we reformulate the extended computation to simplify the proof of its

[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
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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/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.
+
+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/
+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/
+]
+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-.
+
+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/
+]
+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/
+qed-.