*)
inductive lifts: pr_map → relation term ≝
| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: ∀f,i1,i2. @↑❪i1,f❫ ≘ i2 → lifts f (#i1) (#i2)
+| lifts_lref: ∀f,i1,i2. @§❨i1,f❩ ≘ i2 → lifts f (#i1) (#i2)
| lifts_gref: ∀f,l. lifts f (§l) (§l)
| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
lifts f V1 V2 → lifts (⫯f) T1 T2 →
/2 width=4 by lifts_inv_sort1_aux/ qed-.
fact lifts_inv_lref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i1. X = #i1 →
- ∃∃i2. @↑❪i1,f❫ ≘ i2 & Y = #i2.
+ ∃∃i2. @§❨i1,f❩ ≘ i2 & Y = #i2.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: was: lift1_lref *)
(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
lemma lifts_inv_lref1: ∀f,Y,i1. ⇧*[f] #i1 ≘ Y →
- ∃∃i2. @↑❪i1,f❫ ≘ i2 & Y = #i2.
+ ∃∃i2. @§❨i1,f❩ ≘ i2 & Y = #i2.
/2 width=3 by lifts_inv_lref1_aux/ qed-.
fact lifts_inv_gref1_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. X = §l → Y = §l.
/2 width=4 by lifts_inv_sort2_aux/ qed-.
fact lifts_inv_lref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀i2. Y = #i2 →
- ∃∃i1. @↑❪i1,f❫ ≘ i2 & X = #i1.
+ ∃∃i1. @§❨i1,f❩ ≘ i2 & X = #i1.
#f #X #Y * -f -X -Y
[ #f #s #x #H destruct
| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
lemma lifts_inv_lref2: ∀f,X,i2. ⇧*[f] X ≘ #i2 →
- ∃∃i1. @↑❪i1,f❫ ≘ i2 & X = #i1.
+ ∃∃i1. @§❨i1,f❩ ≘ i2 & X = #i1.
/2 width=3 by lifts_inv_lref2_aux/ qed-.
fact lifts_inv_gref2_aux: ∀f,X,Y. ⇧*[f] X ≘ Y → ∀l. Y = §l → X = §l.
lemma lifts_inv_atom1: ∀f,I,Y. ⇧*[f] ⓪[I] ≘ Y →
∨∨ ∃∃s. I = Sort s & Y = ⋆s
- | ∃∃i,j. @↑❪i,f❫ ≘ j & I = LRef i & Y = #j
+ | ∃∃i,j. @§❨i,f❩ ≘ j & I = LRef i & Y = #j
| ∃∃l. I = GRef l & Y = §l.
#f * #n #Y #H
[ lapply (lifts_inv_sort1 … H)
lemma lifts_inv_atom2: ∀f,I,X. ⇧*[f] X ≘ ⓪[I] →
∨∨ ∃∃s. X = ⋆s & I = Sort s
- | ∃∃i,j. @↑❪i,f❫ ≘ j & X = #i & I = LRef j
+ | ∃∃i,j. @§❨i,f❩ ≘ j & X = #i & I = LRef j
| ∃∃l. X = §l & I = GRef l.
#f * #n #X #H
[ lapply (lifts_inv_sort2 … H)
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
-lemma lifts_fwd_isid: â\88\80f,T1,T2. â\87§*[f] T1 â\89\98 T2 â\86\92 ð\9d\90\88â\9dªfâ\9d« → T1 = T2.
+lemma lifts_fwd_isid: â\88\80f,T1,T2. â\87§*[f] T1 â\89\98 T2 â\86\92 ð\9d\90\88â\9d¨fâ\9d© → T1 = T2.
#f #T1 #T2 #H elim H -f -T1 -T2
/4 width=3 by pr_isi_nat_des, pr_isi_push, eq_f2, eq_f/
qed-.
(* Basic_1: includes: lift_r *)
(* Basic_2A1: includes: lift_refl *)
-lemma lifts_refl: â\88\80T,f. ð\9d\90\88â\9dªfâ\9d« → ⇧*[f] T ≘ T.
+lemma lifts_refl: â\88\80T,f. ð\9d\90\88â\9d¨fâ\9d© → ⇧*[f] T ≘ T.
#T elim T -T *
/4 width=3 by lifts_flat, lifts_bind, lifts_lref, pr_isi_inv_pat, pr_isi_push/
qed.
(* Basic_2A1: includes: lift_total *)
-lemma lifts_total: â\88\80T1,f. ð\9d\90\93â\9dªfâ\9d« → ∃T2. ⇧*[f] T1 ≘ T2.
+lemma lifts_total: â\88\80T1,f. ð\9d\90\93â\9d¨fâ\9d© → ∃T2. ⇧*[f] T1 ≘ T2.
#T1 elim T1 -T1 *
/3 width=2 by lifts_sort, lifts_gref, ex_intro/
[ #i #f #Hf elim (Hf (↑i)) -Hf /3 width=2 by ex_intro, lifts_lref/ ]
(* Note: apparently, this was missing in Basic_2A1 *)
lemma lifts_split_div: ∀f1,T1,T2. ⇧*[f1] T1 ≘ T2 →
- â\88\80f2. ð\9d\90\93â\9dªf2â\9d« → ∀f. f2 ⊚ f1 ≘ f →
+ â\88\80f2. ð\9d\90\93â\9d¨f2â\9d© → ∀f. f2 ⊚ f1 ≘ f →
∃∃T. ⇧*[f2] T2 ≘ T & ⇧*[f] T1 ≘ T.
#f1 #T1 #T2 #H elim H -f1 -T1 -T2
[ /3 width=3 by lifts_sort, ex2_intro/
(* Basic_1: includes: dnf_dec2 dnf_dec *)
(* Basic_2A1: includes: is_lift_dec *)
-lemma is_lifts_dec: â\88\80T2,f. ð\9d\90\93â\9dªfâ\9d« → Decidable (∃T1. ⇧*[f] T1 ≘ T2).
+lemma is_lifts_dec: â\88\80T2,f. ð\9d\90\93â\9d¨fâ\9d© → Decidable (∃T1. ⇧*[f] T1 ≘ T2).
#T1 elim T1 -T1
[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
#i2 #f #Hf elim (is_pr_nat_dec f i2) //