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

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  14
  15 include "basic_2/substitution/lleq_ext.ma".
  16 include "basic_2/computation/lpxs_ldrop.ma".
  17 include "basic_2/computation/lpxs_cpxs.ma".
  18
  19 (* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 fact le_repl_sn_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
  24 // qed-.
  25
  26 axiom cpxs_cpys_conf_lpxs: ∀h,g,G,d,e.
  27                            ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h, g] T1 →
  28                            ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
  29                            ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 →
  30                            ∃∃T.  ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡*[h, g] T.
  31
  32 axiom cpxs_conf_lpxs_lpys: ∀h,g,G,d,e.
  33                            ∀I,L0,V0,T0,T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡*[h, g] T1 →
  34                            ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ▶*[d, e] V2 →
  35                            ∃∃T. ⦃G, L1.ⓑ{I}V0⦄  ⊢ T1 ▶*[⫯d, e] T & ⦃G, L0.ⓑ{I}V2⦄ ⊢ T0 ➡*[h, g] T.
  36
  37
  38 include "basic_2/reduction/cpx_cpys.ma".
  39
  40 fact pippo_aux: ∀h,g,G,L1,T,T1,d,e. ⦃G, L1⦄ ⊢ T ▶*[d, e] T1 → e = ∞ →
  41                 ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
  42                 ∃∃T2. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 & ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 &
  43                       L1 ⋕[T, d] L2 ↔ T1 = T2.
  44 #h #g #G #L1 #T #T1 #d #e #H @(cpys_ind_alt … H) -G -L1 -T -T1 -d -e [ * ]
  45 [ /5 width=5 by lpxs_fwd_length, lleq_sort, ex3_intro, conj/
  46 | #i #G #L1 elim (lt_or_ge i (|L1|)) [2: /6 width=6 by lpxs_fwd_length, lleq_free, le_repl_sn_aux, ex3_intro, conj/ ]
  47   #Hi #d elim (ylt_split i d) [ /5 width=5 by lpxs_fwd_length, lleq_skip, ex3_intro, conj/ ]
  48   #Hdi #e #He #L2 elim (lleq_dec (#i) L1 L2 d) [ /4 width=5 by lpxs_fwd_length, ex3_intro, conj/ ]
  49   #HnL12 #HL12 elim (ldrop_O1_lt L1 i) // -Hi #I #K1 #V1 #HLK1
  50   elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  51   elim (lpxs_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
  52   elim (lift_total V2 0 (i+1)) #W2 #HVW2
  53   @(ex3_intro … W2) /2 width=7 by cpxs_delta, cpys_subst/ -I -K1 -V1 -Hdi
  54   @conj #H [ elim HnL12 // | destruct elim (lift_inv_lref2_be … HVW2) // ]
  55 | /5 width=5 by lpxs_fwd_length, lleq_gref, ex3_intro, conj/
  56 | #I #G #L1 #K1 #V #V1 #T1 #i #d #e #Hdi #Hide #HLK1 #HV1 #HVT1 #_ #He #L2 #HL12 destruct
  57   elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  58   elim (lpxs_inv_pair1 … H) -H #K2 #W #HK12 #HVW #H destruct
  59   elim (cpxs_cpys_conf_lpxs … HVW … HV1 … HK12) -HVW -HV1 -HK12 #W1 #HW1 #VW1
  60   elim (lift_total W1 0 (i+1)) #U1 #HWU1
  61   lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1
  62   @(ex3_intro … U1) /2 width=10 by cpxs_lift, cpys_subst/
  63 | #a #I #G #L #V #V1 #T #T1 #d #e #HV1 #_ #IHV1 #IHT1 #He #L2 #HL12
  64   elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
  65   elim (IHT1 … (L2.ⓑ{I}V2)) /4 width=3 by lpxs_cpx_trans, lpxs_pair, cpys_cpx/ -IHT1 -He #T2 #HT2 #HT12 * #H1T #H2T
  66   elim (cpxs_conf_lpxs_lpys … HT12 … HL12 … HV1) -HT12 -HL12 -HV1 #T0 #HT20 #HT10
  67   @(ex3_intro … (ⓑ{a,I}V2.T0))
  68   [ @cpys_bind // @(cpys_trans_eq … T2) /3 width=5 by lsuby_cpys_trans, lsuby_succ/
  69   | /2 width=1 by cpxs_bind/
  70   | @conj #H destruct
  71     [ elim (lleq_inv_bind … H) -H #HV #HT >H1V -H1V //
  72     | @lleq_bind /2 width=1/
  73
  74
  75      /3 width=5 by lsuby_cpys_trans, lsuby_succ/
  76 | #I #G #L #V #V1 #T #T1 #d #e #HV1 #HT1 #IHV1 #IHT1 #He #L2 #HL12
  77   elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
  78   elim (IHT1 … HL12) // -IHT1 #T2 #HT2 #HT12 * #H1T #H2T -He -HL12
  79   @(ex3_intro … (ⓕ{I}V2.T2)) /2 width=1 by cpxs_flat, cpys_flat/
  80   @conj #H destruct [2: /3 width=1 by lleq_flat/ ]
