(* Constructions with pr_basic **********************************************)
lemma pr_nat_basic_lt (m) (n) (l):
- l < m â\86\92 @â\86\91â\9dªl, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d« ≘ l.
+ l < m â\86\92 @â\86\91â\9d¨l, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d© ≘ l.
#m @(nat_ind_succ … m) -m
[ #n #i #H elim (nlt_inv_zero_dx … H)
| #m #IH #n #l @(nat_ind_succ … l) -l
qed.
lemma pr_nat_basic_ge (m) (n) (l):
- m â\89¤ l â\86\92 @â\86\91â\9dªl, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d« ≘ l+n.
+ m â\89¤ l â\86\92 @â\86\91â\9d¨l, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d© ≘ l+n.
#m @(nat_ind_succ … m) -m //
#m #IH #n #l #H
elim (nle_inv_succ_sn … H) -H #Hml #H >H -H
(* Inversions with pr_basic *************************************************)
lemma pr_nat_basic_inv_lt (m) (n) (l) (k):
- l < m â\86\92 @â\86\91â\9dªl, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d« ≘ k → l = k.
+ l < m â\86\92 @â\86\91â\9d¨l, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d© ≘ k → l = k.
/3 width=4 by pr_nat_basic_lt, pr_nat_mono/ qed-.
lemma pr_nat_basic_inv_ge (m) (n) (l) (k):
- m â\89¤ l â\86\92 @â\86\91â\9dªl, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d« ≘ k → l+n = k.
+ m â\89¤ l â\86\92 @â\86\91â\9d¨l, ð\9d\90\9bâ\9d¨m,nâ\9d©â\9d© ≘ k → l+n = k.
/3 width=4 by pr_nat_basic_ge, pr_nat_mono/ qed-.