X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fstatic%2Fssta.ma;h=eede2acc832ec486c2035036d15251bb8f765893;hb=46a8a345410219548128c2533ce32b1a8eca6c06;hp=d7a32e928b65916c34cef0f7acc8af685c68b81a;hpb=8ff4315142253a1a0478b67c07dddf70c36f50cd;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
index d7a32e928..eede2acc8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/ssta.ma
@@ -12,189 +12,162 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/substitution/ldrop.ma".
-include "basic_2/unfold/frsups.ma".
-include "basic_2/static/sd.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-inductive ssta (h:sh) (g:sd h): nat → lenv → relation term ≝
-| ssta_sort: ∀L,k,l. deg h g k l → ssta h g l L (⋆k) (⋆(next h k))
-| ssta_ldef: ∀L,K,V,W,U,i,l. ⇩[0, i] L ≡ K. ⓓV → ssta h g l K V W →
-             ⇧[0, i + 1] W ≡ U → ssta h g l L (#i) U
-| ssta_ldec: ∀L,K,W,V,U,i,l. ⇩[0, i] L ≡ K. ⓛW → ssta h g l K W V →
-             ⇧[0, i + 1] W ≡ U → ssta h g (l+1) L (#i) U
-| ssta_bind: ∀a,I,L,V,T,U,l. ssta h g l (L. ⓑ{I} V) T U →
-             ssta h g l L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
-| ssta_appl: ∀L,V,T,U,l. ssta h g l L T U →
-             ssta h g l L (ⓐV.T) (ⓐV.U)
-| ssta_cast: ∀L,W,T,U,l. ssta h g l L T U → ssta h g l L (ⓝW. T) U
+include "basic_2/notation/relations/statictype_6.ma".
+include "basic_2/static/da.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS ******************************)
+
+(* activate genv *)
+inductive ssta (h) (g): relation4 genv lenv term term ≝
+| ssta_sort: ∀G,L,k. ssta h g G L (⋆k) (⋆(next h k))
+| ssta_ldef: ∀G,L,K,V,U,W,i. ⇩[0, i] L ≡ K.ⓓV → ssta h g G K V W →
+             ⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
+| ssta_ldec: ∀G,L,K,W,U,l,i. ⇩[0, i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W ▪[h, g] l →
+             ⇧[0, i + 1] W ≡ U → ssta h g G L (#i) U
+| ssta_bind: ∀a,I,G,L,V,T,U. ssta h g G (L.ⓑ{I}V) T U →
+             ssta h g G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| ssta_appl: ∀G,L,V,T,U. ssta h g G L T U → ssta h g G L (ⓐV.T) (ⓐV.U)
+| ssta_cast: ∀G,L,W,T,U. ssta h g G L T U → ssta h g G L (ⓝW.T) U
 .
 
 interpretation "stratified static type assignment (term)"
-   'StaticType h g L T U l = (ssta h g l L T U).
-
-definition ssta_step: ∀h. sd h → lenv → relation term ≝ λh,g,L,T,U.
-                      ∃l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄.
+   'StaticType h g G L T U = (ssta h g G L T U).
 
 (* Basic inversion lemmas ************************************************)
 
-fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
-                         deg h g k0 l ∧ U = ⋆(next h k0).
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #Hkl #k0 #H destruct /2 width=1/
-| #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
-| #a #I #L #V #T #U #l #_ #k0 #H destruct
-| #L #V #T #U #l #_ #k0 #H destruct
-| #L #W #T #U #l #_ #k0 #H destruct
-qed.
-
-(* Basic_1: was just: sty0_gen_sort *)
-lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g] ⦃l, U⦄ →
-                      deg h g k l ∧ U = ⋆(next h k).
-/2 width=4/ qed-.
-
-fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀j. T = #j →
-                         (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
+fact ssta_inv_sort1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀k0. T = ⋆k0 →
+                         U = ⋆(next h k0).
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #k0 #H destruct //
+| #G #L #K #V #U #W #i #_ #_ #_ #k0 #H destruct
+| #G #L #K #W #U #l #i #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #_ #k0 #H destruct
+| #G #L #V #T #U #_ #k0 #H destruct
+| #G #L #W #T #U #_ #k0 #H destruct
+]
+qed-.
+
+lemma ssta_inv_sort1: ∀h,g,G,L,U,k. ⦃G, L⦄ ⊢ ⋆k •[h, g] U → U = ⋆(next h k).
+/2 width=6 by ssta_inv_sort1_aux/ qed-.
+
+fact ssta_inv_lref1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀j. T = #j →
+                         (∃∃K,V,W. ⇩[0, j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
                                    ⇧[0, j + 1] W ≡ U
                          ) ∨
-                         (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
-                                      ⇧[0, j + 1] W ≡ U & l = l0 + 1
+                         (∃∃K,W,l. ⇩[0, j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+                                   ⇧[0, j + 1] W ≡ U
                          ).
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #_ #j #H destruct
-| #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6/
-| #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8/
-| #a #I #L #V #T #U #l #_ #j #H destruct
-| #L #V #T #U #l #_ #j #H destruct
-| #L #W #T #U #l #_ #j #H destruct
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #j #H destruct
+| #G #L #K #V #U #W #i #HLK #HVW #HWU #j #H destruct /3 width=6/
+| #G #L #K #W #U #l #i #HLK #HWl #HWU #j #H destruct /3 width=6/
+| #a #I #G #L #V #T #U #_ #j #H destruct
+| #G #L #V #T #U #_ #j #H destruct
+| #G #L #W #T #U #_ #j #H destruct
 ]
-qed.
+qed-.
 
