- degree-based equivalene for terms

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc_2A1 / sta / lstas_alt.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/sta/lstas_alt.etc
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/statictypestaralt_6.ma".
+include "basic_2/unfold/lstas_lift.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+(* alternative definition of lstas *)
+inductive lstasa (h): genv → relation4 lenv nat term term ≝
+| lstasa_O   : ∀G,L,T. lstasa h G L 0 T T
+| lstasa_sort: ∀G,L,l,k. lstasa h G L l (⋆k) (⋆((next h)^l k))
+| lstasa_ldef: ∀G,L,K,V,W,U,i,l. ⇩[i] L ≡ K.ⓓV → lstasa h G K (l+1) V W →
+               ⇧[0, i+1] W ≡ U → lstasa h G L (l+1) (#i) U
+| lstasa_ldec: ∀G,L,K,W,V,V0,U,i,l. ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
+               lstasa h G K l W V → ⇧[0, i+1] V ≡ U → lstasa h G L (l+1) (#i) U
+| lstasa_bind: ∀a,I,G,L,V,T,U,l. lstasa h G (L.ⓑ{I}V) l T U →
+               lstasa h G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| lstasa_appl: ∀G,L,V,T,U,l. lstasa h G L l T U → lstasa h G L l (ⓐV.T) (ⓐV.U)
+| lstasa_cast: ∀G,L,W,T,U,l. lstasa h G L (l+1) T U → lstasa h G L (l+1) (ⓝW.T) U
+.
+
+interpretation "nat-iterated static type assignment (term) alternative"
+   'StaticTypeStarAlt h G L l T U = (lstasa h G L l T U).
+
+(* Base properties **********************************************************)
+
+lemma sta_lstasa: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T ••*[h, 1] U.
+#h #G #L #T #U #H elim H -G -L -T -U
+/2 width=8 by lstasa_O, lstasa_sort, lstasa_ldef, lstasa_ldec, lstasa_bind, lstasa_appl, lstasa_cast/
+qed.
+
+lemma lstasa_step_dx: ∀h,G,L,T1,T,l. ⦃G, L⦄ ⊢ T1 ••*[h, l] T →
+                      ∀T2. ⦃G, L⦄ ⊢ T •[h] T2 → ⦃G, L⦄ ⊢ T1 ••*[h, l+1] T2.
+#h #G #L #T1 #T #l #H elim H -G -L -T1 -T -l
+[ /2 width=1 by sta_lstasa/
+| #G #L #l #k #X #H >(sta_inv_sort1 … H) -X >commutative_plus //
+| #G #L #K #V #W #U #i #l #HLK #_ #HWU #IHVW #U2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H
+  elim (sta_inv_lift1 … HU2 … H … HWU) -H -U /3 width=6 by lstasa_ldef/
+| #G #L #K #W #V #V0 #U #i #l #HLK #HWl0 #_ #HVU #IHWV #U2 #HU2
+  lapply (drop_fwd_drop2 … HLK) #H
+  elim (sta_inv_lift1 … HU2 … H … HVU) -H -U /3 width=8 by lstasa_ldec/
+| #a #I #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
+  elim (sta_inv_bind1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_bind/
+| #G #L #V #T1 #U1 #l #_ #IHTU1 #X #H
+  elim (sta_inv_appl1 … H) -H #U #HU1 #H destruct /3 width=1 by lstasa_appl/
+| /3 width=1 by lstasa_cast/
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lstas_lstasa: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ⦃G, L⦄ ⊢ T ••*[h, l] U.
+#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /2 width=3 by lstasa_step_dx, lstasa_O/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lstasa_inv_lstas: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T ••*[h, l] U → ⦃G, L⦄ ⊢ T •*[h, l] U.
+#h #G #L #T #U #l #H elim H -G -L -T -U -l
+/2 width=8 by lstas_inv_SO, lstas_ldec, lstas_ldef, lstas_cast, lstas_appl, lstas_bind/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lstas_ind_alt: ∀h. ∀R:genv→relation4 lenv nat term term.
+                     (∀G,L,T. R G L O T T) →
+                     (∀G,L,l,k. R G L l (⋆k) (⋆((next h)^l k))) → (
+                        ∀G,L,K,V,W,U,i,l.
+                        ⇩[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V •*[h, l+1] W → ⇧[O, i+1] W ≡ U →
+                        R G K (l+1) V W → R G L (l+1) (#i) U
+                     ) → (
+                        ∀G,L,K,W,V,V0,U,i,l.
+                        ⇩[i] L ≡ K.ⓛW → ⦃G, K⦄ ⊢ W •[h] V0 →
+                        ⦃G, K⦄ ⊢ W •*[h, l]V → ⇧[O, i+1] V ≡ U →
+                        R G K l W V → R G L (l+1) (#i) U
+                     ) → (
+                        ∀a,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
+                        R G (L.ⓑ{I}V) l T U → R G L l (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+                     ) → (
+                        ∀G,L,V,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
+                        R G L l T U → R G L l (ⓐV.T) (ⓐV.U)
+                     ) → (
+                        ∀G,L,W,T,U,l. ⦃G, L⦄⊢ T •*[h, l+1] U →
+                        R G L (l+1) T U → R G L (l+1) (ⓝW.T) U
+                     ) →
+                     ∀G,L,l,T,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R G L l T U.
+#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #G #L #l #T #U #H
+elim (lstas_lstasa … H) /3 width=10 by lstasa_inv_lstas/
+qed-.