(*** at_monotonic *)
theorem pr_pat_monotonic:
- â\88\80j2,i2,f. @â\9dªi2,fâ\9d« â\89\98 j2 â\86\92 â\88\80j1,i1. @â\9dªi1,fâ\9d« ≘ j1 →
+ â\88\80j2,i2,f. @â\9d¨i2,fâ\9d© â\89\98 j2 â\86\92 â\88\80j1,i1. @â\9d¨i1,fâ\9d© ≘ j1 →
i1 < i2 → j1 < j2.
#j2 elim j2 -j2
[ #i2 #f #H2f elim (pr_pat_inv_unit_dx … H2f) -H2f //
(*** at_inv_monotonic *)
theorem pr_pat_inv_monotonic:
- â\88\80j1,i1,f. @â\9dªi1,fâ\9d« â\89\98 j1 â\86\92 â\88\80j2,i2. @â\9dªi2,fâ\9d« ≘ j2 →
+ â\88\80j1,i1,f. @â\9d¨i1,fâ\9d© â\89\98 j1 â\86\92 â\88\80j2,i2. @â\9d¨i2,fâ\9d© ≘ j2 →
j1 < j2 → i1 < i2.
#j1 elim j1 -j1
[ #i1 #f #H1f elim (pr_pat_inv_unit_dx … H1f) -H1f //
(*** at_mono *)
theorem pr_pat_mono (f) (i):
- â\88\80i1. @â\9dªi,fâ\9d« â\89\98 i1 â\86\92 â\88\80i2. @â\9dªi,fâ\9d« ≘ i2 → i2 = i1.
+ â\88\80i1. @â\9d¨i,fâ\9d© â\89\98 i1 â\86\92 â\88\80i2. @â\9d¨i,fâ\9d© ≘ i2 → i2 = i1.
#f #i #i1 #H1 #i2 #H2 elim (pnat_split_lt_eq_gt i2 i1) //
#Hi elim (plt_ge_false i i)
/2 width=6 by pr_pat_inv_monotonic/
(*** at_inj *)
theorem pr_pat_inj (f) (i):
- â\88\80i1. @â\9dªi1,fâ\9d« â\89\98 i â\86\92 â\88\80i2. @â\9dªi2,fâ\9d« ≘ i → i1 = i2.
+ â\88\80i1. @â\9d¨i1,fâ\9d© â\89\98 i â\86\92 â\88\80i2. @â\9d¨i2,fâ\9d© ≘ i → i1 = i2.
#f #i #i1 #H1 #i2 #H2 elim (pnat_split_lt_eq_gt i2 i1) //
#Hi elim (plt_ge_false i i)
/2 width=6 by pr_pat_monotonic/