(**************************************************************************)
include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/rtmap_id.ma".
+include "ground_2/relocation/rtmap_uni.ma".
(* RELOCATION MAP ***********************************************************)
coinductive at: rtmap → relation nat ≝
-| at_refl: â\88\80f,g,j1,j2. â\86\91f = g → 0 = j1 → 0 = j2 → at g j1 j2
-| at_push: â\88\80f,i1,i2. at f i1 i2 â\86\92 â\88\80g,j1,j2. â\86\91f = g â\86\92 ⫯i1 = j1 â\86\92 ⫯i2 = j2 → at g j1 j2
-| at_next: â\88\80f,i1,i2. at f i1 i2 â\86\92 â\88\80g,j2. ⫯f = g â\86\92 ⫯i2 = j2 → at g i1 j2
+| at_refl: â\88\80f,g,j1,j2. ⫯f = g → 0 = j1 → 0 = j2 → at g j1 j2
+| at_push: â\88\80f,i1,i2. at f i1 i2 â\86\92 â\88\80g,j1,j2. ⫯f = g â\86\92 â\86\91i1 = j1 â\86\92 â\86\91i2 = j2 → at g j1 j2
+| at_next: â\88\80f,i1,i2. at f i1 i2 â\86\92 â\88\80g,j2. â\86\91f = g â\86\92 â\86\91i2 = j2 → at g i1 j2
.
interpretation "relational application (rtmap)"
'RAt i1 f i2 = (at f i1 i2).
+definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1.
+ ∀jf,jg,j. @⦃jf,f2⦄ ≘ j → @⦃jg,g2⦄ ≘ j →
+ ∃∃j0. @⦃j0,f1⦄ ≘ jf & @⦃j0,g1⦄ ≘ jg.
+
(* Basic inversion lemmas ***************************************************)
-lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. 0 = i1 → ↑g = f → 0 = i2.
+lemma at_inv_ppx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. 0 = i1 → ⫯g = f → 0 = i2.
#f #i1 #i2 * -f -i1 -i2 //
[ #f #i1 #i2 #_ #g #j1 #j2 #_ * #_ #x #H destruct
| #f #i1 #i2 #_ #g #j2 * #_ #x #_ #H elim (discr_push_next … H)
]
qed-.
-lemma at_inv_npx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,j1. ⫯j1 = i1 → ↑g = f →
- ∃∃j2. @⦃j1, g⦄ ≡ j2 & ⫯j2 = i2.
+lemma at_inv_npx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j1. ↑j1 = i1 → ⫯g = f →
+ ∃∃j2. @⦃j1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 #_ * #_ #x #x1 #H destruct
| #f #i1 #i2 #Hi #g #j1 #j2 * * * #x #x1 #H #Hf >(injective_push … Hf) -g destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xnx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. ⫯g = f →
- ∃∃j2. @⦃i1, g⦄ ≡ j2 & ⫯j2 = i2.
+lemma at_inv_xnx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ↑g = f →
+ ∃∃j2. @⦃i1,g⦄ ≘ j2 & ↑j2 = i2.
#f #i1 #i2 * -f -i1 -i2
[ #f #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
| #f #i1 #i2 #_ #g #j1 #j2 * #_ #_ #x #H elim (discr_next_push … H)
(* Advanced inversion lemmas ************************************************)
-lemma at_inv_ppn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80g,j2. 0 = i1 â\86\92 â\86\91g = f â\86\92 ⫯j2 = i2 → ⊥.
+lemma at_inv_ppn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80g,j2. 0 = i1 â\86\92 ⫯g = f â\86\92 â\86\91j2 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j2 #H1 #H <(at_inv_ppx … Hf … H1 H) -f -g -i1 -i2
#H destruct
qed-.
-lemma at_inv_npp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80g,j1. ⫯j1 = i1 â\86\92 â\86\91g = f → 0 = i2 → ⊥.
+lemma at_inv_npp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80g,j1. â\86\91j1 = i1 â\86\92 ⫯g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #j1 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_npn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80g,j1,j2. ⫯j1 = i1 â\86\92 â\86\91g = f â\86\92 ⫯j2 = i2 â\86\92 @â¦\83j1, gâ¦\84 â\89¡ j2.
