matita/matita/contribs/lambdadelta/static_2/etc/scl/scl.etc

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  14
  15 include "static_2/notation/relations/clearsn_3.ma".
  16 include "static_2/syntax/cext2.ma".
  17 include "static_2/relocation/sex.ma".
  18
  19 (* CLEAR FOR LOCAL ENVIRONMENTS ON SELECTED ENTRIES *************************)
  20
  21 definition ccl: relation3 lenv bind bind ≝ λL,I1,I2. BUnit Void = I2.
  22
  23 definition scl: rtmap → relation lenv ≝ sex ccl (cext2 ceq).
  24
  25 interpretation
  26   "clear (local environment)"
  27   'ClearSn f L1 L2 = (scl f L1 L2).
  28
  29 (* Basic eliminators ********************************************************)
  30
  31 lemma scl_ind (Q:rtmap→relation lenv):
  32       (∀f. Q f (⋆) (⋆)) →
  33       (∀f,I,K1,K2. K1 ⊐ⓧ[f] K2 → Q f K1 K2 → Q (⫯f) (K1.ⓘ{I}) (K2.ⓘ{I})) →
  34       (∀f,I,K1,K2. K1 ⊐ⓧ[f] K2 → Q f K1 K2 → Q (↑f) (K1.ⓘ{I}) (K2.ⓧ)) →
  35       ∀f,L1,L2. L1 ⊐ⓧ[f] L2 → Q f L1 L2.
  36 #Q #IH1 #IH2 #IH3 #f #L1 #L2 #H elim H -f -L1 -L2
  37 [ //
  38 | #f #I1 #I2 #K1 #K2 #HK #H #IH destruct /2 by/
  39 | #f #I1 #I2 #K1 #K2 #HK * #I [| #V1 #V2 #H ] #IH destruct /2 by/
  40 ]
  41 qed-.
  42
  43 (* Basic inversion lemmas ***************************************************)
  44
  45 lemma scl_inv_atom_sn: ∀g,L2. ⋆ ⊐ⓧ[g] L2 → L2 = ⋆.
  46 /2 width=4 by sex_inv_atom1/ qed-.
  47
  48 lemma scl_inv_push_sn: ∀f,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[⫯f] L2 →
  49                        ∃∃K2. K1 ⊐ⓧ[f] K2 & L2 = K2.ⓘ{I}.
  50 #f #I #K1 #L2 #H
  51 elim (sex_inv_push1 … H) -H #J #K2 #HK12 *
  52 /2 width=3 by ex2_intro/
  53 qed-.
  54
  55 lemma scl_inv_next_sn: ∀f,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[↑f] L2 →
  56                        ∃∃K2. K1 ⊐ⓧ[f] K2 & L2 = K2.ⓧ.
  57 #f #I #K1 #L2 #H
  58 elim (sex_inv_next1 … H) -H
  59 /2 width=3 by ex2_intro/
  60 qed-.
  61
  62 (* Advanced inversion lemmas ************************************************)
  63
  64 lemma scl_inv_bind_sn_gen: ∀g,I,K1,L2. K1.ⓘ{I} ⊐ⓧ[g] L2 →
  65                            ∨∨ ∃∃f,K2. K1 ⊐ⓧ[f] K2 & g = ⫯f & L2 = K2.ⓘ{I}
  66                             | ∃∃f,K2. K1 ⊐ⓧ[f] K2 & g = ↑f & L2 = K2.ⓧ.
  67 #g #I #K1 #L2 #H
  68 elim (pn_split g) * #f #Hf destruct
  69 [ elim (scl_inv_push_sn … H) -H
  70 | elim (scl_inv_next_sn … H) -H
  71 ]
  72 /3 width=5 by ex3_2_intro, or_intror, or_introl/
  73 qed-.
  74
  75 (* Advanced forward lemmas **************************************************)
  76
  77 lemma scl_fwd_bind_sn: ∀g,I1,K1,L2. K1.ⓘ{I1} ⊐ⓧ[g] L2 →
  78                        ∃∃I2,K2. K1 ⊐ⓧ[⫱g] K2 & L2 = K2.ⓘ{I2}.
  79 #g #I1 #K1 #L2
  80 elim (pn_split g) * #f #Hf destruct #H
  81 [ elim (scl_inv_push_sn … H) -H
  82 | elim (scl_inv_next_sn … H) -H
  83 ]
  84 /2 width=4 by ex2_2_intro/
  85 qed-.
  86
  87 (* Basic properties *********************************************************)
  88
  89 lemma scl_atom: ∀f. ⋆ ⊐ⓧ[f] ⋆.
  90 /by sex_atom/ qed.
  91
  92 lemma scl_push: ∀f,K1,K2. K1 ⊐ⓧ[f] K2 → ∀I. K1.ⓘ{I} ⊐ⓧ[⫯f] K2.ⓘ{I}.
  93 #f #K1 #K2 #H * /3 width=1 by sex_push, ext2_unit, ext2_pair/
  94 qed.
  95
  96 lemma scl_next: ∀f,K1,K2. K1 ⊐ⓧ[f] K2 → ∀I. K1.ⓘ{I} ⊐ⓧ[↑f] K2.ⓧ.
  97 /2 width=1 by sex_next/ qed.
  98
  99 lemma scl_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ⊐ⓧ[f] L2).
 100 /2 width=3 by sex_eq_repl_back/ qed-.
 101
 102 lemma scl_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ⊐ⓧ[f] L2).
 103 /2 width=3 by sex_eq_repl_fwd/ qed-.
 104
 105 (* Advanced properties ******************************************************)
 106
 107 lemma scl_refl: ∀f. 𝐈⦃f⦄ → reflexive … (scl f).
 108 #f #Hf #L elim L -L
 109 /3 width=3 by scl_eq_repl_back, scl_push, eq_push_inv_isid/
 110 qed.