(* sort degree specification *)
record sd (h:sh): Type[0] ≝ {
deg : relation nat; (* degree of the sort *)
- deg_total: ∀k. ∃l. deg k l; (* functional relation axioms *)
- deg_mono : ∀k,l1,l2. deg k l1 → deg k l2 → l1 = l2;
- deg_next : ∀k,l. deg k l → deg (next h k) (l - 1) (* compatibility condition *)
+ deg_total: ∀k. ∃d. deg k d; (* functional relation axioms *)
+ deg_mono : ∀k,d1,d2. deg k d1 → deg k d2 → d1 = d2;
+ deg_next : ∀k,d. deg k d → deg (next h k) (d - 1) (* compatibility condition *)
}.
(* Notable specifications ***************************************************)
-definition deg_O: relation nat ≝ λk,l. l = 0.
+definition deg_O: relation nat ≝ λk,d. d = 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)
-| deg_SO_zero: ((∃l0. (next h)^l0 k0 = k) → ⊥) → deg_SO h k k0 0
+| deg_SO_pos : ∀d0. (next h)^d0 k0 = k → deg_SO h k k0 (d0 + 1)
+| deg_SO_zero: ((∃d0. (next h)^d0 k0 = k) → ⊥) → deg_SO h k k0 0
.
-fact deg_SO_inv_pos_aux: ∀h,k,k0,l0. deg_SO h k k0 l0 → ∀l. l0 = l + 1 →
- (next h)^l k0 = k.
-#h #k #k0 #l0 * -l0
-[ #l0 #Hl0 #l #H
+fact deg_SO_inv_pos_aux: ∀h,k,k0,d0. deg_SO h k k0 d0 → ∀d. d0 = d + 1 →
+ (next h)^d k0 = k.
+#h #k #k0 #d0 * -d0
+[ #d0 #Hd0 #d #H
lapply (injective_plus_l … H) -H #H destruct //
-| #_ #
l0 <plus_n_Sm #H destruct
+| #_ #
d0 <plus_n_Sm #H destruct
]
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-.
+lemma deg_SO_inv_pos: ∀h,k,k0,d0. deg_SO h k k0 (d0 + 1) → (next h)^d0 k0 = k.
+/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 ?) //
qed.
lemma deg_SO_gt: ∀h,k1,k2. k1 < k2 → deg_SO h k1 k2 0.
-#h #k1 #k2 #HK12 @deg_SO_zero * #l elim l -l normalize
+#h #k1 #k2 #HK12 @deg_SO_zero * #d elim d -d normalize
[ #H destruct
elim (lt_refl_false … HK12)
-| #l #_ #H
- lapply (next_lt h ((next h)^l k2)) >H -H #H
+| #d #_ #H
+ lapply (next_lt h ((next h)^d k2)) >H -H #H
lapply (transitive_lt … H HK12) -k1 #H1
- lapply (nexts_le h k2 l) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -l #H
+ lapply (nexts_le h k2 d) #H2
+ lapply (le_to_lt_to_lt … H2 H1) -h -d #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/ ]
-| #K0 #l1 #l2 * [ #l01 ] #H1 * [1,3: #l02 ] #H2 //
+ 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 #d1 #d2 * [ #d01 ] #H1 * [1,3: #d02 ] #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/
- | #H1 @deg_SO_zero * #l #H2 destruct
- @H1 -H1 @(ex_intro … (S l)) /2 width=1/ (**) (* explicit constructor *)
+| #k0 #d0 *
+ [ #d #H destruct elim d -d normalize
+ /2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
+ | #H1 @deg_SO_zero * #d #H2 destruct
+ @H1 -H1 @(ex_intro … (S d)) /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
+rec definition sd_d (h:sh) (k:nat) (d:nat) on d : sd h ≝
+ match d with
[ O ⇒ sd_O h
- | S l ⇒ match l with
+ | S d ⇒ match d with
[ O ⇒ sd_SO h k
- | _ ⇒ sd_l h (next h k) l
+ | _ ⇒ sd_d h (next h k) d
]
].
-(* 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).
-#h #g #k #l #H1
-elim (deg_total h g k) #l0 #H0
+lemma deg_inv_pred: ∀h,g,k,d. deg h g (next h k) (d+1) → deg h g k (d+2).
+#h #g #k #d #H1
+elim (deg_total h g k) #d0 #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 d 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+qed-.
+
+lemma deg_inv_prec: ∀h,g,k,d,d0. deg h g ((next h)^d k) (d0+1) → deg h g k (d+d0+1).
+#h #g #k #d @(nat_ind_plus … d) -d //
+#d #IHd #d0 >iter_SO #H
+lapply (deg_inv_pred … H) -H <(associative_plus d0 1 1) #H
+lapply (IHd … H) -IHd -H //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter: ∀h,g,k,d1,d2. deg h g k d1 → deg h g ((next h)^d2 k) (d1-d2).
+#h #g #k #d1 #d2 @(nat_ind_plus … d2) -d2 [ <minus_n_O // ]
+#d2 #IHd2 #Hkd1 >iter_SO <minus_plus /3 width=1 by deg_next/
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 //
+lemma deg_next_SO: ∀h,g,k,d. deg h g k (d+1) → deg h g (next h k) d.
+#h #g #k #d #Hkd
+lapply (deg_next … Hkd) -Hkd <minus_plus_m_m //
+qed-.
+
+lemma sd_d_SS: ∀h,k,d. sd_d h k (d + 2) = sd_d h (next h k) (d + 1).
+#h #k #d <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/
+lemma sd_d_correct: ∀h,d,k. deg h (sd_d h k d) k d.
+#h #d @(nat_ind_plus … d) -d // #d @(nat_ind_plus … d) -d /3 width=1 by deg_inv_pred/
qed.
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