(* Main properties **********************************************************)
-theorem lsubf_sor: āK,L,g1,f1. ā¦K, g1ā¦ ā«š
* ā¦L, f1ā¦ ā
- āg2,f2. ā¦K, g2ā¦ ā«š
* ā¦L, f2ā¦ ā
- āg. g1 ā g2 ā g ā āf. f1 ā f2 ā f ā ā¦K, gā¦ ā«š
* ā¦L, fā¦.
+theorem lsubf_sor:
+ āK,L,g1,f1. āŖK,g1ā« ā«š
+ āŖL,f1ā« ā
+ āg2,f2. āŖK,g2ā« ā«š
+ āŖL,f2ā« ā
+ āg. g1 ā g2 ā g ā āf. f1 ā f2 ā f ā āŖK,gā« ā«š
+ āŖL,fā«.
#K elim K -K
[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
elim (lsubf_inv_atom1 ā¦ H1) -H1 #H1 #H destruct
lapply (lsubf_inv_atom ā¦ H2) -H2 #H2
- /5 width=4 by lsubf_atom, sor_mono, sor_eq_repl_back2, sor_eq_repl_back1/
+ /5 width=4 by lsubf_atom, pr_sor_mono, pr_sor_eq_repl_back_dx, pr_sor_eq_repl_back_sn/
| #K #J #IH #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
- elim (pn_split g1) * #y1 #H destruct
- elim (pn_split g2) * #y2 #H destruct
- [ elim (sor_inv_ppx ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (pr_map_split_tl g1) * #y1 #H destruct
+ elim (pr_map_split_tl g2) * #y2 #H destruct
+ [ elim (pr_sor_inv_push_bi ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 ā¦ H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
elim (lsubf_inv_push_sn ā¦ H2) -H2 #x2 #H2 #H destruct
- elim (sor_inv_ppx ā¦ Hf) -Hf [|*: // ] #x #Hx #H destruct
+ elim (pr_sor_inv_push_bi ā¦ Hf) -Hf [|*: // ] #x #Hx #H destruct
/3 width=8 by lsubf_push/
- | elim (sor_inv_pnx ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
+ | elim (pr_sor_inv_push_next ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 ā¦ H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
generalize in match H2; -H2 cases J -J #J [| #V ] #H2
[ elim (lsubf_inv_unit1 ā¦ H2) -H2 #x2 #Y2 #H2 #H #H0 destruct
| #y3 #y4 #x2 #Z2 #Y2 #H2 #Hy3 #Hy2 #H #H0 destruct
]
]
- elim (sor_inv_pnx ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_assoc_sn/
- | elim (sor_inv_npx ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (pr_sor_inv_push_next ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, pr_sor_assoc_sn/
+ | elim (pr_sor_inv_next_push ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 ā¦ H2) -H2 #x2 #Z2 #Y2 #H2 #H #H0 destruct
generalize in match H1; -H1 cases J -J #J [| #V ] #H1
[ elim (lsubf_inv_unit1 ā¦ H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
| #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
]
]
- elim (sor_inv_npx ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_comm_23/
- | elim (sor_inv_nnx ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (pr_sor_inv_next_push ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, pr_sor_comm_23/
+ | elim (pr_sor_inv_next_bi ā¦ Hg) -Hg [|*: // ] #y #Hy #H destruct
generalize in match H2; generalize in match H1; -H1 -H2 cases J -J #J [| #V ] #H1 #H2
[ elim (lsubf_inv_unit1 ā¦ H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
elim (lsubf_inv_bind_sn ā¦ H2) -H2 #x2 #H2 #H destruct
| #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
elim (lsubf_inv_beta_sn ā¦ H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
lapply (frees_mono ā¦ Hz3 ā¦ Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 ā¦ Hy2 ā¦ H3) -z3 #Hy2
+ lapply (pr_sor_eq_repl_back_dx ā¦ Hy2 ā¦ H3) -z3 #Hy2
| #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
elim (lsubf_inv_unit_sn ā¦ H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
lapply (frees_mono ā¦ Hz3 ā¦ Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 ā¦ Hy2 ā¦ H3) -z3 #Hy2
+ lapply (pr_sor_eq_repl_back_dx ā¦ Hy2 ā¦ H3) -z3 #Hy2
]
]
- elim (sor_inv_nnx ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_coll_dx/
+ elim (pr_sor_inv_next_bi ā¦ Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, pr_sor_coll_dx/
]
]
qed-.