definition deg_O: relation nat ≝ λk,l. l = 0.
definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
-// /2 width=1/ /2 width=2/ qed.
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat ≝
| deg_SO_pos : ∀l0. (next h)^l0 k0 = k → deg_SO h k k0 (l0 + 1)
qed.
lemma deg_SO_inv_pos: ∀h,k,k0,l0. deg_SO h k k0 (l0 + 1) → (next h)^l0 k0 = k.
-/2 width=3/ qed-.
+/2 width=3 by deg_SO_inv_pos_aux/ qed-.
lemma deg_SO_refl: ∀h,k. deg_SO h k k 1.
#h #k @(deg_SO_pos … 0 ?) //
lapply (nexts_le h k2 l) #H2
lapply (le_to_lt_to_lt … H2 H1) -h -l #H
elim (lt_refl_false … H)
+]
qed.
definition sd_SO: ∀h. nat → sd h ≝ λh,k. mk_sd h (deg_SO h k) ….
[ #k0
- lapply (nexts_dec h k0 k) * [ * /3 width=2/ | /4 width=2/ ]
+ lapply (nexts_dec h k0 k) *
+ [ * /3 width=2 by deg_SO_pos, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
| #K0 #l1 #l2 * [ #l01 ] #H1 * [1,3: #l02 ] #H2 //
[ < H2 in H1; -H2 #H
lapply (nexts_inj … H) -H #H destruct //
- | elim (H1 ?) /2 width=2/
- | elim (H2 ?) /2 width=2/
+ | elim H1 /2 width=2 by ex_intro/
+ | elim H2 /2 width=2 by ex_intro/
]
| #k0 #l0 *
- [ #l #H destruct elim l -l normalize /2 width=1/
+ [ #l #H destruct elim l -l normalize
+ /2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
| #H1 @deg_SO_zero * #l #H2 destruct
- @H1 -H1 @(ex_intro … (S l)) /2 width=1/ (**) (* explicit constructor *)
+ @H1 -H1 @(ex_intro … (S l)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
]
]
-qed.
+defined.
let rec sd_l (h:sh) (k:nat) (l:nat) on l : sd h ≝
- match l with
+ match l with
[ O ⇒ sd_O h
| S l ⇒ match l with
[ O ⇒ sd_SO h k
]
].
-(* Basic properties *********************************************************)
+(* Basic inversion lemmas ***************************************************)
-lemma deg_pred: ∀h,g,k,l. deg h g (next h k) (l + 1) → deg h g k (l + 2).
+lemma deg_inv_pred: ∀h,g,k,l. deg h g (next h k) (l+1) → deg h g k (l+2).
#h #g #k #l #H1
elim (deg_total h g k) #l0 #H0
lapply (deg_next … H0) #H2
lapply (deg_mono … H1 H2) -H1 -H2 #H
-<(associative_plus l 1 1) >H <plus_minus_m_m // /2 width=3 by transitive_le/
+<(associative_plus l 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+qed-.
+
+lemma deg_inv_prec: ∀h,g,k,l,l0. deg h g ((next h)^l k) (l0+1) → deg h g k (l+l0+1).
+#h #g #k #l @(nat_ind_plus … l) -l //
+#l #IHl #l0 >iter_SO #H
+lapply (deg_inv_pred … H) -H <(associative_plus l0 1 1) #H
+lapply (IHl … H) -IHl -H //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter: ∀h,g,k,l1,l2. deg h g k l1 → deg h g ((next h)^l2 k) (l1-l2).
+#h #g #k #l1 #l2 @(nat_ind_plus … l2) -l2 [ <minus_n_O // ]
+#l2 #IHl2 #Hkl1 >iter_SO <minus_plus /3 width=1 by deg_next/
qed.
+lemma deg_next_SO: ∀h,g,k,l. deg h g k (l+1) → deg h g (next h k) l.
+#h #g #k #l #Hkl
+lapply (deg_next … Hkl) -Hkl <minus_plus_m_m //
+qed-.
+
lemma sd_l_SS: ∀h,k,l. sd_l h k (l + 2) = sd_l h (next h k) (l + 1).
#h #k #l <plus_n_Sm <plus_n_Sm //
qed.
lemma sd_l_correct: ∀h,l,k. deg h (sd_l h k l) k l.
-#h #l @(nat_ind_plus … l) -l // #l @(nat_ind_plus … l) -l // /3 width=1/
+#h #l @(nat_ind_plus … l) -l // #l @(nat_ind_plus … l) -l /3 width=1 by deg_inv_pred/
qed.