(*** sle_px_tl *)
lemma gr_sle_push_sn_tl:
- â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. ⫯f1 = g1 â\86\92 f1 â\8a\86 ⫱g2.
+ â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f1. ⫯f1 = g1 â\86\92 f1 â\8a\86 â«°g2.
#g1 #g2 #H #f1 #H1
elim (gr_sle_inv_push_sn … H … H1) -H -H1 * //
qed.
(*** sle_xn_tl *)
lemma gr_sle_next_dx_tl:
- â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. â\86\91f2 = g2 â\86\92 ⫱g1 ⊆ f2.
+ â\88\80g1,g2. g1 â\8a\86 g2 â\86\92 â\88\80f2. â\86\91f2 = g2 â\86\92 â«°g1 ⊆ f2.
#g1 #g2 #H #f2 #H2
elim (gr_sle_inv_next_dx … H … H2) -H -H2 * //
qed.
(*** sle_tl *)
lemma gr_sle_tl:
- â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 ⫱f1 â\8a\86 ⫱f2.
+ â\88\80f1,f2. f1 â\8a\86 f2 â\86\92 â«°f1 â\8a\86 â«°f2.
#f1 elim (gr_map_split_tl f1) #H1 #f2 #H
[ lapply (gr_sle_push_sn_tl … H … H1) -H //
| elim (gr_sle_inv_next_sn … H … H1) -H //
(*** sle_inv_tl_sn *)
lemma gr_sle_inv_tl_sn:
- â\88\80f1,f2. ⫱f1 ⊆ f2 → f1 ⊆ ↑f2.
+ â\88\80f1,f2. â«°f1 ⊆ f2 → f1 ⊆ ↑f2.
#f1 elim (gr_map_split_tl f1) #H #f2 #Hf12
/2 width=5 by gr_sle_next, gr_sle_weak/
qed-.
(*** sle_inv_tl_dx *)
lemma gr_sle_inv_tl_dx:
- â\88\80f1,f2. f1 â\8a\86 ⫱f2 → ⫯f1 ⊆ f2.
+ â\88\80f1,f2. f1 â\8a\86 â«°f2 → ⫯f1 ⊆ f2.
#f1 #f2 elim (gr_map_split_tl f2) #H #Hf12
/2 width=5 by gr_sle_push, gr_sle_weak/
qed-.