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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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11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/static/ssta.ma".
17 (* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************)
19 inductive sstas (h:sh) (g:sd h) (L:lenv): relation term ≝
20 | sstas_refl: ∀T,U. ⦃h, L⦄ ⊢ T •[g, 0] U → sstas h g L T T
21 | sstas_step: ∀T,U1,U2,l. ⦃h, L⦄ ⊢ T •[g, l+1] U1 → sstas h g L U1 U2 →
24 interpretation "stratified unwind (term)"
25 'StaticTypeStar h g L T U = (sstas h g L T U).
27 (* Basic eliminators ********************************************************)
29 fact sstas_ind_alt_aux: ∀h,g,L,U2. ∀R:predicate term.
30 (∀T. ⦃h, L⦄ ⊢ U2 •[g , 0] T → R U2) →
31 (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
32 ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
34 ∀T,U. ⦃h, L⦄ ⊢ T •*[g] U → U = U2 → R T.
35 #h #g #L #U2 #R #H1 #H2 #T #U #H elim H -H -T -U /2 width=2/ /3 width=5/
38 lemma sstas_ind_alt: ∀h,g,L,U2. ∀R:predicate term.
39 (∀T. ⦃h, L⦄ ⊢ U2 •[g , 0] T → R U2) →
40 (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
41 ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
43 ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
44 /3 width=9 by sstas_ind_alt_aux/ qed-.
46 (* Basic inversion lemmas ***************************************************)
48 fact sstas_inv_sort1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀k. T = ⋆k →
49 ∀l. deg h g k l → U = ⋆((next h)^l k).
50 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
51 [ #U0 #HU0 #k #H #l #Hkl destruct
52 elim (ssta_inv_sort1 … HU0) -L #HkO #_ -U0
53 >(deg_mono … Hkl HkO) -g -l //
54 | #T0 #U0 #l0 #HTU0 #_ #IHU0 #k #H #l #Hkl destruct
55 elim (ssta_inv_sort1 … HTU0) -L #HkS #H destruct
56 lapply (deg_mono … Hkl HkS) -Hkl #H destruct
57 >(IHU0 (next h k) ? l0) -IHU0 // /2 width=1/ >iter_SO >iter_n_Sm //
61 lemma sstas_inv_sort1: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k •*[g] U → ∀l. deg h g k l →
65 fact sstas_inv_bind1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
67 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z.
68 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
69 [ #U0 #HU0 #J #X #Y #H destruct
70 elim (ssta_inv_bind1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
71 | #T0 #U0 #l #HTU0 #_ #IHU0 #J #X #Y #H destruct
72 elim (ssta_inv_bind1 … HTU0) -HTU0 #X0 #HX0 #H destruct
73 elim (IHU0 J X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
77 lemma sstas_inv_bind1: ∀h,g,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X •*[g] U →
78 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z.
81 fact sstas_inv_appl1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀X,Y. T = ⓐY.X →
82 ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
83 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
84 [ #U0 #HU0 #X #Y #H destruct
85 elim (ssta_inv_appl1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
86 | #T0 #U0 #l #HTU0 #_ #IHU0 #X #Y #H destruct
87 elim (ssta_inv_appl1 … HTU0) -HTU0 #X0 #HX0 #H destruct
88 elim (IHU0 X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
92 lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
93 ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
96 (* Basic forward lemmas *****************************************************)
98 lemma sstas_fwd_correct: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
99 ∃∃W. ⦃h, L⦄ ⊢ U •[g, 0] W & ⦃h, L⦄ ⊢ U •*[g] U.
100 #h #g #L #T #U #H @(sstas_ind_alt … H) -T /2 width=1/ /3 width=2/
103 (* Basic_1: removed theorems 7:
104 sty1_bind sty1_abbr sty1_appl sty1_cast2
105 sty1_lift sty1_correct sty1_trans