matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg_drops.ma

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   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "ground_2/xoa/ex_5_5.ma".
  16 include "static_2/relocation/drops_drops.ma".
  17 include "static_2/s_computation/fqup_weight.ma".
  18 include "static_2/s_computation/fqup_drops.ma".
  19 include "basic_2/rt_transition/cpg.ma".
  20
  21 (* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
  22
  23 (* Advanced properties ******************************************************)
  24
  25 lemma cpg_delta_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⇩*[i] L ≘ K.ⓓV → ❪G,K❫ ⊢ V ⬈[Rt,c,h] V2 →
  26                        ⇧*[↑i] V2 ≘ T2 →  ❪G,L❫ ⊢ #i ⬈[Rt,c,h] T2.
  27 #Rt #c #h #G #K #V #V2 #i elim i -i
  28 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
  29 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
  30   elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
  31   elim (lifts_split_trans … HVT2 (𝐔❨↑i❩) (𝐔❨1❩) ?) -HVT2 /3 width=3 by cpg_lref/
  32 ]
  33 qed.
  34
  35 lemma cpg_ell_drops: ∀Rt,c,h,G,K,V,V2,i,L,T2. ⇩*[i] L ≘ K.ⓛV → ❪G,K❫ ⊢ V ⬈[Rt,c,h] V2 →
  36                      ⇧*[↑i] V2 ≘ T2 →  ❪G,L❫ ⊢ #i ⬈[Rt,c+𝟘𝟙,h] T2.
  37 #Rt #c #h #G #K #V #V2 #i elim i -i
  38 [ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
  39 | #i #IH #L0 #T0 #H0 #HV2 #HVT2
  40   elim (drops_inv_succ … H0) -H0 #I #L #HLK #H destruct
  41   elim (lifts_split_trans … HVT2 (𝐔❨↑i❩) (𝐔❨1❩) ?) -HVT2 /3 width=3 by cpg_lref/
  42 ]
  43 qed.
  44
  45 (* Advanced inversion lemmas ************************************************)
  46
  47 lemma cpg_inv_lref1_drops: ∀Rt,c,h,G,i,L,T2. ❪G,L❫ ⊢ #i ⬈[Rt,c,h] T2 →
  48                            ∨∨ T2 = #i ∧ c = 𝟘𝟘
  49                             | ∃∃cV,K,V,V2. ⇩*[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
  50                                            ⇧*[↑i] V2 ≘ T2 & c = cV
  51                             | ∃∃cV,K,V,V2. ⇩*[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
  52                                            ⇧*[↑i] V2 ≘ T2 & c = cV + 𝟘𝟙.
  53 #Rt #c #h #G #i elim i -i
  54 [ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
  55   /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
  56 | #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
  57   #I #K #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H
  58   [ * #H1 #H2 destruct lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct /3 width=1 by or3_intro0, conj/ ] *
  59   #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct
  60   lapply (lifts_trans … HWV2 … HVT2 ??) -V2
  61   /4 width=8 by drops_drop, ex4_4_intro, or3_intro2, or3_intro1/
  62 ]
  63 qed-.
  64
  65 lemma cpg_inv_atom1_drops: ∀Rt,c,h,I,G,L,T2. ❪G,L❫ ⊢ ⓪[I] ⬈[Rt,c,h] T2 →
  66                            ∨∨ T2 = ⓪[I] ∧ c = 𝟘𝟘
  67                             | ∃∃s. T2 = ⋆(⫯[h]s) & I = Sort s & c = 𝟘𝟙
  68                             | ∃∃cV,i,K,V,V2. ⇩*[i] L ≘ K.ⓓV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
  69                                              ⇧*[↑i] V2 ≘ T2 & I = LRef i & c = cV
  70                             | ∃∃cV,i,K,V,V2. ⇩*[i] L ≘ K.ⓛV & ❪G,K❫ ⊢ V ⬈[Rt,cV,h] V2 &
  71                                              ⇧*[↑i] V2 ≘ T2 & I = LRef i & c = cV + 𝟘𝟙.
