matita/matita/contribs/lambdadelta/basic_2/static/frees_drops.ma

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   2 (*       ___                                                              *)
   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/relocation/nstream_coafter.ma".
  16 include "basic_2/relocation/drops_drops.ma".
  17 include "basic_2/static/frees_fqup.ma".
  18
  19 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
  20
  21 (* Advanced properties ******************************************************)
  22
  23 lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
  24                         ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
  25 #b #L elim L -L /2 width=1 by frees_atom/
  26 #L #I #IH *
  27 [ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
  28 | /4 width=3 by frees_lref, drops_inv_drop1/
  29 ]
  30 qed.
  31
  32 lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
  33                         ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
  34 #f #K #V #Hf #i elim i -i
  35 [ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
  36 | #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
  37 ]
  38 qed.
  39
  40 lemma frees_unit_drops: ∀f.  𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
  41                        L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
  42 #f #Hf #I #K #i elim i -i
  43 [ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
  44 | #i #IH #Y #H elim (drops_inv_succ … H) -H
  45   #J #L #HLK #H destruct /3 width=1 by frees_lref/
  46 ]
  47 qed.
  48 (*
  49 lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
  50                         ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
  51 #f #K #s #Hf #i elim i -i
  52 [ #L #H lapply (drops_fwd_isid … H ?) -H //
  53 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
  54 ]
  55 qed.
  56 *)
  57 lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
  58                         ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
  59 #f #K #j #Hf #i elim i -i
  60 [ #L #H lapply (drops_fwd_isid … H ?) -H //
  61 | #i #IH #L #H elim (drops_inv_succ … H) -H
  62   #I #Y #HYK #H destruct /3 width=1 by frees_lref/
  63 ]
  64 qed.
  65 (*
  66 lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
  67                         ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
  68 #f #K #l #Hf #i elim i -i
  69 [ #L #H lapply (drops_fwd_isid … H ?) -H //
  70 | #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
  71 ]
  72 qed.
  73 *)
  74 (* Advanced inversion lemmas ************************************************)
  75
  76 lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
  77                             ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
  78                              | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
  79                                           ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
  80                              | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
  81 #L elim L -L
  82 [ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
  83 [ elim (frees_inv_atom … H) -H #f #Hf #H destruct
  84   /3 width=3 by or3_intro0, ex3_intro/
  85 | elim (frees_inv_unit … H) -H #f #Hf #H destruct
  86   /4 width=3 by drops_refl, or3_intro2, ex3_3_intro/
  87 | elim (frees_inv_pair … H) -H #f #Hf #H destruct
  88   /4 width=7 by drops_refl, or3_intro1, ex3_4_intro/
  89 | elim (frees_inv_lref … H) -H #f #Hf #H destruct
  90   elim (IH … Hf) -IH -Hf *
  91   [ /4 width=3 by drops_drop, or3_intro0, ex3_intro/
  92   | /4 width=7 by drops_drop, or3_intro1, ex3_4_intro/
  93   | /4 width=3 by drops_drop, or3_intro2, ex3_3_intro/
  94   ]
  95 ]
  96 qed-.
  97
  98 (* Properties with generic slicing for local environments *******************)
  99
 100 lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
 101                    ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
 102                    ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
 103 #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
 104 [ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
 105   lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
 106   >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
 107 | #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
 108   elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
 109   elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
 110   lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
 111   elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
 112   elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
 113   lapply (at_inv_uni … Hj) -Hj #H destruct
 114   /3 width=8 by frees_atom_drops, drops_trans/
 115 | #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
 116   lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
 117   lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
 118   elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
 119   elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
 120   elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
 121   lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
 122   /2 width=5 by frees_pair_drops/
 123 | #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
 124   lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
 125   elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
 126   lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
 127   elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
 128   lapply (liftsb_inv_unit_sn … H) -H #H destruct
 129   /2 width=3 by frees_unit_drops/
 130 | #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
 131   lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
 132   lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
 133   elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
 134   elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
 135   elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
 136   lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
 137   lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
 138   >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
 139 | #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
 140   lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
 141   >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
 142 | #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
 143   elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
 144   lapply (isfin_inv_tl … H) -H
 145   elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
 146   elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
 147   elim (coafter_inv_tl1 … H) -H
 148   /5 width=5 by frees_bind, drops_skip, ext2_pair/
 149 | #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
 150   elim (sor_inv_isfin3 … H1f1) //
 151   elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
 152   elim (coafter_sor … H3 … H1f1)
 153   /3 width=5 by coafter_isfin2_fwd, frees_flat/
 154 ]
 155 qed-.
 156
 157 lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
 158                       ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
 159 #b #L #K #HLK #T #U #HTU #f #Hf
 160 @(frees_lifts b … Hf … HTU) //  (**) (* auto fails *)
 161 qed.
 162
 163 (* Forward lemmas with generic slicing for local environments ***************)
 164
 165 lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
 166                          ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
 167                          ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
 168 /4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
 169
 170 (* Inversion lemmas with generic slicing for local environments *************)
 171
 172 lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
 173                           ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
 174                           ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
 175 #b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
 176 /3 width=9 by frees_fwd_coafter, ex2_intro/
 177 qed-.
 178
 179 lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
 180                           ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
 181                           K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
 182 #b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
 183 #f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
 184 /3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
 185 qed-.
 186
 187 lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
 188                        ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
 189                        ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
 190 #b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
 191 /3 width=7 by frees_eq_repl_back, coafter_inj/
 192 qed-.
 193
 194 (* Note: this is used by rex_conf and might be modified *)
 195 lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
 196                             ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
 197                             ∀g1. ↑g1 = ⫱*[n] f1 →
 198                             ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
 199 #f1 #L1 #T1 #H elim H -f1 -L1 -T1
 200 [ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
 201   lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
 202   elim (isid_inv_next … Hf1) -Hf1 //
 203 | #f1 #i #_ #I2 #L2 #V2 #n #H
 204   elim (drops_inv_atom1 … H) -H #H destruct
 205 | #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
 206   [ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
 207     #H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
 208     /2 width=3 by ex2_intro/
 209   | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
 210     #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
 211     /2 width=3 by ex2_intro/
 212   ]
 213 | #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
 214   [ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
 215   | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
 216   ]
 217 | #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
 218   [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
 219   | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
 220     #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
 221     /2 width=3 by ex2_intro/
 222   ]
 223 | #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
 224   lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
 225   elim (isid_inv_next … Hf1) -Hf1 //
 226 | #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
 227   lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
 228   elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
 229   #gV1 #gT1 #Hg1
 230   [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
 231     /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
 232   | -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
 233     /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
 234   ]
 235 | #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
 236   lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
 237   elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
 238   #gV1 #gT1 #Hg1
 239   [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
 240     /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
 241   | -IHV1 #_ #H2 elim (IHT1 … HL12 … H2) -IHT1 -HL12 -H2
 242     /3 width=6 by sor_inv_sle_dx_trans, ex2_intro/
 243   ]
 244 ]
 245 qed-.