+lemma at_inv_npn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80g,j1,j2. â\86\91j1 = i1 â\86\92 ⫯g = f â\86\92 â\86\91j2 = i2 â\86\92 @â¦\83j1,gâ¦\84 â\89\98 j2.
#f #i1 #i2 #Hf #g #j1 #j2 #H1 #H elim (at_inv_npx … Hf … H1 H) -f -i1
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_xnp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80g. ⫯g = f → 0 = i2 → ⊥.
+lemma at_inv_xnp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80g. â\86\91g = f → 0 = i2 → ⊥.
#f #i1 #i2 #Hf #g #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct
qed-.
-lemma at_inv_xnn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80g,j2. ⫯g = f â\86\92 ⫯j2 = i2 â\86\92 @â¦\83i1, gâ¦\84 â\89¡ j2.
+lemma at_inv_xnn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80g,j2. â\86\91g = f â\86\92 â\86\91j2 = i2 â\86\92 @â¦\83i1,gâ¦\84 â\89\98 j2.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xnx … Hf … H) -f
#x2 #Hg * -i2 #H destruct //
qed-.
-lemma at_inv_pxp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → 0 = i1 → 0 = i2 → ∃g. ↑g = f.
+lemma at_inv_pxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i1 → 0 = i2 → ∃g. ⫯g = f.
#f elim (pn_split … f) * /2 width=2 by ex_intro/
#g #H #i1 #i2 #Hf #H1 #H2 cases (at_inv_xnp … Hf … H H2)
qed-.
-lemma at_inv_pxn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀j2. 0 = i1 → ⫯j2 = i2 →
- ∃∃g. @⦃i1, g⦄ ≡ j2 & ⫯g = f.
+lemma at_inv_pxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. 0 = i1 → ↑j2 = i2 →
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split … f) *
#g #H #i1 #i2 #Hf #j2 #H1 #H2
[ elim (at_inv_ppn … Hf … H1 H H2)
]
qed-.
-lemma at_inv_nxp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 →
- â\88\80j1. ⫯j1 = i1 → 0 = i2 → ⊥.
+lemma at_inv_nxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ â\88\80j1. â\86\91j1 = i1 → 0 = i2 → ⊥.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j1 #H1 #H2
[ elim (at_inv_npp … Hf … H1 H H2)
]
qed-.
-lemma at_inv_nxn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀j1,j2. ⫯j1 = i1 → ⫯j2 = i2 →
- (∃∃g. @⦃j1, g⦄ ≡ j2 & ↑g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≡ j2 & ⫯g = f.
+lemma at_inv_nxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j1,j2. ↑j1 = i1 → ↑j2 = i2 →
+ (∃∃g. @⦃j1,g⦄ ≘ j2 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
/4 width=7 by at_inv_xnn, at_inv_npn, ex2_intro, or_intror, or_introl/
qed-.
(* Note: the following inversion lemmas must be checked *)
-lemma at_inv_xpx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. ↑g = f →
+lemma at_inv_xpx: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f →
(0 = i1 ∧ 0 = i2) ∨
- ∃∃j1,j2. @⦃j1, g⦄ ≡ j2 & ⫯j1 = i1 & ⫯j2 = i2.
+ ∃∃j1,j2. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ↑j2 = i2.
#f * [2: #i1 ] #i2 #Hf #g #H
[ elim (at_inv_npx … Hf … H) -f /3 width=5 by or_intror, ex3_2_intro/
| >(at_inv_ppx … Hf … H) -f /3 width=1 by conj, or_introl/
]
qed-.
-lemma at_inv_xpp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. ↑g = f → 0 = i2 → 0 = i1.
+lemma at_inv_xpp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g. ⫯g = f → 0 = i2 → 0 = i1.
#f #i1 #i2 #Hf #g #H elim (at_inv_xpx … Hf … H) -f * //
#j1 #j2 #_ #_ * -i2 #H destruct
qed-.
-lemma at_inv_xpn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g,j2. ↑g = f → ⫯j2 = i2 →
- ∃∃j1. @⦃j1, g⦄ ≡ j2 & ⫯j1 = i1.
+lemma at_inv_xpn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀g,j2. ⫯g = f → ↑j2 = i2 →
+ ∃∃j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1.
#f #i1 #i2 #Hf #g #j2 #H elim (at_inv_xpx … Hf … H) -f *
[ #_ * -i2 #H destruct
| #x1 #x2 #Hg #H1 * -i2 #H destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma at_inv_xxp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → 0 = i2 →
- â\88\83â\88\83g. 0 = i1 & â\86\91g = f.
