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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/static/ssta.ma".
17 (* ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *********************)
19 (* Note: includes: stass_refl *)
20 definition sstas: ∀h. sd h → lenv → relation term ≝
21 λh,g,L. star … (ssta_step h g L).
23 interpretation "iterated stratified static type assignment (term)"
24 'StaticTypeStar h g L T U = (sstas h g L T U).
26 (* Basic eliminators ********************************************************)
28 lemma sstas_ind: ∀h,g,L,T. ∀R:predicate term.
30 ∀U1,U2,l. ⦃h, L⦄ ⊢ T •* [g] U1 → ⦃h, L⦄ ⊢ U1 •[g, l + 1] U2 →
33 ∀U. ⦃h, L⦄ ⊢ T •*[g] U → R U.
34 #h #g #L #T #R #IH1 #IH2 #U #H elim H -U //
35 #U1 #U2 #H * /2 width=5/
38 lemma sstas_ind_dx: ∀h,g,L,U2. ∀R:predicate term.
40 ∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 → ⦃h, L⦄ ⊢ U1 •* [g] U2 →
43 ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
44 #h #g #L #U2 #R #IH1 #IH2 #T #H @(star_ind_l … T H) -T //
48 (* Basic properties *********************************************************)
50 lemma ssta_sstas: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l+1] U → ⦃h, L⦄ ⊢ T •*[g] U.
53 lemma sstas_strap1: ∀h,g,L,T1,T2,U2,l. ⦃h, L⦄ ⊢ T1 •*[g] T2 → ⦃h, L⦄ ⊢ T2 •[g,l+1] U2 →
55 /3 width=4 by sstep, ex_intro/ (**) (* auto fails without trace *)
58 lemma sstas_strap2: ∀h,g,L,T1,U1,U2,l. ⦃h, L⦄ ⊢ T1 •[g, l+1] U1 → ⦃h, L⦄ ⊢ U1 •*[g] U2 →
60 /3 width=3 by star_compl, ex_intro/ (**) (* auto fails without trace *)
63 (* Basic inversion lemmas ***************************************************)
65 lemma sstas_inv_bind1: ∀h,g,a,I,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •*[g] U →
66 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •*[g] Z & U = ⓑ{a,I}Y.Z.
67 #h #g #a #I #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
68 #T #U #l #_ #HTU * #Z #HXZ #H destruct
69 elim (ssta_inv_bind1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/
72 lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
73 ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
74 #h #g #L #Y #X #U #H @(sstas_ind … H) -U /2 width=3/
75 #T #U #l #_ #HTU * #Z #HXZ #H destruct
76 elim (ssta_inv_appl1 … HTU) -HTU #Z0 #HZ0 #H destruct /3 width=4/