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ⋕[T2, 0] L2.
  90 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
  91 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
  92   /3 width=4 by fqu_fqup, ex3_intro/
  93 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
  94   #K #HT1 #H1KL #H2KL elim (lpx_lleq_fqu_trans … HT2 … H1KL H2KL) -L
  95   /3 width=5 by fqup_strap1, ex3_intro/
  96 ]
  97 qed-.
  98
  99 lemma lpx_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
 100                            ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ⋕[T1, 0] L1 →
 101                            ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ⋕[T2, 0] L2.
 102 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
 103 elim (fqus_inv_gen … H) -H
 104 [ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
 105   /3 width=4 by fqup_fqus, ex3_intro/
 106 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
 107 ]
 108 qed-.