matita/matita/contribs/lambdadelta/basic_2/etc/cpr/tpss_lift.etc

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  14
  15 include "basic_2/substitution/tps_lift.ma".
  16 include "basic_2/unfold/tpss.ma".
  17
  18 (* PARTIAL UNFOLD ON TERMS **************************************************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma tpss_subst: ∀L,K,V,U1,i,d,e.
  23                   d ≤ i → i < d + e →
  24                   ⇩[0, i] L ≡ K. ⓓV → K ⊢ V ▶* [0, d + e - i - 1] U1 →
  25                   ∀U2. ⇧[0, i + 1] U1 ≡ U2 → L ⊢ #i ▶* [d, e] U2.
  26 #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(tpss_ind … H) -U1
  27 [ /3 width=4/
  28 | #U #U1 #_ #HU1 #IHU #U2 #HU12
  29   elim (lift_total U 0 (i+1)) #U0 #HU0
  30   lapply (IHU … HU0) -IHU #H
  31   lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
  32   lapply (tps_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
  33   lapply (tps_weak … HU02 d e ? ?) -HU02 [ >minus_plus >commutative_plus /2 width=1/ | /2 width=1/ | /2 width=3/ ]
  34 ]
  35 qed.
  36
  37 (* Advanced inverion lemmas *************************************************)
  38
  39 lemma tpss_inv_atom1: ∀L,T2,I,d,e. L ⊢ ⓪{I} ▶* [d, e] T2 →
  40                       T2 = ⓪{I} ∨
  41                       ∃∃K,V1,V2,i. d ≤ i & i < d + e &
  42                                    ⇩[O, i] L ≡ K. ⓓV1 &
  43                                    K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
  44                                    ⇧[O, i + 1] V2 ≡ T2 &
  45                                    I = LRef i.
  46 #L #T2 #I #d #e #H @(tpss_ind … H) -T2
  47 [ /2 width=1/
  48 | #T #T2 #_ #HT2 *
  49   [ #H destruct
  50     elim (tps_inv_atom1 … HT2) -HT2 [ /2 width=1/ | * /3 width=10/ ]
  51   | * #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
  52     lapply (ldrop_fwd_ldrop2 … HLK) #H
  53     elim (tps_inv_lift1_ge_up … HT2 … H … HVT ? ? ?) normalize -HT2 -H -HVT [2,3,4: /2 width=1/ ] #V2 <minus_plus #HV2 #HVT2
  54     @or_intror @(ex6_4_intro … Hdi Hide HLK … HVT2 HI) /2 width=3/ (**) (* /4 width=10/ is too slow *)
  55   ]
  56 ]
  57 qed-.
  58
  59 lemma tpss_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i ▶* [d, e] T2 →
  60                       T2 = #i ∨
  61                       ∃∃K,V1,V2. d ≤ i & i < d + e &
  62                                  ⇩[O, i] L ≡ K. ⓓV1 &
  63                                  K ⊢ V1 ▶* [0, d + e - i - 1] V2 &
  64                                  ⇧[O, i + 1] V2 ≡ T2.
  65 #L #T2 #i #d #e #H
  66 elim (tpss_inv_atom1 … H) -H /2 width=1/
  67 * #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=6/
  68 qed-.
  69
  70 lemma tpss_inv_S2: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e + 1] T2 →
  71                    ∀K,V. ⇩[0, d] L ≡ K. ⓛV → L ⊢ T1 ▶* [d + 1, e] T2.
  72 #L #T1 #T2 #d #e #H #K #V #HLK @(tpss_ind … H) -T2 //
  73 #T #T2 #_ #HT2 #IHT
  74 lapply (tps_inv_S2 … HT2 … HLK) -HT2 -HLK /2 width=3/
  75 qed-.
  76
  77 lemma tpss_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 →
  78                          ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
  79 #L #T1 #T2 #d #H #K #V #HLK @(tpss_ind … H) -T2 //
  80 #T #T2 #_ #HT2 #IHT <(tps_inv_refl_SO2 … HT2 … HLK) //
  81 qed-.