-(* Basic_1: was just: sty0_gen_lref *)
-lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g] ⦃l, U⦄ →
-                      (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
+lemma ssta_inv_lref1: ∀h,g,G,L,U,i. ⦃G, L⦄ ⊢ #i •[h, g] U →
+                      (∃∃K,V,W. ⇩[0, i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •[h, g] W &
                                 ⇧[0, i + 1] W ≡ U
                       ) ∨
-                      (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
-                                   ⇧[0, i + 1] W ≡ U & l = l0 + 1
+                      (∃∃K,W,l. ⇩[0, i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] l &
+                                ⇧[0, i + 1] W ≡ U
                       ).
-/2 width=3/ qed-.
-
-fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #_ #p0 #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
-| #a #I #L #V #T #U #l #_ #p0 #H destruct
-| #L #V #T #U #l #_ #p0 #H destruct
-| #L #W #T #U #l #_ #p0 #H destruct
-qed.
-
-lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g] ⦃l, U⦄ → ⊥.
-/2 width=9/ qed-.
-
-fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
-                         ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
-                         ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #_ #a #I #X #Y #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
-| #b #J #L #V #T #U #l #HTU #a #I #X #Y #H destruct /2 width=3/
-| #L #V #T #U #l #_ #a #I #X #Y #H destruct
-| #L #W #T #U #l #_ #a #I #X #Y #H destruct
-]
-qed.
-
-(* Basic_1: was just: sty0_gen_bind *)
-lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g] ⦃l, U⦄ →
-                      ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
-/2 width=3/ qed-.
-
-fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
-                         ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #_ #X #Y #H destruct
-| #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
-| #a #I #L #V #T #U #l #_ #X #Y #H destruct
-| #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3/
-| #L #W #T #U #l #_ #X #Y #H destruct
+/2 width=3 by ssta_inv_lref1_aux/ qed-.
+
+fact ssta_inv_gref1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀p0. T = §p0 → ⊥.
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #p0 #H destruct
+| #G #L #K #V #U #W #i #_ #_ #_ #p0 #H destruct
+| #G #L #K #W #U #l #i #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #_ #p0 #H destruct
+| #G #L #V #T #U #_ #p0 #H destruct
+| #G #L #W #T #U #_ #p0 #H destruct
 ]
-qed.
-
-(* Basic_1: was just: sty0_gen_appl *)
-lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g] ⦃l, U⦄ →
-                      ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
-/2 width=3/ qed-.
-
-fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
-                         ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
-#h #g #L #T #U #l * -L -T -U -l
-[ #L #k #l #_ #X #Y #H destruct
-| #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
-| #L #K #W #V #U #l #i #_ #_ #_ #X #Y #H destruct
-| #a #I #L #V #T #U #l #_ #X #Y #H destruct
-| #L #V #T #U #l #_ #X #Y #H destruct
-| #L #W #T #U #l #HTU #X #Y #H destruct //
+qed-.
+
+lemma ssta_inv_gref1: ∀h,g,G,L,U,p. ⦃G, L⦄ ⊢ §p •[h, g] U → ⊥.
+/2 width=9 by ssta_inv_gref1_aux/ qed-.
+
+fact ssta_inv_bind1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
+                         ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
+                         ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h, g] Z & U = ⓑ{b,J}Y.Z.
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #b #J #X #Y #H destruct
+| #G #L #K #V #U #W #i #_ #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #U #l #i #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #HTU #b #J #X #Y #H destruct /2 width=3/
+| #G #L #V #T #U #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #_ #b #J #X #Y #H destruct
 ]
-qed.
-
-(* Basic_1: was just: sty0_gen_cast *)
-lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g] ⦃l, U⦄ →
-                      ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
-/2 width=4/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
-  elim (frsupp_inv_atom1_frsups … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
-  elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
-  elim (lift_inv_lref2_be … H ? ?) -H //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
-  elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
-  elim (lift_inv_lref2_be … H ? ?) -H //
-| #a #I #L #V #T #U #l #_ #IHTU #H
-  elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
-  lapply (frsups_fwd_fw … H) -H normalize
-  <associative_plus <associative_plus #H
-  elim (le_plus_xySz_x_false … H)
-| #L #V #T #U #l #_ #IHTU #H
-  elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
-  lapply (frsups_fwd_fw … H) -H normalize
-  <associative_plus <associative_plus #H
-  elim (le_plus_xySz_x_false … H)
-| /3 width=4/
+qed-.
+
+lemma ssta_inv_bind1: ∀h,g,b,J,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X •[h, g] U →
+                      ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •[h, g] Z & U = ⓑ{b,J}Y.Z.
+/2 width=3 by ssta_inv_bind1_aux/ qed-.
+
+fact ssta_inv_appl1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀X,Y. T = ⓐY.X →
+                         ∃∃Z. ⦃G, L⦄ ⊢ X •[h, g] Z & U = ⓐY.Z.
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #X #Y #H destruct
+| #G #L #K #V #U #W #i #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #U #l #i #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #V #T #U #HTU #X #Y #H destruct /2 width=3/
+| #G #L #W #T #U #_ #X #Y #H destruct
 ]
 qed-.
 
-fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → T = U → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
-  lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
-  <e1 -e1 #H elim (lt_refl_false … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
-  elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
-  elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #W #T #U #l #HTU #_ #H destruct
-  elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
+lemma ssta_inv_appl1: ∀h,g,G,L,Y,X,U. ⦃G, L⦄ ⊢ ⓐY.X •[h, g] U →
+                      ∃∃Z. ⦃G, L⦄ ⊢ X •[h, g] Z & U = ⓐY.Z.
+/2 width=3 by ssta_inv_appl1_aux/ qed-.
+
+fact ssta_inv_cast1_aux: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ∀X,Y. T = ⓝY.X →
+                         ⦃G, L⦄ ⊢ X •[h, g] U.
+#h #g #G #L #T #U * -G -L -T -U
+[ #G #L #k #X #Y #H destruct
+| #G #L #K #V #U #W #i #_ #_ #_ #X #Y #H destruct
+| #G #L #K #W #U #l #i #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #V #T #U #_ #X #Y #H destruct
+| #G #L #W #T #U #HTU #X #Y #H destruct //
 ]
 qed-.
 
-lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, T⦄ → ⊥.
-/2 width=8 by ssta_inv_refl_aux/ qed-.
+lemma ssta_inv_cast1: ∀h,g,G,L,X,Y,U. ⦃G, L⦄ ⊢ ⓝY.X •[h, g] U → ⦃G, L⦄ ⊢ X •[h, g] U.
+/2 width=4 by ssta_inv_cast1_aux/ qed-.
+
+(* Inversion lemmas on degree assignment for terms **************************)
 
-lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
-[ * #_ #H destruct /2 width=6 by ssta_inv_refl/
-| /2 width=8 by ssta_inv_frsupp/
+lemma ssta_inv_da: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U →
+                   ∃l. ⦃G, L⦄ ⊢ T ▪[h, g] l.
+#h #g #G #L #T #U #H elim H -G -L -T -U
+[ #G #L #k elim (deg_total h g k) /3 width=2/
+| #G #L #K #V #U #W #i #HLK #_ #_ * /3 width=5/
+| #G #L #K #W #U #l #i #HLK #HWl #_ /3 width=5/
+| #a #I #G #L #V #T #U #_ * /3 width=2/
+| #G #L #V #T #U #_ * /3 width=2/
+| #G #L #W #T #U #_ * /3 width=2/
 ]
 qed-.
+
+(* Properties on degree assignment for terms ********************************)
+
+lemma da_ssta: ∀h,g,G,L,T,l. ⦃G, L⦄ ⊢ T ▪[h, g] l →
+               ∃U. ⦃G, L⦄ ⊢ T •[h, g] U.
+#h #g #G #L #T #l #H elim H -G -L -T -l
+[ /2 width=2/
+| #G #L #K #V #i #l #HLK #_ * #W #HVW
+  elim (lift_total W 0 (i+1)) /3 width=7/
+| #G #L #K #W #i #l #HLK #HW #_
+  elim (lift_total W 0 (i+1)) /3 width=7/
+| #a #I #G #L #V #T #l #_ * /3 width=2/
+| * #G #L #V #T #l #_ * /3 width=2/
+]
+qed-.
+
+(* Basic_1: removed theorems 7:
+            sty0_gen_sort sty0_gen_lref sty0_gen_bind sty0_gen_appl sty0_gen_cast
+            sty0_lift sty0_correct
+*)