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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/substitution/ldrop.ma".
16 include "basic_2/static/sh.ma".
18 (* STATIC TYPE ASSIGNMENT ON TERMS ******************************************)
20 inductive sta (h:sh): lenv → relation term ≝
21 | sta_sort: ∀L,k. sta h L (⋆k) (⋆(next h k))
22 | sta_ldef: ∀L,K,V,W,U,i. ⇩[0, i] L ≡ K. ⓓV → sta h K V W →
23 ⇧[0, i + 1] W ≡ U → sta h L (#i) U
24 | sta_ldec: ∀L,K,W,V,U,i. ⇩[0, i] L ≡ K. ⓛW → sta h K W V →
25 ⇧[0, i + 1] W ≡ U → sta h L (#i) U
26 | sta_bind: ∀I,L,V,T,U. sta h (L. ⓑ{I} V) T U →
27 sta h L (ⓑ{I}V.T) (ⓑ{I}V.U)
28 | sta_appl: ∀L,V,T,U. sta h L T U →
30 | sta_cast: ∀L,W,T,U. sta h L T U → sta h L (ⓝW. T) U
33 interpretation "static type assignment (term)"
34 'StaticType h L T U = (sta h L T U).
36 (* Basic inversion lemmas ************************************************)
38 fact sta_inv_sort1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀k0. T = ⋆k0 →
40 #h #L #T #U * -L -T -U
41 [ #L #k #k0 #H destruct //
42 | #L #K #V #W #U #i #_ #_ #_ #k0 #H destruct
43 | #L #K #W #V #U #i #_ #_ #_ #k0 #H destruct
44 | #I #L #V #T #U #_ #k0 #H destruct
45 | #L #V #T #U #_ #k0 #H destruct
46 | #L #W #T #U #_ #k0 #H destruct
49 (* Basic_1: was: sty0_gen_sort *)
50 lemma sta_inv_sort1: ∀h,L,U,k. ⦃h, L⦄ ⊢ ⋆k • U → U = ⋆(next h k).
53 fact sta_inv_lref1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀j. T = #j →
54 (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W &
57 (∃∃K,W,V. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V &
60 #h #L #T #U * -L -T -U
61 [ #L #k #j #H destruct
62 | #L #K #V #W #U #i #HLK #HVW #HWU #j #H destruct /3 width=6/
63 | #L #K #W #V #U #i #HLK #HWV #HWU #j #H destruct /3 width=6/
64 | #I #L #V #T #U #_ #j #H destruct
65 | #L #V #T #U #_ #j #H destruct
66 | #L #W #T #U #_ #j #H destruct
70 (* Basic_1: was sty0_gen_lref *)
71 lemma sta_inv_lref1: ∀h,L,U,i. ⦃h, L⦄ ⊢ #i • U →
72 (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V • W &
75 (∃∃K,W,V. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W • V &
80 fact sta_inv_bind1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀J,X,Y. T = ⓑ{J}Y.X →
81 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z.
82 #h #L #T #U * -L -T -U
83 [ #L #k #J #X #Y #H destruct
84 | #L #K #V #W #U #i #_ #_ #_ #J #X #Y #H destruct
85 | #L #K #W #V #U #i #_ #_ #_ #J #X #Y #H destruct
86 | #I #L #V #T #U #HTU #J #X #Y #H destruct /2 width=3/
87 | #L #V #T #U #_ #J #X #Y #H destruct
88 | #L #W #T #U #_ #J #X #Y #H destruct
92 (* Basic_1: was: sty0_gen_bind *)
93 lemma sta_inv_bind1: ∀h,J,L,Y,X,U. ⦃h, L⦄ ⊢ ⓑ{J}Y.X • U →
94 ∃∃Z. ⦃h, L.ⓑ{J}Y⦄ ⊢ X • Z & U = ⓑ{J}Y.Z.
97 fact sta_inv_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓐY.X →
98 ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z.
99 #h #L #T #U * -L -T -U
100 [ #L #k #X #Y #H destruct
101 | #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
102 | #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
103 | #I #L #V #T #U #_ #X #Y #H destruct
104 | #L #V #T #U #HTU #X #Y #H destruct /2 width=3/
105 | #L #W #T #U #_ #X #Y #H destruct
109 (* Basic_1: was: sty0_gen_appl *)
110 lemma sta_inv_appl1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X • U →
111 ∃∃Z. ⦃h, L⦄ ⊢ X • Z & U = ⓐY.Z.
114 fact sta_inv_cast1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T • U → ∀X,Y. T = ⓝY.X →
116 #h #L #T #U * -L -T -U
117 [ #L #k #X #Y #H destruct
118 | #L #K #V #W #U #i #_ #_ #_ #X #Y #H destruct
119 | #L #K #W #V #U #i #_ #_ #_ #X #Y #H destruct
120 | #I #L #V #T #U #_ #X #Y #H destruct
121 | #L #V #T #U #_ #X #Y #H destruct
122 | #L #W #T #U #HTU #X #Y #H destruct //
126 (* Basic_1: was: sty0_gen_cast *)
127 lemma sta_inv_cast1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓝY.X • U → ⦃h, L⦄ ⊢ X • U.