we rephrased sstas in terms of star

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / unwind / sstas.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
index 3928e49cc551fe2e13adeaecc8980bb9b95b972d..c504e4edc1dfd5fbf99745a33b82ade43434e1ac 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unwind/sstas.ma
@@ -16,69 +16,62 @@ include "basic_2/static/ssta.ma".
 
 (* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
 
-inductive sstas (h:sh) (g:sd h) (L:lenv): relation term ≝
-| sstas_refl: ∀T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → sstas h g L T T
-| sstas_step: ∀T,U1,U2,l. ⦃h, L⦄ ⊢ T •[g, l+1] U1 → sstas h g L U1 U2 →
-              sstas h g L T U2.
+(* Note: includes: stass_refl *)
+definition sstas: ∀h. sd h → lenv → relation term ≝
+                  λh,g,L. star … (ssta_step h g L).
 
 interpretation "iterated stratified static type assignment (term)"
    'StaticTypeStar h g L T U = (sstas h g L T U).
 
 (* Basic eliminators ********************************************************)
 
-fact sstas_ind_alt_aux: ∀h,g,L,U2. ∀R:predicate term.
-                        (∀T,l. ⦃h, L⦄ ⊢ U2 •[g , l] T → R U2) →
-                        (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
-                                  ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
-                        ) →
-                        ∀T,U. ⦃h, L⦄ ⊢ T •*[g] U → U = U2 → R T.
-#h #g #L #U2 #R #H1 #H2 #T #U #H elim H -H -T -U /2 width=3/ /3 width=5/
+lemma sstas_ind: ∀h,g,L,T. ∀R:predicate term.
+                 R T → (
+                    ∀U1,U2,l. ⦃h, L⦄ ⊢ T •* [g] U1 →  ⦃h, L⦄ ⊢ U1 •[g, l + 1] U2 →
+                    R U1 → R U2
+                 ) →
+                 ∀U. ⦃h, L⦄ ⊢ T •*[g] U → R U.
+#h #g #L #T #R #IH1 #IH2 #U #H elim H -U //
+#U1 #U2 #H * /2 width=5/
 qed-.
 
-lemma sstas_ind_alt: ∀h,g,L,U2. ∀R:predicate term.
-                     (∀T,l. ⦃h, L⦄ ⊢ U2 •[g , l] T → R U2) →
-                     (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
-                               ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
-                     ) →
-                     ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
-/3 width=9 by sstas_ind_alt_aux/ qed-.
-                         
-(* Basic inversion lemmas ***************************************************)
-
-fact sstas_inv_bind1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
-                          ∀a,J,X,Y. T = ⓑ{a,J}Y.X →
-                          ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,J}Y.Z.
-#h #g #L #T #U #H @(sstas_ind_alt … H) -T
-[ #U0 #l #HU0 #a #J #X #Y #H destruct
-  elim (ssta_inv_bind1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
-| #T0 #U0 #l #HTU0 #_ #IHU0 #a #J #X #Y #H destruct
-  elim (ssta_inv_bind1 … HTU0) -HTU0 #X0 #HX0 #H destruct
-  elim (IHU0 a J X0 Y …) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
-]
+lemma sstas_ind_dx: ∀h,g,L,U2. ∀R:predicate term.
+                    R U2 → (
+                       ∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 → ⦃h, L⦄ ⊢ U1 •* [g] U2 →
+                       R U1 → R T
+                    ) →
+                    ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
+#h #g #L #U2 #R #IH1 #IH2 #T #H @(star_ind_l … T H) -T //
+#T #T0 * /2 width=5/
 qed-.
 
-lemma sstas_inv_bind1: ∀h,g,a,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{a,J}Y.X •*[g] U →
-                       ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,J}Y.Z.
-/2 width=3 by sstas_inv_bind1_aux/ qed-.
+(* Basic properties *********************************************************)
+
+lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l+1] U → ⦃h, L⦄ ⊢ T •*[g] U.
+/3 width=2/ qed.
+
+lemma sstas_strap1: ∀h,g,L,T1,T2,U2,l. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T2 •[g,l+1] U2 →
+                    ⦃h, L⦄ ⊢ T1 •*[g] U2.
+/3 width=4 by sstep, ex_intro/ (**) (* auto fails without trace *)
+qed.
+
+lemma sstas_strap2: ∀h,g,L,T1,U1,U2,l. ⦃h, L⦄ ⊢ T1 •[g, l+1] U1 → ⦃h, L⦄ ⊢ U1 •*[g] U2 →
+                    ⦃h, L⦄ ⊢ T1 •*[g] U2.
+/3 width=3 by star_compl, ex_intro/ (**) (* auto fails without trace *)
+qed.
 
-fact sstas_inv_appl1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀X,Y. T = ⓐY.X →
-                          ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
-#h #g #L #T #U #H @(sstas_ind_alt … H) -T
-[ #U0 #l #HU0 #X #Y #H destruct
-  elim (ssta_inv_appl1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
-| #T0 #U0 #l #HTU0 #_ #IHU0 #X #Y #H destruct
-  elim (ssta_inv_appl1 … HTU0) -HTU0 #X0 #HX0 #H destruct
-  elim (IHU0 X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
-]
+(* Basic inversion lemmas ***************************************************)
+
+lemma sstas_inv_bind1: ∀h,g,a,I,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •*[g] U →
+                       ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,I}Y.Z.
+#h #g #a #I #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
+#T #U #l #_ #HTU * #Z #HXZ #H destruct
+elim (ssta_inv_bind1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
 qed-.
 
 lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
                        ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
-/2 width=3 by sstas_inv_appl1_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma sstas_fwd_correct: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
-                         ∃∃l,W. ⦃h, L⦄ ⊢ U •[g, l] W.
-#h #g #L #T #U #H @(sstas_ind_alt … H) -T // /2 width=3/
+#h #g #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
+#T #U #l #_ #HTU * #Z #HXZ #H destruct
+elim (ssta_inv_appl1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
 qed-.