+lemma at_inv_xxp: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → 0 = i2 →
+ â\88\83â\88\83g. 0 = i1 & ⫯g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #H2
[ /3 width=6 by at_inv_xpp, ex2_intro/
]
qed-.
-lemma at_inv_xxn: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀j2. ⫯j2 = i2 →
- (∃∃g,j1. @⦃j1, g⦄ ≡ j2 & ⫯j1 = i1 & ↑g = f) ∨
- ∃∃g. @⦃i1, g⦄ ≡ j2 & ⫯g = f.
+lemma at_inv_xxn: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 → ∀j2. ↑j2 = i2 →
+ (∃∃g,j1. @⦃j1,g⦄ ≘ j2 & ↑j1 = i1 & ⫯g = f) ∨
+ ∃∃g. @⦃i1,g⦄ ≘ j2 & ↑g = f.
#f elim (pn_split f) *
#g #H #i1 #i2 #Hf #j2 #H2
[ elim (at_inv_xpn … Hf … H H2) -i2 /3 width=5 by or_introl, ex3_2_intro/
(* Basic forward lemmas *****************************************************)
-lemma at_increasing: ∀i2,i1,f. @⦃i1, f⦄ ≡ i2 → i1 ≤ i2.
+lemma at_increasing: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → i1 ≤ i2.
#i2 elim i2 -i2
[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf //
| #i2 #IH * //
]
qed-.
-lemma at_increasing_strict: ∀g,i1,i2. @⦃i1, g⦄ ≡ i2 → ∀f. ⫯f = g →
- i1 < i2 ∧ @⦃i1, f⦄ ≡ ⫰i2.
+lemma at_increasing_strict: ∀g,i1,i2. @⦃i1,g⦄ ≘ i2 → ∀f. ↑f = g →
+ i1 < i2 ∧ @⦃i1,f⦄ ≘ ↓i2.
#g #i1 #i2 #Hg #f #H elim (at_inv_xnx … Hg … H) -Hg -H
/4 width=2 by conj, at_increasing, le_S_S/
qed-.
-lemma at_fwd_id_ex: ∀f,i. @⦃i, f⦄ ≡ i → ∃g. ↑g = f.
+lemma at_fwd_id_ex: ∀f,i. @⦃i,f⦄ ≘ i → ∃g. ⫯g = f.
#f elim (pn_split f) * /2 width=2 by ex_intro/
#g #H #i #Hf elim (at_inv_xnx … Hf … H) -Hf -H
#j2 #Hg #H destruct lapply (at_increasing … Hg) -Hg
(* Basic properties *********************************************************)
-let corec at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≡ i2) ≝ ?.
+corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 #f1 #H1 cases H1 -f1 -i1 -i2
[ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /2 width=2 by at_refl/
| #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /3 width=7 by at_push/
]
qed-.
-lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1, f⦄ ≡ i2).
+lemma at_eq_repl_fwd: ∀i1,i2. eq_repl_fwd (λf. @⦃i1,f⦄ ≘ i2).
#i1 #i2 @eq_repl_sym /2 width=3 by at_eq_repl_back/
qed-.
-lemma at_le_ex: ∀j2,i2,f. @⦃i2, f⦄ ≡ j2 → ∀i1. i1 ≤ i2 →
- ∃∃j1. @⦃i1, f⦄ ≡ j1 & j1 ≤ j2.
+lemma at_le_ex: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀i1. i1 ≤ i2 →
+ ∃∃j1. @⦃i1,f⦄ ≘ j1 & j1 ≤ j2.
#j2 elim j2 -j2 [2: #j2 #IH ] #i2 #f #Hf
[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
#g [ #x2 ] #Hg [ #H2 ] #H0
]
qed-.
-lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2, f⦄ ≡ i2 → @⦃i1, f⦄ ≡ i1.
+lemma at_id_le: ∀i1,i2. i1 ≤ i2 → ∀f. @⦃i2,f⦄ ≘ i2 → @⦃i1,f⦄ ≘ i1.
#i1 #i2 #H @(le_elim … H) -i1 -i2 [ #i2 | #i1 #i2 #IH ]
#f #Hf elim (at_fwd_id_ex … Hf) /4 width=7 by at_inv_npn, at_push, at_refl/
qed-.
