matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_tps.ma

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   3 (*      ||M||                                                             *)
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   5 (*      ||T||                                                             *)
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  14
  15 include "Basic-2/substitution/tps_split.ma".
  16
  17 (* PARTIAL SUBSTITUTION ON TERMS ********************************************)
  18
  19 (* Main properties **********************************************************)
  20 (*
  21 theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
  22                    L ⊢ T1 [d, e] ≫ T2.
  23 #L #T1 #T #d #e #H elim H -L T1 T d e
  24 [ //
  25 | //
  26 | #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
  27   lapply (drop_fwd_drop2 … HLK) #H
  28   elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
  29   @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
  30 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
  31   elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
  32   @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
  33 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
  34   elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
  35 ]
  36 qed.
  37 *)
  38
  39 axiom tps_conf_subst_subst_lt: ∀L,K1,V1,T1,i1,d,e,T2,K2,V2,i2.
  40    ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
  41    ↑[O, i1 + 1] V1 ≡ T1 → ↑[O, i2 + 1] V2 ≡ T2 →
  42    d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
  43    ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
  44 (*
  45 #L #K1 #V1 #T1 #i1 #d #e #T2 #K2 #V2 #i2
  46 #HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
  47 *)
  48
  49 theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
  50                   ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
  51 #L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
  52 [ /2/
  53 | /2/
  54 | #L #K1 #V1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVT1 #T2 #H
  55   elim (tps_inv_lref1 … H) -H
  56   [ #HX destruct -T2 /4/
  57   | * #K2 #V2 #i2 #Hdi2 #Hi2de #HLK2 #HVT2
  58     elim (lt_or_eq_or_gt i1 i2) #Hi12
  59     [ @tps_conf_subst_subst_lt /width=15/
  60     | -Hdi2 Hi2de; destruct -i2;
  61       lapply (drop_mono … HLK1 … HLK2) -HLK1 #H destruct -V1 K1
  62       >(lift_mono … HVT1 … HVT2) -HVT1 /2/
  63     | @ex2_1_comm @tps_conf_subst_subst_lt /width=15/
  64     ]
  65   ]
  66 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
  67   elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
  68   elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
  69   elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
  70   @ex2_1_intro
  71   [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
  72   |1: skip
  73   |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
  74   ] // (**) (* /5/ is too slow *)
  75 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
  76   elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
  77   elim (IHV01 … HV02) -IHV01 HV02;
  78   elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
  79 ]
  80 qed.
  81
  82 (*
  83       Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
  84                              (u1,u:?; i:?) (subst0 i u u1 u2) ->
  85                              (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
  86
  87       Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
  88                                   (u1,u:?; i:?) (subst0 i u u2 u1) ->
  89                                   (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
  90
  91 *)