(* sort degree specification *)
record sd (h:sh): Type[0] ≝ {
deg : relation nat; (* degree of the sort *)
- 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 *)
+ deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
+ deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
+ deg_next : ∀s,d. deg s d → deg (next h s) (⫰d) (* compatibility condition *)
}.
(* Notable specifications ***************************************************)
-definition deg_O: relation nat ≝ λk,d. d = 0.
+definition deg_O: relation nat ≝ λs,d. d = 0.
definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
/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 : ∀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
+(* Basic_2A1: includes: deg_SO_pos *)
+inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
+| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (⫯n)
+| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
.
-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 //
-| #_ #d0 <plus_n_Sm #H destruct
+fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ⫯n →
+ (next h)^n s0 = s.
+#h #s #s0 #n0 * -n0
+[ #n #Hn #x #H destruct //
+| #_ #x #H destruct
]
-qed.
+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-.
+(* Basic_2A1: was: deg_SO_inv_pos *)
+lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (⫯n) → (next h)^n s0 = s.
+/2 width=3 by deg_SO_inv_succ_aux/ qed-.
-lemma deg_SO_refl: ∀h,k. deg_SO h k k 1.
-#h #k @(deg_SO_pos … 0 ?) //
+lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
+#h #s @(deg_SO_succ … 0 ?) //
qed.
-lemma deg_SO_gt: ∀h,k1,k2. k1 < k2 → deg_SO h k1 k2 0.
-#h #k1 #k2 #HK12 @deg_SO_zero * #d elim d -d normalize
+lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
+#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
[ #H destruct
elim (lt_refl_false … HK12)
-| #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 d) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -d #H
+| #n #_ #H
+ lapply (next_lt h ((next h)^n s2)) >H -H #H
+ lapply (transitive_lt … H HK12) -s1 #H1
+ lapply (nexts_le h s2 n) #H2
+ lapply (le_to_lt_to_lt … H2 H1) -h -n #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 by deg_SO_pos, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #d1 #d2 * [ #d01 ] #H1 * [1,3: #d02 ] #H2 //
+definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
+[ #s0
+ lapply (nexts_dec h s0 s) *
+ [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
+| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
[ < H2 in H1; -H2 #H
lapply (nexts_inj … H) -H #H destruct //
| elim H1 /2 width=2 by ex_intro/
| elim H2 /2 width=2 by ex_intro/
]
-| #k0 #d0 *
+| #s0 #n *
[ #d #H destruct elim d -d normalize
- /2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
+ /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
| #H1 @deg_SO_zero * #d #H2 destruct
- @H1 -H1 @(ex_intro … (S d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+ @H1 -H1 @(ex_intro … (⫯d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
]
]
defined.
-rec definition sd_d (h:sh) (k:nat) (d:nat) on d : sd h ≝
+rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
match d with
[ O ⇒ sd_O h
| S d ⇒ match d with
- [ O ⇒ sd_SO h k
- | _ ⇒ sd_d h (next h k) d
+ [ O ⇒ sd_SO h s
+ | _ ⇒ sd_d h (next h s) d
]
].
(* Basic inversion lemmas ***************************************************)
-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
+lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (⫯d) → deg h o s (⫯⫯d).
+#h #o #s #d #H1
+elim (deg_total h o s) #n #H0
lapply (deg_next … H0) #H2
-lapply (deg_mono … H1 H2) -H1 -H2 #H
-<(associative_plus d 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
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 //
+lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (⫯d) → deg h o s (⫯(d+n)).
+#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
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/
+lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
+#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
qed.
-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 deg_next_SO: ∀h,o,s,d. deg h o s (⫯d) → deg h o (next h s) d.
+/2 width=1 by deg_next/ 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_d_SS: ∀h,s,d. sd_d h s (⫯⫯d) = sd_d h (next h s) (⫯d).
+// qed.
-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/
+lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
+#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
qed.
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