(* Main properties **********************************************************)
theorem lsubf_sor:
- â\88\80K,L,g1,f1. â¦\83K,g1â¦\84 â«\83ð\9d\90\85+ â¦\83L,f1â¦\84 →
- â\88\80g2,f2. â¦\83K,g2â¦\84 â«\83ð\9d\90\85+ â¦\83L,f2â¦\84 →
- â\88\80g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. f1 â\8b\93 f2 â\89\98 f â\86\92 â¦\83K,gâ¦\84 â«\83ð\9d\90\85+ â¦\83L,fâ¦\84.
+ â\88\80K,L,g1,f1. â\9d¨K,g1â\9d© â«\83ð\9d\90\85+ â\9d¨L,f1â\9d© →
+ â\88\80g2,f2. â\9d¨K,g2â\9d© â«\83ð\9d\90\85+ â\9d¨L,f2â\9d© →
+ â\88\80g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. f1 â\8b\93 f2 â\89\98 f â\86\92 â\9d¨K,gâ\9d© â«\83ð\9d\90\85+ â\9d¨L,fâ\9d©.
#K elim K -K
[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
lapply (lsubf_inv_atom … 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 … 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 … Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
elim (lsubf_inv_push_sn … H2) -H2 #x2 #H2 #H destruct
- elim (sor_inv_ppx … Hf) -Hf [|*: // ] #x #Hx #H destruct
+ elim (pr_sor_inv_push_bi … Hf) -Hf [|*: // ] #x #Hx #H destruct
/3 width=8 by lsubf_push/
- | elim (sor_inv_pnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ | elim (pr_sor_inv_push_next … Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
generalize in match H2; -H2 cases J -J #J [| #V ] #H2
[ elim (lsubf_inv_unit1 … H2) -H2 #x2 #Y2 #H2 #H #H0 destruct
| #y3 #y4 #x2 #Z2 #Y2 #H2 #Hy3 #Hy2 #H #H0 destruct
]
]
- elim (sor_inv_pnx … 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 … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (pr_sor_inv_push_next … 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 … Hg) -Hg [|*: // ] #y #Hy #H destruct
elim (lsubf_inv_push1 … H2) -H2 #x2 #Z2 #Y2 #H2 #H #H0 destruct
generalize in match H1; -H1 cases J -J #J [| #V ] #H1
[ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
| #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
]
]
- elim (sor_inv_npx … 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 … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (pr_sor_inv_next_push … 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 … 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 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
elim (lsubf_inv_bind_sn … 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 … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
+ lapply (pr_sor_eq_repl_back_dx … Hy2 … H3) -z3 #Hy2
| #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
elim (lsubf_inv_unit_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
+ lapply (pr_sor_eq_repl_back_dx … Hy2 … H3) -z3 #Hy2
]
]
- elim (sor_inv_nnx … 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 … 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-.