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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "static_2/static/frees_frees.ma".
16 include "static_2/static/lsubf.ma".
18 (* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
20 (* Main properties **********************************************************)
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⦄.
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
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
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
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/