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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 •* break [ g ] break term 46 T2 )"
16 non associative with precedence 45
17 for @{ 'StaticTypeStar $h $g $L $T1 $T2 }.
19 include "basic_2/static/ssta.ma".
21 (* ITERATED STRATIFIED STATIC TYPE ASSIGNMENTON TERMS ***********************)
23 inductive sstas (h:sh) (g:sd h) (L:lenv): relation term ≝
24 | sstas_refl: ∀T,U. ⦃h, L⦄ ⊢ T •[g, 0] U → sstas h g L T T
25 | sstas_step: ∀T,U1,U2,l. ⦃h, L⦄ ⊢ T •[g, l+1] U1 → sstas h g L U1 U2 →
28 interpretation "stratified unwind (term)"
29 'StaticTypeStar h g L T U = (sstas h g L T U).
31 (* Basic eliminators ********************************************************)
33 fact sstas_ind_alt_aux: ∀h,g,L,U2. ∀R:predicate term.
34 (∀T. ⦃h, L⦄ ⊢ U2 •[g , 0] T → R U2) →
35 (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
36 ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
38 ∀T,U. ⦃h, L⦄ ⊢ T •*[g] U → U = U2 → R T.
39 #h #g #L #U2 #R #H1 #H2 #T #U #H elim H -H -T -U /2 width=2/ /3 width=5/
42 lemma sstas_ind_alt: ∀h,g,L,U2. ∀R:predicate term.
43 (∀T. ⦃h, L⦄ ⊢ U2 •[g , 0] T → R U2) →
44 (∀T,U1,l. ⦃h, L⦄ ⊢ T •[g, l + 1] U1 →
45 ⦃h, L⦄ ⊢ U1 •* [g] U2 → R U1 → R T
47 ∀T. ⦃h, L⦄ ⊢ T •*[g] U2 → R T.
48 /3 width=9 by sstas_ind_alt_aux/ qed-.
50 (* Basic inversion lemmas ***************************************************)
52 fact sstas_inv_sort1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀k. T = ⋆k →
53 ∀l. deg h g k l → U = ⋆((next h)^l k).
54 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
55 [ #U0 #HU0 #k #H #l #Hkl destruct
56 elim (ssta_inv_sort1 … HU0) -L #HkO #_ -U0
57 >(deg_mono … Hkl HkO) -g -l //
58 | #T0 #U0 #l0 #HTU0 #_ #IHU0 #k #H #l #Hkl destruct
59 elim (ssta_inv_sort1 … HTU0) -L #HkS #H destruct
60 lapply (deg_mono … Hkl HkS) -Hkl #H destruct
61 >(IHU0 (next h k) ? l0) -IHU0 // /2 width=1/ >iter_SO >iter_n_Sm //
65 lemma sstas_inv_sort1: ∀h,g,L,U,k. ⦃h, L⦄ ⊢ ⋆k •*[g] U → ∀l. deg h g k l →
69 fact sstas_inv_bind1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
71 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z.
72 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
73 [ #U0 #HU0 #J #X #Y #H destruct
74 elim (ssta_inv_bind1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
75 | #T0 #U0 #l #HTU0 #_ #IHU0 #J #X #Y #H destruct
76 elim (ssta_inv_bind1 … HTU0) -HTU0 #X0 #HX0 #H destruct
77 elim (IHU0 J X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
81 lemma sstas_inv_bind1: ∀h,g,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X •*[g] U →
82 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X •*[g] Z & U = ⓑ{J}Y.Z.
85 fact sstas_inv_appl1_aux: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U → ∀X,Y. T = ⓐY.X →
86 ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
87 #h #g #L #T #U #H @(sstas_ind_alt … H) -T
88 [ #U0 #HU0 #X #Y #H destruct
89 elim (ssta_inv_appl1 … HU0) -HU0 #X0 #HX0 #H destruct /3 width=3/
90 | #T0 #U0 #l #HTU0 #_ #IHU0 #X #Y #H destruct
91 elim (ssta_inv_appl1 … HTU0) -HTU0 #X0 #HX0 #H destruct
92 elim (IHU0 X0 Y ?) -IHU0 // #X1 #HX01 #H destruct /3 width=4/
96 lemma sstas_inv_appl1: ∀h,g,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X •*[g] U →
97 ∃∃Z. ⦃h, L⦄ ⊢ X •*[g] Z & U = ⓐY.Z.
100 (* Basic forward lemmas *****************************************************)
102 lemma sstas_fwd_correct: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T •*[g] U →
103 ∃∃W. ⦃h, L⦄ ⊢ U •[g, 0] W & ⦃h, L⦄ ⊢ U •*[g] U.
104 #h #g #L #T #U #H @(sstas_ind_alt … H) -T /2 width=1/ /3 width=2/
107 (* Basic_1: removed theorems 7:
108 sty1_bind sty1_abbr sty1_appl sty1_cast2
109 sty1_lift sty1_correct sty1_trans