  72 #Rt #c #h * #n #G #L #T2 #H
  73 [ elim (cpg_inv_sort1 … H) -H *
  74   /3 width=3 by or4_intro0, or4_intro1, ex3_intro, conj/
  75 | elim (cpg_inv_lref1_drops … H) -H *
  76   /3 width=10 by or4_intro0, or4_intro2, or4_intro3, ex5_5_intro, conj/
  77 | elim (cpg_inv_gref1 … H) -H
  78   /3 width=1 by or4_intro0, conj/
  79 ]
  80 qed-.
  81
  82 (* Properties with generic slicing for local environments *******************)
  83
  84 (* Note: it should use drops_split_trans_pair2 *)
  85 lemma cpg_lifts_sn: ∀Rt. reflexive … Rt →
  86                     ∀c,h,G. d_liftable2_sn … lifts (cpg Rt h c G).
  87 #Rt #HRt #c #h #G #K #T generalize in match c; -c
  88 @(fqup_wf_ind_eq (Ⓣ) … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
  89 [ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
  90   lapply (lifts_inv_sort1 … H1) -H1 #H destruct
  91   elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
  92   /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
  93 | #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
  94   elim (cpg_inv_lref1_drops … H2) -H2 *
  95   [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
  96   #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
  97   elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
  98   lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
  99   elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
 100   lapply (drops_tls_at … Hf … HY) -HY #HY
 101   elim (drops_inv_skip2 … HY) -HY #Z #L0 #HLK0 #HZ #H destruct
 102   elim (liftsb_inv_pair_sn … HZ) -HZ #W #HVW #H destruct
 103   elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
 104   elim (lifts_total W2 (𝐔❨↑i2❩)) #U2 #HWU2
 105   lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
 106   lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
 107   /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
 108 | #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
 109   lapply (lifts_inv_gref1 … H1) -H1 #H destruct
 110   elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
 111   /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
 112 | #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
 113   elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
 114   elim (cpg_inv_bind1 … H2) -H2 *
 115   [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 116     elim (IH … HV12 … HLK … HVW1) -HV12 //
 117     elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 118     /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
 119   | #cT #T2 #HT21 #HTX2 #H1 #H2 #H3 destruct
 120     elim (lifts_trans4_one … HT21 … HTU1) -HTU1 #U2 #HTU2 #HU21
 121     elim (IH … HTX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -K -V1 -T1 -T2
 122     /3 width=5 by cpg_zeta, ex2_intro/
 123   ]
 124 | * #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
 125   elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
 126   [ elim (cpg_inv_appl1 … H2) -H2 *
 127     [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 128       elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
 129       elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
 130       /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
 131     | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 destruct
 132       elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
 133       elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
 134       elim (IH … HY12 … HLK … HYZ1) -HY12 //
 135       elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 136       /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
 137     | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 destruct
 138       elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
 139       elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
 140       elim (IH … HY12 … HLK … HYZ1) -HY12 //
 141       elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 142       elim (lifts_total W2 (𝐔❨1❩)) #W20 #HW20
 143       lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
 144       lapply (lifts_conf … HV20 … H (⫯f) ?) -V2 /2 width=3 by after_uni_one_sn/
 145       /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
 146     ]
 147   | elim (cpg_inv_cast1 … H2) -H2 *
 148     [ #cV #cT #V2 #T2 #HV12 #HT12 #HcVT #H1 #H2 destruct
 149       elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
 150       elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
 151       /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
 152     | #cT #HT12 #H destruct
 153       elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
 154       /3 width=3 by cpg_eps, ex2_intro/
 155     | #cV #HV12 #H destruct
 156       elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
 157       /3 width=3 by cpg_ee, ex2_intro/
 158     ]
 159   ]
 160 ]
 161 qed-.
