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

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/static/frees_frees.ma".
  16 include "basic_2/static/lsubf.ma".
  17
  18 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
  19
  20 (* Main properties **********************************************************)
  21
  22 theorem lsubf_sor: ∀K,L,g1,f1. ⦃K, g1⦄ ⫃𝐅* ⦃L, f1⦄ →
  23                    ∀g2,f2. ⦃K, g2⦄ ⫃𝐅* ⦃L, f2⦄ →
  24                    ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K, g⦄ ⫃𝐅* ⦃L, f⦄.
  25 #K elim K -K
  26 [ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
  27   elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
  28   lapply (lsubf_inv_atom … H2) -H2 #H2
  29   /5 width=4 by lsubf_atom, sor_mono, sor_eq_repl_back2, sor_eq_repl_back1/
  30 | #K #J #IH #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
  31   elim (pn_split g1) * #y1 #H destruct
  32   elim (pn_split g2) * #y2 #H destruct
  33   [ elim (sor_inv_ppx … Hg) -Hg [|*: // ] #y #Hy #H destruct
  34     elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
  35     elim (lsubf_inv_push_sn … H2) -H2 #x2 #H2 #H destruct
  36     elim (sor_inv_ppx … Hf) -Hf [|*: // ] #x #Hx #H destruct
  37     /3 width=8 by lsubf_push/
  38   | elim (sor_inv_pnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
  39     elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
  40     generalize in match H2; -H2 cases J -J #J [| #V ] #H2
  41     [ elim (lsubf_inv_unit1 … H2) -H2 #x2 #Y2 #H2 #H #H0 destruct
  42     | elim (lsubf_inv_pair1 … H2) -H2 *
  43       [ #x2 #Z2 #H2 #H #H0 destruct
  44       | #y3 #y4 #x2 #Y2 #W #U #H2 #Hy3 #Hy2 #H #H0 #H3 #H4 destruct
  45       | #y3 #y4 #x2 #Z2 #Y2 #H2 #Hy3 #Hy2 #H #H0 destruct
  46       ]
  47     ]
  48     elim (sor_inv_pnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
  49     /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_assoc_sn/
  50   | elim (sor_inv_npx … Hg) -Hg [|*: // ] #y #Hy #H destruct
  51     elim (lsubf_inv_push1 … H2) -H2 #x2 #Z2 #Y2 #H2 #H #H0 destruct
  52     generalize in match H1; -H1 cases J -J #J [| #V ] #H1
  53     [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
  54     | elim (lsubf_inv_pair1 … H1) -H1 *
  55       [ #x1 #Z1 #H1 #H #H0 destruct
  56       | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
  57       | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
  58       ]
  59     ]
  60     elim (sor_inv_npx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
  61     /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_comm_23/
  62   | elim (sor_inv_nnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
  63     generalize in match H2; generalize in match H1; -H1 -H2 cases J -J #J [| #V ] #H1 #H2
  64     [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
  65       elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
  66     | elim (lsubf_inv_pair1 … H1) -H1 *
  67       [ #x1 #Z1 #H1 #H #H0 destruct
  68         elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
  69       | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
  70         elim (lsubf_inv_beta_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
  71         lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
  72         lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
  73       | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
  74         elim (lsubf_inv_unit_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
  75         lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
  76         lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
  77       ]
  78     ]
  79     elim (sor_inv_nnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
  80     /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_coll_dx/
  81   ]
  82 ]
  83 qed-.