X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fsyntax%2Fsd.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fsyntax%2Fsd.ma;h=533a79e05bc91ce73cce3c3391f94d6088c76f14;hb=647b419e96770d90a82d7a9e5e8843566a9f93ee;hp=8a9ae71837c6618253a13358fa63df5b635c2227;hpb=f308429a0fde273605a2330efc63268b4ac36c99;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma
index 8a9ae7183..533a79e05 100644
--- a/matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/syntax/sd.ma
@@ -12,113 +12,57 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "static_2/syntax/item_sh.ma".
+include "static_2/syntax/sh.ma".
 
 (* SORT DEGREE **************************************************************)
 
 (* sort degree specification *)
-record sd (h:sh): Type[0] ≝ {
-   deg      : relation nat;                            (* degree of the sort *)
-   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 *)
+record sd: Type[0] ≝ {
+(* degree of the sort *)
+   deg: relation nat
+}.
+
+(* sort degree postulates *)
+record sd_props (h) (o): Prop ≝ {
+(* functional relation axioms *)
+   deg_total: ∀s. ∃d. deg o s d;
+   deg_mono : ∀s,d1,d2. deg o s d1 → deg o s d2 → d1 = d2;
+(* compatibility condition *)
+   deg_next : ∀s,d. deg o s d → deg o (⫯[h]s) (↓d)
 }.
 
 (* Notable specifications ***************************************************)
 
 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.
-
-(* 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_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-.
-
-(* 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,s. deg_SO h s s 1.
-#h #s @(deg_SO_succ … 0 ?) //
-qed.
-
-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)
-| #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,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/
-  ]
-| #s0 #n *
-  [ #d #H destruct elim d -d normalize
-    /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
-  | #H1 @deg_SO_zero * #d #H2 destruct
-    @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
-  ]
-]
-defined.
+definition sd_O: sd ≝ mk_sd deg_O.
 
-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 s
-           | _ ⇒ sd_d h (next h s) d
-           ]
-   ].
+lemma sd_O_props (h): sd_props h sd_O ≝ mk_sd_props ….
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-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 >H >S_pred /2 width=2 by ltn_to_ltO/
+lemma deg_inv_pred (h) (o): sd_props h o →
+      ∀s,d. deg o (⫯[h]s) (↑d) → deg o s (↑↑d).
+#h #o #Ho #s #d #H1
+elim (deg_total … Ho s) #d0 #H0
+lapply (deg_next … Ho … H0) #H2
+lapply (deg_mono … Ho … H1 H2) -H1 -H2 #H >H >S_pred
+/2 width=2 by ltn_to_ltO/
 qed-.
 
-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/
+lemma deg_inv_prec (h) (o): sd_props h o →
+      ∀s,n,d. deg o ((next h)^n s) (↑d) → deg o s (↑(d+n)).
+#h #o #H0 #s #n elim n -n normalize /3 width=3 by deg_inv_pred/
 qed-.
 
 (* Basic properties *********************************************************)
 
-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/
+lemma deg_iter (h) (o): sd_props h o →
+      ∀s,d,n. deg o s d → deg o ((next h)^n s) (d-n).
+#h #o #Ho #s #d #n elim n -n normalize /3 width=1 by deg_next/
 qed.
 
-lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
+lemma deg_next_SO (h) (o): sd_props h o →
+      ∀s,d. deg o s (↑d) → deg o (next h s) d.
 /2 width=1 by deg_next/ 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,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.