interpretation "test for totality (rtmap)"
'IsTotal f = (istot f).
(* Basic inversion lemmas ***************************************************)
interpretation "test for totality (rtmap)"
'IsTotal f = (istot f).
(* Basic inversion lemmas ***************************************************)
#g #Hg #f #H #i elim (Hg i) -Hg
#j #Hg elim (at_inv_xnx … Hg … H) -Hg -H /2 width=2 by ex_intro/
qed-.
(* Properties on tl *********************************************************)
#g #Hg #f #H #i elim (Hg i) -Hg
#j #Hg elim (at_inv_xnx … Hg … H) -Hg -H /2 width=2 by ex_intro/
qed-.
(* Properties on tl *********************************************************)
#f cases (pn_split f) *
#g * -f /2 width=3 by istot_inv_next, istot_inv_push/
qed.
(* Properties on tls ********************************************************)
#f cases (pn_split f) *
#g * -f /2 width=3 by istot_inv_next, istot_inv_push/
qed.
(* Properties on tls ********************************************************)
#n elim n -n /3 width=1 by istot_tl/
qed.
(* Main forward lemmas on at ************************************************)
#n elim n -n /3 width=1 by istot_tl/
qed.
(* Main forward lemmas on at ************************************************)
-corec theorem at_ext: â\88\80f1,f2. ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84 →
- (â\88\80i,i1,i2. @â¦\83i,f1â¦\84 â\89\98 i1 â\86\92 @â¦\83i,f2â¦\84 ≘ i2 → i1 = i2) →
+corec theorem at_ext: â\88\80f1,f2. ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d« →
+ (â\88\80i,i1,i2. @â\9dªi,f1â\9d« â\89\98 i1 â\86\92 @â\9dªi,f2â\9d« ≘ i2 → i1 = i2) →
-lemma at_dec: â\88\80f,i1,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (@â¦\83i1,fâ¦\84 ≘ i2).
+lemma at_dec: â\88\80f,i1,i2. ð\9d\90\93â\9dªfâ\9d« â\86\92 Decidable (@â\9dªi1,fâ\9d« ≘ i2).
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
#j2 #Hf elim (eq_nat_dec i2 j2)
[ #H destruct /2 width=1 by or_introl/
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
#j2 #Hf elim (eq_nat_dec i2 j2)
[ #H destruct /2 width=1 by or_introl/
-lemma is_at_dec_le: â\88\80f,i2,i. ð\9d\90\93â¦\83fâ¦\84 â\86\92 (â\88\80i1. i1 + i â\89¤ i2 â\86\92 @â¦\83i1,fâ¦\84 ≘ i2 → ⊥) →
- Decidable (â\88\83i1. @â¦\83i1,fâ¦\84 ≘ i2).
+lemma is_at_dec_le: â\88\80f,i2,i. ð\9d\90\93â\9dªfâ\9d« â\86\92 (â\88\80i1. i1 + i â\89¤ i2 â\86\92 @â\9dªi1,fâ\9d« ≘ i2 → ⊥) →
+ Decidable (â\88\83i1. @â\9dªi1,fâ\9d« ≘ i2).
#f #i2 #i #Hf elim i -i
[ #Ht @or_intror * /3 width=3 by at_increasing/
| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/
#f #i2 #i #Hf elim i -i
[ #Ht @or_intror * /3 width=3 by at_increasing/
| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/
-lemma is_at_dec: â\88\80f,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (â\88\83i1. @â¦\83i1,fâ¦\84 ≘ i2).
+lemma is_at_dec: â\88\80f,i2. ð\9d\90\93â\9dªfâ\9d« â\86\92 Decidable (â\88\83i1. @â\9dªi1,fâ\9d« ≘ i2).
#f #i2 #Hf @(is_at_dec_le ?? (↑i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on isid **********************************************)
#f #i2 #Hf @(is_at_dec_le ?? (↑i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on isid **********************************************)
-lemma isid_at_total: â\88\80f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 (â\88\80i1,i2. @â¦\83i1,fâ¦\84 â\89\98 i2 â\86\92 i1 = i2) â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma isid_at_total: â\88\80f. ð\9d\90\93â\9dªfâ\9d« â\86\92 (â\88\80i1,i2. @â\9dªi1,fâ\9d« â\89\98 i2 â\86\92 i1 = i2) â\86\92 ð\9d\90\88â\9dªfâ\9d«.