  81   elim (lleq_inv_flat … H) -H /3 width=1 by eq_f2/
  82 ]
  83
  84
  85
  86     [
  87     | @cpxs_bind //
  88       @(lpx_cpxs_trans … HT12)
  89 |
  90 ]
  91
  92 axiom lleq_lpxs_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ∀K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
  93                        ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2.
  94 (*
  95 #h #g #G #L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
  96 [
  97 |
  98 |
  99 |
 100 |
 101 | #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #K2 #HLK2
 102   elim (IHV … HLK2) -IHV #KV #HLKV #HV
 103   elim (IHT (K2.ⓑ{I}V)) -IHT /2 width=1 by lpxs_pair_refl/ -HLK2 #Y #H #HT
 104   elim (lpxs_inv_pair1 … H) -H #KT #VT #HLKT #_ #H destruct
 105
 106 #h #g #G #L1 #L2 #T #d * #HL12 #IH #K2 #HLK2
 107 *)
 108
 109 (* Properties on lazy equivalence for local environments ********************)
 110
 111 lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
 112                            ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
 113                            ∃∃K2. ⦃G1, K1, T1⦄ ⊃ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
 114 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
 115 [ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
 116   #K0 #V0 #H1KL1 #_ #H destruct
 117   elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
 118   #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 … H) -H #H destruct
 119   /2 width=4 by fqu_lref_O, ex3_intro/
 120 | * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
 121   [ elim (lleq_inv_bind … H)
 122   | elim (lleq_inv_flat … H)
 123   ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
 124 | #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
 125   /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
 126 | #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
 127   /2 width=4 by fqu_flat_dx, ex3_intro/
 128 | #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
 129   elim (ldrop_O1_le (e+1) K1)
 130   [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
 131     #H2KL elim (lpxs_ldrop_trans_O1 … H1KL1 … HL1) -L1
 132     #K0 #HK10 #H1KL lapply (ldrop_mono … HK10 … HK1) -HK10 #H destruct
 133     /3 width=4 by fqu_drop, ex3_intro/
 134   | lapply (ldrop_fwd_length_le2 … HL1) -L -T1 -g
 135     lapply (lleq_fwd_length … H2KL1) //
 136   ]
 137 ]
 138 qed-.
 139
 140 lemma lpxs_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
 141                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
 142                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
 143 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
 144 elim (fquq_inv_gen … H) -H
 145 [ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
 146   /3 width=4 by fqu_fquq, ex3_intro/
 147 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
 148 ]
 149 qed-.
 150
 151 lemma lpxs_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
 152                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
 153                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
 154 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
 155 [ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
 156   /3 width=4 by fqu_fqup, ex3_intro/
 157 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
 158   #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
 159   /3 width=5 by fqup_strap1, ex3_intro/
 160 ]
 161 qed-.
 162
 163 lemma lpxs_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
 164                             ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
 165                             ∃∃K2. ⦃G1, K1, T1⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2 & K2 ⋕[T2, 0] L2.
 166 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
 167 elim (fqus_inv_gen … H) -H
 168 [ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
 169   /3 width=4 by fqup_fqus, ex3_intro/
 170 | * #HG #HL #HT destruct /2 width=4 by ex3_intro/
 171 ]
 172 qed-.