(* Main properties **********************************************************)
-theorem at_monotonic: ∀j2,i2,f. @⦃i2, f⦄ ≡ j2 → ∀j1,i1. @⦃i1, f⦄ ≡ j1 →
+theorem at_monotonic: ∀j2,i2,f. @⦃i2,f⦄ ≘ j2 → ∀j1,i1. @⦃i1,f⦄ ≘ j1 →
i1 < i2 → j1 < j2.
#j2 elim j2 -j2
[ #i2 #f #H2f elim (at_inv_xxp … H2f) -H2f //
]
qed-.
-theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1, f⦄ ≡ j1 → ∀j2,i2. @⦃i2, f⦄ ≡ j2 →
+theorem at_inv_monotonic: ∀j1,i1,f. @⦃i1,f⦄ ≘ j1 → ∀j2,i2. @⦃i2,f⦄ ≘ j2 →
j1 < j2 → i1 < i2.
#j1 elim j1 -j1
[ #i1 #f #H1f elim (at_inv_xxp … H1f) -H1f //
]
qed-.
-theorem at_mono: ∀f,i,i1. @⦃i, f⦄ ≡ i1 → ∀i2. @⦃i, f⦄ ≡ i2 → i2 = i1.
+theorem at_mono: ∀f,i,i1. @⦃i,f⦄ ≘ i1 → ∀i2. @⦃i,f⦄ ≘ i2 → i2 = i1.
#f #i #i1 #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_inv_monotonic, eq_sym/
qed-.
-theorem at_inj: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀i2. @⦃i2, f⦄ ≡ i → i1 = i2.
+theorem at_inj: ∀f,i1,i. @⦃i1,f⦄ ≘ i → ∀i2. @⦃i2,f⦄ ≘ i → i1 = i2.
#f #i1 #i #H1 #i2 #H2 elim (lt_or_eq_or_gt i2 i1) //
#Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/
qed-.
+theorem at_div_comm: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div g2 f2 g1 f1.
+#f2 #g2 #f1 #g1 #IH #jg #jf #j #Hg #Hf
+elim (IH … Hf Hg) -IH -j /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_pp: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (⫯g2) (⫯f1) (⫯g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xpx … Hf) -Hf [1,2: * |*: // ]
+[ #H1 #H2 destruct -IH
+ lapply (at_inv_xpp … Hg ???) -Hg [4: |*: // ] #H destruct
+ /3 width=3 by at_refl, ex2_intro/
+| #xf #i #Hf2 #H1 #H2 destruct
+ lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+ elim (IH … Hf2 Hg2) -IH -i /3 width=9 by at_push, ex2_intro/
+]
+qed-.
+
+theorem at_div_nn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (↑g2) (f1) (g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xnn … Hg ????) -Hg [5: |*: // ] #Hg2
+elim (IH … Hf2 Hg2) -IH -i /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_np: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (⫯g2) (f1) (↑g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+elim (IH … Hf2 Hg2) -IH -i /3 width=7 by at_next, ex2_intro/
+qed-.
+
+theorem at_div_pn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (↑g2) (↑f1) (g1).
+/4 width=6 by at_div_np, at_div_comm/ qed-.
+
(* Properties on tls ********************************************************)
-lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≡ n → @⦃0, ⫱*[n]f⦄ ≡ 0.
+lemma at_pxx_tls: ∀n,f. @⦃0,f⦄ ≘ n → @⦃0,⫱*[n]f⦄ ≘ 0.
#n elim n -n //
-#n #IH #f #Hf cases (at_inv_pxn … Hf) -Hf /2 width=3 by/
+#n #IH #f #Hf
+cases (at_inv_pxn … Hf) -Hf [ |*: // ] #g #Hg #H0 destruct
+<tls_xn /2 width=1 by/
qed.
+lemma at_tls: ∀i2,f. ⫯⫱*[↑i2]f ≡ ⫱*[i2]f → ∃i1. @⦃i1,f⦄ ≘ i2.
+#i2 elim i2 -i2
+[ /4 width=4 by at_eq_repl_back, at_refl, ex_intro/
+| #i2 #IH #f <tls_xn <tls_xn in ⊢ (??%→?); #H
+ elim (IH … H) -IH -H #i1 #Hf
+ elim (pn_split f) * #g #Hg destruct /3 width=8 by at_push, at_next, ex_intro/
+]
+qed-.
