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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/notation/relations/statictypestar_7.ma".
16 include "basic_2/static/ssta.ma".
18 (* NAT-ITERATED STRATIFIED STATIC TYPE ASSIGNMENT FOR TERMS *****************)
20 definition lsstas: ∀h. sd h → genv → lenv → nat → relation term ≝
21 λh,g,G,L. lstar … (ssta h g G L).
23 interpretation "nat-iterated stratified static type assignment (term)"
24 'StaticTypeStar h g G L l T U = (lsstas h g G L l T U).
26 (* Basic eliminators ********************************************************)
28 lemma lsstas_ind_sn: ∀h,g,G,L,U2. ∀R:relation2 nat term.
30 ∀l,T,U1. ⦃G, L⦄ ⊢ T •[h, g] U1 → ⦃G, L⦄ ⊢ U1 •* [h, g, l] U2 →
33 ∀l,T. ⦃G, L⦄ ⊢ T •*[h, g, l] U2 → R l T.
34 /3 width=5 by lstar_ind_l/ qed-.
36 lemma lsstas_ind_dx: ∀h,g,G,L,T. ∀R:relation2 nat term.
38 ∀l,U1,U2. ⦃G, L⦄ ⊢ T •* [h, g, l] U1 → ⦃G, L⦄ ⊢ U1 •[h, g] U2 →
41 ∀l,U. ⦃G, L⦄ ⊢ T •*[h, g, l] U → R l U.
42 /3 width=5 by lstar_ind_r/ qed-.
44 (* Basic inversion lemmas ***************************************************)
46 lemma lsstas_inv_O: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, g, 0] U → T = U.
47 /2 width=4 by lstar_inv_O/ qed-.
49 lemma lsstas_inv_SO: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •*[h, g, 1] U → ⦃G, L⦄ ⊢ T •[h, g] U.
50 /2 width=1 by lstar_inv_step/ qed-.
52 lemma lsstas_inv_step_sn: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, g, l+1] T2 →
53 ∃∃T. ⦃G, L⦄ ⊢ T1 •[h, g] T & ⦃G, L⦄ ⊢ T •*[h, g, l] T2.
54 /2 width=3 by lstar_inv_S/ qed-.
56 lemma lsstas_inv_step_dx: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •*[h, g, l+1] T2 →
57 ∃∃T. ⦃G, L⦄ ⊢ T1 •*[h, g, l] T & ⦃G, L⦄ ⊢ T •[h, g] T2.
58 /2 width=3 by lstar_inv_S_dx/ qed-.
60 lemma lsstas_inv_sort1: ∀h,g,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, g, l] X → X = ⋆((next h)^l k).
61 #h #g #G #L #X #k #l #H @(lsstas_ind_dx … H) -X -l //
62 #l #X #X0 #_ #H #IHX destruct
63 lapply (ssta_inv_sort1 … H) -H #H destruct
67 lemma lsstas_inv_gref1: ∀h,g,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, g, l+1] X → ⊥.
68 #h #g #G #L #X #p #l #H elim (lsstas_inv_step_sn … H) -H
69 #U #H #HUX elim (ssta_inv_gref1 … H)
72 lemma lsstas_inv_bind1: ∀h,g,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, g, l] X →
73 ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, g, l] U & X = ⓑ{a,I}V.U.
74 #h #g #a #I #G #L #V #T #X #l #H @(lsstas_ind_dx … H) -X -l [ /2 width=3/ ]
75 #l #X #X0 #_ #HX0 * #U #HTU #H destruct
76 elim (ssta_inv_bind1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3/
79 lemma lsstas_inv_appl1: ∀h,g,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, g, l] X →
80 ∃∃U. ⦃G, L⦄ ⊢ T •*[h, g, l] U & X = ⓐV.U.
81 #h #g #G #L #V #T #X #l #H @(lsstas_ind_dx … H) -X -l [ /2 width=3/ ]
82 #l #X #X0 #_ #HX0 * #U #HTU #H destruct
83 elim (ssta_inv_appl1 … HX0) -HX0 #U0 #HU0 #H destruct /3 width=3/
86 lemma lsstas_inv_cast1: ∀h,g,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, g, l+1] U → ⦃G, L⦄ ⊢ T •*[h, g, l+1] U.
87 #h #g #G #L #W #T #X #l #H elim (lsstas_inv_step_sn … H) -H
88 #U #H #HUX lapply (ssta_inv_cast1 … H) -H /2 width=3/
91 (* Basic properties *********************************************************)
93 lemma lsstas_refl: ∀h,g,G,L. reflexive … (lsstas h g G L 0).
96 lemma ssta_lsstas: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T •[h, g] U → ⦃G, L⦄ ⊢ T •*[h, g, 1] U.
99 lemma lsstas_step_sn: ∀h,g,G,L,T1,U1,U2,l. ⦃G, L⦄ ⊢ T1 •[h, g] U1 → ⦃G, L⦄ ⊢ U1 •*[h, g, l] U2 →
100 ⦃G, L⦄ ⊢ T1 •*[h, g, l+1] U2.
103 lemma lsstas_step_dx: ∀h,g,G,L,T1,T2,U2,l. ⦃G, L⦄ ⊢ T1 •*[h, g, l] T2 → ⦃G, L⦄ ⊢ T2 •[h, g] U2 →
104 ⦃G, L⦄ ⊢ T1 •*[h, g, l+1] U2.
107 lemma lsstas_split: ∀h,g,G,L. inv_ltransitive … (lsstas h g G L).
108 /2 width=1 by lstar_inv_ltransitive/ qed-.
110 lemma lsstas_sort: ∀h,g,G,L,l,k. ⦃G, L⦄ ⊢ ⋆k •*[h, g, l] ⋆((next h)^l k).
111 #h #g #G #L #l @(nat_ind_plus … l) -l //
112 #l #IHl #k >iter_SO /2 width=3/
115 lemma lsstas_bind: ∀h,g,I,G,L,V,T,U,l. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, g, l] U →
116 ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, g, l] ⓑ{a,I}V.U.
117 #h #g #I #G #L #V #T #U #l #H @(lsstas_ind_dx … H) -U -l // /3 width=3/
120 lemma lsstas_appl: ∀h,g,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, g, l] U →
121 ∀V.⦃G, L⦄ ⊢ ⓐV.T •*[h, g, l] ⓐV.U.
122 #h #g #G #L #T #U #l #H @(lsstas_ind_dx … H) -U -l // /3 width=3/
125 lemma lsstas_cast: ∀h,g,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, g, l+1] U →
126 ∀W. ⦃G, L⦄ ⊢ ⓝW.T •*[h, g, l+1] U.
127 #h #g #G #L #T #U #l #H elim (lsstas_inv_step_sn … H) -H /3 width=3/
130 (* Basic_1: removed theorems 7:
131 sty1_abbr sty1_appl sty1_bind sty1_cast2
132 sty1_correct sty1_lift sty1_trans