--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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/statictypestar_6.ma".
+include "basic_2/static/sta.ma".
+
+(* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
+
+definition lstas: ∀h. genv → lenv → nat → relation term ≝
+ λh,G,L. lstar … (sta h G L).
+
+interpretation "nat-iterated static type assignment (term)"
+ 'StaticTypeStar h G L l T U = (lstas h G L l T U).
+
+(* Basic eliminators ********************************************************)
+
+lemma lstas_ind_sn: ∀h,G,L,U2. ∀R:relation2 nat term.
+ R 0 U2 → (
+ ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h] U1 → ⦃G, L⦄ ⊢ U1 •* [h, l] U2 →
+ R l U1 → R (l+1) T
+ ) →
+ ∀l,T. ⦃G, L⦄ ⊢ T •*[h, l] U2 → R l T.
+/3 width=5 by lstar_ind_l/ qed-.
+
+lemma lstas_ind_dx: ∀h,G,L,T. ∀R:relation2 nat term.
+ R 0 T → (
+ ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, l] U1 → ⦃G, L⦄ ⊢ U1 •[h] U2 →
+ R l U1 → R (l+1) U2
+ ) →
+ ∀l,U. ⦃G, L⦄ ⊢ T •*[h, l] U → R l U.
+/3 width=5 by lstar_ind_r/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lstas_inv_O: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 0] U → T = U.
+/2 width=4 by lstar_inv_O/ qed-.
+
+lemma lstas_inv_SO: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, 1] U → ⦃G, L⦄ ⊢ T •[h] U.
+/2 width=1 by lstar_inv_step/ qed-.
+
+lemma lstas_inv_step_sn: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •[h] T & ⦃G, L⦄ ⊢ T •*[h, l] T2.
+/2 width=3 by lstar_inv_S/ qed-.
+
+lemma lstas_inv_step_dx: ∀h,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, l+1] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, l] T & ⦃G, L⦄ ⊢ T •[h] T2.
+/2 width=3 by lstar_inv_S_dx/ qed-.
+
+lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
+#h #G #L #X #k #l #H @(lstas_ind_dx … H) -X -l //
+#l #X #X0 #_ #H #IHX destruct
+lapply (sta_inv_sort1 … H) -H #H destruct
+>iter_SO //
+qed-.
+
+lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l+1] X → ⊥.
+#h #G #L #X #p #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX elim (sta_inv_gref1 … H)
+qed-.
+
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
+ ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
+#h #a #I #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
+ ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
+#h #G #L #V #T #X #l #H @(lstas_ind_dx … H) -X -l /2 width=3 by ex2_intro/
+#l #X #X0 #_ #HX0 * #U #HTU #H destruct
+elim (sta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3 by lstar_dx, ex2_intro/
+qed-.
+
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U → ⦃G, L⦄ ⊢ T •*[h, l+1] U.
+#h #G #L #W #T #X #l #H elim (lstas_inv_step_sn … H) -H
+#U #H #HUX lapply (sta_inv_cast1 … H) -H /2 width=3 by lstar_S/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lstas_refl: ∀h,G,L. reflexive … (lstas h G L 0).
+// qed.
+
+lemma sta_lstas: ∀h,G,L,T,U. ⦃G, L⦄ ⊢ T •[h] U → ⦃G, L⦄ ⊢ T •*[h, 1] U.
+/2 width=1 by lstar_step/ qed.
+
+lemma lstas_step_sn: ∀h,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h] U1 → ⦃G, L⦄ ⊢ U1 •*[h, l] U2 →
+ ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_S/ qed.
+
+lemma lstas_step_dx: ∀h,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, l] T2 → ⦃G, L⦄ ⊢ T2 •[h] U2 →
+ ⦃G, L⦄ ⊢ T1 •*[h, l+1] U2.
+/2 width=3 by lstar_dx/ qed.
+
+lemma lstas_split: ∀h,G,L. inv_ltransitive … (lstas h G L).
+/2 width=1 by lstar_inv_ltransitive/ qed-.
+
+lemma lstas_sort: ∀h,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, l] ⋆((next h)^l k).
+#h #G #L #l @(nat_ind_plus … l) -l //
+#l #IHl #k >iter_SO /2 width=3 by sta_sort, lstas_step_dx/
+qed.
+
+lemma lstas_bind: ∀h,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] ⓑ{a,I}V.U.
+#h #I #G #L #V #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_bind, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_appl: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U →
+ ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, l] ⓐV.U.
+#h #G #L #T #U #l #H @(lstas_ind_dx … H) -U -l /3 width=3 by sta_appl, lstar_O, lstas_step_dx/
+qed.
+
+lemma lstas_cast: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l+1] U →
+ ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l+1] U.
+#h #G #L #T #U #l #H elim (lstas_inv_step_sn … H) -H /3 width=3 by sta_cast, lstas_step_sn/
+qed.
+
+(* Basic_1: removed theorems 7:
+ sty1_abbr sty1_appl sty1_bind sty1_cast2
+ sty1_correct sty1_lift sty1_trans
+*)
projects / helm.git / blobdiff