  82
  83 (* Relocation properties ****************************************************)
  84
  85 lemma tpss_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
  86                     ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
  87                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
  88                     L ⊢ U1 ▶* [dt, et] U2.
  89 #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(tpss_ind … H) -T2
  90 [ #U2 #H >(lift_mono … HTU1 … H) -H //
  91 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
  92   elim (lift_total T d e) #U #HTU
  93   lapply (IHT … HTU) -IHT #HU1
  94   lapply (tps_lift_le … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
  95 ]
  96 qed.
  97
  98 lemma tpss_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
  99                     ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
 100                     ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
 101                     ∀U2. ⇧[d, e] T2 ≡ U2 → L ⊢ U1 ▶* [dt, et + e] U2.
 102 #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(tpss_ind … H) -T2
 103 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 104 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
 105   elim (lift_total T d e) #U #HTU
 106   lapply (IHT … HTU) -IHT #HU1
 107   lapply (tps_lift_be … HT2 … HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
 108 ]
 109 qed.
 110
 111 lemma tpss_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 ▶* [dt, et] T2 →
 112                     ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
 113                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
 114                     L ⊢ U1 ▶* [dt + e, et] U2.
 115 #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(tpss_ind … H) -T2
 116 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 117 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
 118   elim (lift_total T d e) #U #HTU
 119   lapply (IHT … HTU) -IHT #HU1
 120   lapply (tps_lift_ge … HT2 … HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 // /2 width=3/
 121 ]
 122 qed.
 123
 124 lemma tpss_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 125                          ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 126                          dt + et ≤ d →
 127                          ∃∃T2. K ⊢ T1 ▶* [dt, et] T2 & ⇧[d, e] T2 ≡ U2.
 128 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(tpss_ind … H) -U2
 129 [ /2 width=3/
 130 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 131   elim (tps_inv_lift1_le … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
 132 ]
 133 qed.
 134
 135 lemma tpss_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 136                          ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 137                          dt ≤ d → d + e ≤ dt + et →
 138                          ∃∃T2. K ⊢ T1 ▶* [dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
 139 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(tpss_ind … H) -U2
 140 [ /2 width=3/
 141 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 142   elim (tps_inv_lift1_be … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
 143 ]
 144 qed.
 145
 146 lemma tpss_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 147                          ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 148                          d + e ≤ dt →
 149                          ∃∃T2. K ⊢ T1 ▶* [dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
 150 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(tpss_ind … H) -U2
 151 [ /2 width=3/
 152 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 153   elim (tps_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU // /3 width=3/
 154 ]
 155 qed.
 156
 157 lemma tpss_inv_lift1_eq: ∀L,U1,U2,d,e.
 158                          L ⊢ U1 ▶* [d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
 159 #L #U1 #U2 #d #e #H #T1 #HTU1 @(tpss_ind … H) -U2 //
 160 #U #U2 #_ #HU2 #IHU destruct
 161 <(tps_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
 162 qed.
 163
 164 lemma tpss_inv_lift1_ge_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 165                             ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 166                             d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
 167                             ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
 168                                  ⇧[d, e] T2 ≡ U2.
 169 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(tpss_ind … H) -U2
 170 [ /2 width=3/
 171 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 172   elim (tps_inv_lift1_ge_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
 173 ]
 174 qed.
 175
 176 lemma tpss_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 177                             ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 178                             dt ≤ d → dt + et ≤ d + e →
 179                             ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
 180 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(tpss_ind … H) -U2
 181 [ /2 width=3/
 182 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 183   elim (tps_inv_lift1_be_up … HU2 … HLK … HTU ? ?) -HU2 -HLK -HTU // /3 width=3/
 184 ]
 185 qed.
 186
 187 lemma tpss_inv_lift1_le_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
 188                             ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
 189                             dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
 190                             ∃∃T2. K ⊢ T1 ▶* [dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
 191 #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(tpss_ind … H) -U2
 192 [ /2 width=3/
 193 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
 194   elim (tps_inv_lift1_le_up … HU2 … HLK … HTU ? ? ?) -HU2 -HLK -HTU // /3 width=3/
 195 ]
 196 qed.