matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma

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  14
  15 include "basic_2/relocation/ldrop.ma".
  16 include "basic_2/relocation/lpx_sn.ma".
  17
  18 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
  19
  20 (* alternative definition of lpx_sn *)
  21 inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
  22 | lpx_sn_alt_intro: ∀L1,L2.
  23                     (∀I1,I2,K1,K2,V1,V2,i.
  24                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
  25                     ) →
  26                     (∀I1,I2,K1,K2,V1,V2,i.
  27                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → lpx_sn_alt R K1 K2
  28                     ) → |L1| = |L2| → lpx_sn_alt R L1 L2
  29 .
  30
  31 (* compact definition of lpx_sn_alt *****************************************)
  32
  33 lemma lpx_sn_alt_ind_alt: ∀R. ∀S:relation lenv.
  34                           (∀L1,L2. |L1| = |L2| → (
  35                              ∀I1,I2,K1,K2,V1,V2,i.
  36                              ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
  37                              ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2 & S K1 K2
  38                           ) → S L1 L2) →
  39                           ∀L1,L2. lpx_sn_alt R L1 L2 → S L1 L2.
  40 #R #S #IH #L1 #L2 #H elim H -L1 -L2
  41 #L1 #L2 #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
  42 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (H1 … HLK1 HLK2) -H1
  43 /3 width=7 by and4_intro/
  44 qed-.
  45
  46 lemma lpx_sn_alt_inv_alt: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
  47                           |L1| = |L2| ∧
  48                           ∀I1,I2,K1,K2,V1,V2,i.
  49                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
  50                           ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2.
  51 #R #L1 #L2 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
  52 #L1 #L2 #HL12 #IH @conj // -HL12
  53 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
  54 /2 width=1 by and3_intro/
  55 qed-.
  56
  57 lemma lpx_sn_alt_intro_alt: ∀R,L1,L2. |L1| = |L2| →
  58                             (∀I1,I2,K1,K2,V1,V2,i.
  59                                ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
  60                                ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2
  61                             ) → lpx_sn_alt R L1 L2.
  62 #R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
  63 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
  64 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
  65 qed.
  66
  67 (* Basic forward lemmas ******************************************************)
  68
  69 lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
  70 #R #L1 #L2 #H elim (lpx_sn_alt_inv_alt … H) //
  71 qed-.
  72
  73 (* Basic inversion lemmas ***************************************************)
  74
  75 lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
  76 #R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
  77 normalize /2 width=1 by length_inv_zero_sn/
  78 qed-.
  79
  80 lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
  81                             ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
  82 #R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_alt … H) -H
  83 #H #IH elim (length_inv_pos_sn … H) -H
  84 #I2 #K2 #V2 #HK12 #H destruct
  85 elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
  86 qed-.
  87
  88 lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
  89 #R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
  90 normalize /2 width=1 by length_inv_zero_dx/
  91 qed-.
  92
  93 lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
  94                             ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
  95 #R #I2 #L1 #K2 #V2 #H elim (lpx_sn_alt_inv_alt … H) -H
  96 #H #IH elim (length_inv_pos_dx … H) -H
  97 #I1 #K1 #V1 #HK12 #H destruct
  98 elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
  99 qed-.
 100
 101 (* Basic properties *********************************************************)
 102
 103 lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
 104 #R @lpx_sn_alt_intro_alt //
 105 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
 106 #H destruct
 107 qed.
 108
 109 lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
 110                        lpx_sn_alt R L1 L2 → R L1 V1 V2 →
 111                        lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
 112 #R #I #L1 #L2 #V1 #V2 #H #HV12 elim (lpx_sn_alt_inv_alt … H) -H
 113 #HL12 #IH @lpx_sn_alt_intro_alt normalize //
 114 #I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
 115 [ #HLK1 #HLK2
 116   lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
 117   lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
 118   /4 width=3 by lpx_sn_alt_intro_alt, and3_intro/
 119 | -HL12 -HV12 /3 width=5 by ldrop_inv_drop1/
 120 ]
 121 qed.
 122
 123 (* Main properties **********************************************************)
 124
 125 theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
 126 #R #L1 #L2 #H elim H -L1 -L2
 127 /2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
 128 qed.
 129
 130 (* Main inversion lemmas ****************************************************)
 131
 132 theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
 133 #R #L1 elim L1 -L1
 134 [ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
 135 | #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
 136   #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
 137 ]
 138 qed-.
 139
 140 (* alternative definition of lpx_sn *****************************************)
 141
 142 lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
 143                         (∀I1,I2,K1,K2,V1,V2,i.
 144                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
 145                            ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2
 146                         ) → lpx_sn R L1 L2.
 147 #R #L1 #L2 #HL12 #IH @lpx_sn_alt_inv_lpx_sn
 148 @lpx_sn_alt_intro_alt // -HL12
 149 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
 150 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_lpx_sn_alt, and3_intro/
 151 qed.
 152
 153 lemma lpx_sn_ind_alt: ∀R. ∀S:relation lenv.
 154                       (∀L1,L2. |L1| = |L2| → (
 155                          ∀I1,I2,K1,K2,V1,V2,i.
 156                          ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
 157                          ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2 & S K1 K2
 158                       ) → S L1 L2) →
 159                       ∀L1,L2. lpx_sn R L1 L2 → S L1 L2.
 160 #R #S #IH1 #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
 161 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
 162 #L1 #L2 #HL12 #IH2 @IH1 -IH1 // -HL12
 163 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH2 … HLK1 HLK2) -IH2 -HLK1 -HLK2
 164 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and4_intro/
 165 qed-.
 166
 167 lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
 168                       |L1| = |L2| ∧
 169                       ∀I1,I2,K1,K2,V1,V2,i.
 170                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
 171                       ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2.
 172 #R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
 173 #H elim (lpx_sn_alt_inv_alt … H) -H
 174 #HL12 #IH @conj //
 175 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
 176 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and3_intro/
 177 qed-.