matita/matita/contribs/lambdadelta/basic_2/etc/lsubss/lsubss.etc

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  14
  15 include "basic_2/static/ssta.ma".
  16
  17 (* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED STATIC TYPE ASSIGNMENT *******)
  18
  19 inductive lsubss (h:sh) (g:sd h): relation lenv ≝
  20 | lsubss_atom: lsubss h g (⋆) (⋆)
  21 | lsubss_pair: ∀I,L1,L2,W. lsubss h g L1 L2 →
  22                lsubss h g (L1. ⓑ{I} W) (L2. ⓑ{I} W)
  23 | lsubss_abbr: ∀L1,L2,V,W,l. ⦃h, L1⦄ ⊢ V •[g, l+1] W → ⦃h, L2⦄ ⊢ V •[g, l+1] W →
  24                lsubss h g L1 L2 → lsubss h g (L1. ⓓV) (L2. ⓛW)
  25 .
  26
  27 interpretation
  28   "local environment refinement (stratified static type assigment)"
  29   'CrSubEqS h g L1 L2 = (lsubss h g L1 L2).
  30
  31 (* Basic inversion lemmas ***************************************************)
  32
  33 fact lsubss_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 = ⋆ → L2 = ⋆.
  34 #h #g #L1 #L2 * -L1 -L2
  35 [ //
  36 | #I #L1 #L2 #V #_ #H destruct
  37 | #L1 #L2 #V #W #l #_ #_ #_ #H destruct
  38 ]
  39 qed.
  40
  41 lemma lsubss_inv_atom1: ∀h,g,L2. h ⊢ ⋆ •⊑[g] L2 → L2 = ⋆.
  42 /2 width=5/ qed-.
  43
  44 fact lsubss_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
  45                            ∀I,K1,V. L1 = K1. ⓑ{I} V →
  46                            (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
  47                            ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
  48                                      h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
  49 #h #g #L1 #L2 * -L1 -L2
  50 [ #I #K1 #V #H destruct
  51 | #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3/
  52 | #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K1 #V1 #H destruct /3 width=7/
  53 ]
  54 qed.
  55
  56 lemma lsubss_inv_pair1: ∀h,g,I,K1,L2,V. h ⊢ K1. ⓑ{I} V •⊑[g] L2 →
  57                         (∃∃K2. h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓑ{I} V) ∨
  58                         ∃∃K2,W,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
  59                                   h ⊢ K1 •⊑[g] K2 & L2 = K2. ⓛW & I = Abbr.
  60 /2 width=3/ qed-.
  61
  62 fact lsubss_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L2 = ⋆ → L1 = ⋆.
  63 #h #g #L1 #L2 * -L1 -L2
  64 [ //
  65 | #I #L1 #L2 #V #_ #H destruct
  66 | #L1 #L2 #V #W #l #_ #_ #_ #H destruct
  67 ]
  68 qed.
  69
  70 lemma lsubss_inv_atom2: ∀h,g,L1. h ⊢ L1 •⊑[g] ⋆ → L1 = ⋆.
  71 /2 width=5/ qed-.
  72
  73 fact lsubss_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 →
  74                            ∀I,K2,W. L2 = K2. ⓑ{I} W →
  75                            (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
  76                            ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
  77                                      h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
  78 #h #g #L1 #L2 * -L1 -L2
  79 [ #I #K2 #W #H destruct
  80 | #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3/
  81 | #L1 #L2 #V #W #l #H1VW #H2VW #HL12 #I #K2 #W2 #H destruct /3 width=7/
  82 ]
  83 qed.
  84
  85 lemma lsubss_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 •⊑[g] K2. ⓑ{I} W →
  86                         (∃∃K1. h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓑ{I} W) ∨
  87                         ∃∃K1,V,l. ⦃h, K1⦄ ⊢ V •[g,l+1] W & ⦃h, K2⦄ ⊢ V •[g,l+1] W &
  88                                   h ⊢ K1 •⊑[g] K2 & L1 = K1. ⓓV & I = Abst.
  89 /2 width=3/ qed-.
  90
  91 (* Basic_forward lemmas *****************************************************)
  92
  93 lemma lsubss_fwd_lsubs1: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L1|] L2.
  94 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
  95 qed-.
  96
  97 lemma lsubss_fwd_lsubs2: ∀h,g,L1,L2. h ⊢ L1 •⊑[g] L2 → L1 ≼[0, |L2|] L2.
  98 #h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
  99 qed-.
 100
 101 (* Basic properties *********************************************************)
 102
 103 lemma lsubss_refl: ∀h,g,L. h ⊢ L •⊑[g] L.
 104 #h #g #L elim L -L // /2 width=1/
 105 qed.