 162
 163 lemma cpg_lifts_bi: ∀Rt. reflexive … Rt →
 164                     ∀c,h,G. d_liftable2_bi … lifts (cpg Rt h c G).
 165 /3 width=12 by cpg_lifts_sn, d_liftable2_sn_bi, lifts_mono/ qed-.
 166
 167 (* Inversion lemmas with generic slicing for local environments *************)
 168
 169 lemma cpg_inv_lifts_sn: ∀Rt. reflexive … Rt →
 170                         ∀c,h,G. d_deliftable2_sn … lifts (cpg Rt h c G).
 171 #Rt #HRt #c #h #G #L #U generalize in match c; -c
 172 @(fqup_wf_ind_eq (Ⓣ) … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
 173 [ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
 174   lapply (lifts_inv_sort2 … H1) -H1 #H destruct
 175   elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
 176   /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
 177 | #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
 178   elim (cpg_inv_lref1_drops … H2) -H2 *
 179   [ #H1 #H2 destruct /3 width=3 by cpg_refl, ex2_intro/ ]
 180   #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
 181   elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
 182   lapply (drops_split_div … HLK (𝐔❨i1❩) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
 183   lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
 184   lapply (drops_tls_at … Hf … HLY0) -HLY0 #HLY0
 185   elim (drops_inv_skip1 … HLY0) -HLY0 #Z #K0 #HLK0 #HZ #H destruct
 186   elim (liftsb_inv_pair_dx … HZ) -HZ #V #HVW #H destruct
 187   elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
 188   lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
 189   elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
 190   /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
 191 | #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
 192   lapply (lifts_inv_gref2 … H1) -H1 #H destruct
 193   elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
 194   /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
 195 | #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
 196   elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
 197   elim (cpg_inv_bind1 … H2) -H2 *
 198   [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
 199     elim (IH … HW12 … HLK … HVW1) -HW12 //
 200     elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 201     /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
 202   | #cU #U2 #HU21 #HUX2 #H1 #H2 #H3 destruct
 203     elim (lifts_div4_one … HTU1 … HU21) -HTU1 #T2 #HT21 #HTU2
 204     elim (IH … HUX2 … HLK … HTU2) [| /3 width=1 by fqup_zeta/ ] -L -W1 -U1 -U2
 205     /3 width=5 by cpg_zeta, ex2_intro/
 206   ]
 207 | * #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
 208   elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
 209   [ elim (cpg_inv_appl1 … H2) -H2 *
 210     [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
 211       elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
 212       elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
 213       /3 width=5 by cpg_appl, lifts_flat, ex2_intro/
 214     | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 destruct
 215       elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
 216       elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
 217       elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
 218       elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 219       /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
 220     | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 destruct
 221       elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
 222       elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
 223       elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
 224       elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip, ext2_pair/ ]
 225       lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
 226       elim (lifts_split_trans … H ? (⫯f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
 227       /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
 228     ]
 229   | elim (cpg_inv_cast1 … H2) -H2 *
 230     [ #cW #cU #W2 #U2 #HW12 #HU12 #HcWU #H1 #H2 destruct
 231       elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
 232       elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
 233       /3 width=5 by cpg_cast, lifts_flat, ex2_intro/
 234     | #cU #HU12 #H destruct
 235       elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
 236       /3 width=3 by cpg_eps, ex2_intro/
 237     | #cW #HW12 #H destruct
 238       elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
 239       /3 width=3 by cpg_ee, ex2_intro/
 240     ]
 241   ]
 242 ]
 243 qed-.
 244
 245 lemma cpg_inv_lifts_bi: ∀Rt. reflexive … Rt →
 246                         ∀c,h,G. d_deliftable2_bi … lifts (cpg Rt h c G).
 247 /3 width=12 by cpg_inv_lifts_sn, d_deliftable2_sn_bi, lifts_inj/ qed-.