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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 include "basic_2/relocation/ldrop.ma".
16 include "basic_2/static/sd.ma".
18 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
20 inductive ssta (h:sh) (g:sd h): nat → lenv → relation term ≝
21 | ssta_sort: ∀L,k,l. deg h g k l → ssta h g l L (⋆k) (⋆(next h k))
22 | ssta_ldef: ∀L,K,V,W,U,i,l. ⇩[0, i] L ≡ K. ⓓV → ssta h g l K V W →
23 ⇧[0, i + 1] W ≡ U → ssta h g l L (#i) U
24 | ssta_ldec: ∀L,K,W,V,U,i,l. ⇩[0, i] L ≡ K. ⓛW → ssta h g l K W V →
25 ⇧[0, i + 1] W ≡ U → ssta h g (l+1) L (#i) U
26 | ssta_bind: ∀a,I,L,V,T,U,l. ssta h g l (L. ⓑ{I} V) T U →
27 ssta h g l L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
28 | ssta_appl: ∀L,V,T,U,l. ssta h g l L T U →
29 ssta h g l L (ⓐV.T) (ⓐV.U)
30 | ssta_cast: ∀L,W,T,U,l. ssta h g l L T U → ssta h g l L (ⓝW. T) U
33 interpretation "stratified static type assignment (term)"
34 'StaticType h g L T U l = (ssta h g l L T U).
36 definition ssta_step: ∀h. sd h → lenv → relation term ≝ λh,g,L,T,U.
37 ∃l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄.
39 (* Basic inversion lemmas ************************************************)
41 fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
42 deg h g k0 l ∧ U = ⋆(next h k0).
43 #h #g #L #T #U #l * -L -T -U -l
44 [ #L #k #l #Hkl #k0 #H destruct /2 width=1/
45 | #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
46 | #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
47 | #a #I #L #V #T #U #l #_ #k0 #H destruct
48 | #L #V #T #U #l #_ #k0 #H destruct
49 | #L #W #T #U #l #_ #k0 #H destruct
52 (* Basic_1: was just: sty0_gen_sort *)
53 lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g] ⦃l, U⦄ →
54 deg h g k l ∧ U = ⋆(next h k).
57 fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀j. T = #j →
58 (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
61 (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
62 ⇧[0, j + 1] W ≡ U & l = l0 + 1
64 #h #g #L #T #U #l * -L -T -U -l
65 [ #L #k #l #_ #j #H destruct
66 | #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6/
67 | #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8/
68 | #a #I #L #V #T #U #l #_ #j #H destruct
69 | #L #V #T #U #l #_ #j #H destruct
70 | #L #W #T #U #l #_ #j #H destruct
74 (* Basic_1: was just: sty0_gen_lref *)
75 lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g] ⦃l, U⦄ →
76 (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
79 (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
80 ⇧[0, i + 1] W ≡ U & l = l0 + 1
84 fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
85 #h #g #L #T #U #l * -L -T -U -l
86 [ #L #k #l #_ #p0 #H destruct
87 | #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
88 | #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
89 | #a #I #L #V #T #U #l #_ #p0 #H destruct
90 | #L #V #T #U #l #_ #p0 #H destruct
91 | #L #W #T #U #l #_ #p0 #H destruct
94 lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g] ⦃l, U⦄ → ⊥.
97 fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
98 ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
99 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
100 #h #g #L #T #U #l * -L -T -U -l
101 [ #L #k #l #_ #a #I #X #Y #H destruct
102 | #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
103 | #L #K #W #V #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
104 | #b #J #L #V #T #U #l #HTU #a #I #X #Y #H destruct /2 width=3/
105 | #L #V #T #U #l #_ #a #I #X #Y #H destruct
106 | #L #W #T #U #l #_ #a #I #X #Y #H destruct
110 (* Basic_1: was just: sty0_gen_bind *)
111 lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g] ⦃l, U⦄ →
112 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
115 fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
116 ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
117 #h #g #L #T #U #l * -L -T -U -l
118 [ #L #k #l #_ #X #Y #H destruct
119 | #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
120 | #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
121 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
122 | #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3/
123 | #L #W #T #U #l #_ #X #Y #H destruct
127 (* Basic_1: was just: sty0_gen_appl *)
128 lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g] ⦃l, U⦄ →
129 ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
132 fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
133 ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
134 #h #g #L #T #U #l * -L -T -U -l
135 [ #L #k #l #_ #X #Y #H destruct
136 | #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
137 | #L #K #W #V #U #l #i #_ #_ #_ #X #Y #H destruct
138 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
139 | #L #V #T #U #l #_ #X #Y #H destruct
140 | #L #W #T #U #l #HTU #X #Y #H destruct //
144 (* Basic_1: was just: sty0_gen_cast *)
145 lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g] ⦃l, U⦄ →
146 ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.