+
+(* Inversion lemmas with tls ************************************************)
+
+lemma at_inv_nxx: ∀n,g,i1,j2. @⦃↑i1,g⦄ ≘ j2 → @⦃0,g⦄ ≘ n →
+ ∃∃i2. @⦃i1,⫱*[↑n]g⦄ ≘ i2 & ↑(n+i2) = j2.
+#n elim n -n
+[ #g #i1 #j2 #Hg #H
+ elim (at_inv_pxp … H) -H [ |*: // ] #f #H0
+ elim (at_inv_npx … Hg … H0) -Hg [ |*: // ] #x2 #Hf #H2 destruct
+ /2 width=3 by ex2_intro/
+| #n #IH #g #i1 #j2 #Hg #H
+ elim (at_inv_pxn … H) -H [ |*: // ] #f #Hf2 #H0
+ elim (at_inv_xnx … Hg … H0) -Hg #x2 #Hf1 #H2 destruct
+ elim (IH … Hf1 Hf2) -IH -Hf1 -Hf2 #i2 #Hf #H2 destruct
+ /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma at_inv_tls: ∀i2,i1,f. @⦃i1,f⦄ ≘ i2 → ⫯⫱*[↑i2]f ≡ ⫱*[i2]f.
+#i2 elim i2 -i2
+[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf // #g #H1 #H destruct
+ /2 width=1 by eq_refl/
+| #i2 #IH #i1 #f #Hf
+ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ [ #g #j1 #Hg #H1 #H2 | #g #Hg #Ho ] destruct
+ <tls_xn /2 width=2 by/
+]
+qed-.
+
(* Advanced inversion lemmas on isid ****************************************)
-lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≡ i.
+lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i,f⦄ ≘ i.
#i elim i -i
[ #f #H elim (isid_inv_gen … H) -H /2 width=2 by at_refl/
| #i #IH #f #H elim (isid_inv_gen … H) -H /3 width=7 by at_push/
]
qed.
-lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1, f⦄ ≡ i2 → i1 = i2.
+lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1,f⦄ ≘ i2 → i1 = i2.
/3 width=6 by isid_inv_at, at_mono/ qed-.
-(* Advancedd properties on isid *********************************************)
+(* Advanced properties on isid **********************************************)
-let corec isid_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → 𝐈⦃f⦄ ≝ ?.
+corec lemma isid_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈⦃f⦄.
#f #Hf lapply (Hf 0)
#H cases (at_fwd_id_ex … H) -H
#g #H @(isid_push … H) /3 width=7 by at_inv_npn/
(* Advanced properties on id ************************************************)
-lemma id_inv_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → 𝐈𝐝 ≗ f.
+lemma id_inv_at: ∀f. (∀i. @⦃i,f⦄ ≘ i) → 𝐈𝐝 ≡ f.
/3 width=1 by isid_at, eq_id_inv_isid/ qed-.
-lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≡ i.
+lemma id_at: ∀i. @⦃i,𝐈𝐝⦄ ≘ i.
/2 width=1 by isid_inv_at/ qed.
+
+(* Advanced forward lemmas on id ********************************************)
+
+lemma at_id_fwd: ∀i1,i2. @⦃i1,𝐈𝐝⦄ ≘ i2 → i1 = i2.
+/2 width=4 by at_mono/ qed.
+
+(* Main properties on id ****************************************************)
+
+theorem at_div_id_dx: ∀f. H_at_div f 𝐈𝐝 𝐈𝐝 f.
+#f #jf #j0 #j #Hf #H0
+lapply (at_id_fwd … H0) -H0 #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_id_sn: ∀f. H_at_div 𝐈𝐝 f f 𝐈𝐝.
+/3 width=6 by at_div_id_dx, at_div_comm/ qed-.
+
+(* Properties with uniform relocations **************************************)
+
+lemma at_uni: ∀n,i. @⦃i,𝐔❴n❵⦄ ≘ n+i.
+#n elim n -n /2 width=5 by at_next/
+qed.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma at_inv_uni: ∀n,i,j. @⦃i,𝐔❴n❵⦄ ≘ j → j = n+i.
+/2 width=4 by at_mono/ qed-.
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