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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/substitution/ldrop.ma".
16 include "basic_2/unfold/frsups.ma".
17 include "basic_2/static/sd.ma".
19 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
21 inductive ssta (h:sh) (g:sd h): nat → lenv → relation term ≝
22 | ssta_sort: ∀L,k,l. deg h g k l → ssta h g l L (⋆k) (⋆(next h k))
23 | ssta_ldef: ∀L,K,V,W,U,i,l. ⇩[0, i] L ≡ K. ⓓV → ssta h g l K V W →
24 ⇧[0, i + 1] W ≡ U → ssta h g l L (#i) U
25 | ssta_ldec: ∀L,K,W,V,U,i,l. ⇩[0, i] L ≡ K. ⓛW → ssta h g l K W V →
26 ⇧[0, i + 1] W ≡ U → ssta h g (l+1) L (#i) U
27 | ssta_bind: ∀a,I,L,V,T,U,l. ssta h g l (L. ⓑ{I} V) T U →
28 ssta h g l L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
29 | ssta_appl: ∀L,V,T,U,l. ssta h g l L T U →
30 ssta h g l L (ⓐV.T) (ⓐV.U)
31 | ssta_cast: ∀L,W,T,U,l. ssta h g l L T U → ssta h g l L (ⓝW. T) U
34 interpretation "stratified static type assignment (term)"
35 'StaticType h g l L T U = (ssta h g l L T U).
37 (* Basic inversion lemmas ************************************************)
39 fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀k0. T = ⋆k0 →
40 deg h g k0 l ∧ U = ⋆(next h k0).
41 #h #g #L #T #U #l * -L -T -U -l
42 [ #L #k #l #Hkl #k0 #H destruct /2 width=1/
43 | #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
44 | #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
45 | #a #I #L #V #T #U #l #_ #k0 #H destruct
46 | #L #V #T #U #l #_ #k0 #H destruct
47 | #L #W #T #U #l #_ #k0 #H destruct
50 (* Basic_1: was just: sty0_gen_sort *)
51 lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g, l] U →
52 deg h g k l ∧ U = ⋆(next h k).
55 fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀j. T = #j →
56 (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g, l] W &
59 (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g, l0] V &
60 ⇧[0, j + 1] W ≡ U & l = l0 + 1
62 #h #g #L #T #U #l * -L -T -U -l
63 [ #L #k #l #_ #j #H destruct
64 | #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6/
65 | #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8/
66 | #a #I #L #V #T #U #l #_ #j #H destruct
67 | #L #V #T #U #l #_ #j #H destruct
68 | #L #W #T #U #l #_ #j #H destruct
72 (* Basic_1: was just: sty0_gen_lref *)
73 lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g, l] U →
74 (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g, l] W &
77 (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g, l0] V &
78 ⇧[0, i + 1] W ≡ U & l = l0 + 1
82 fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀p0. T = §p0 → ⊥.
83 #h #g #L #T #U #l * -L -T -U -l
84 [ #L #k #l #_ #p0 #H destruct
85 | #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
86 | #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
87 | #a #I #L #V #T #U #l #_ #p0 #H destruct
88 | #L #V #T #U #l #_ #p0 #H destruct
89 | #L #W #T #U #l #_ #p0 #H destruct
92 lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g, l] U → ⊥.
95 fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U →
96 ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
97 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
98 #h #g #L #T #U #l * -L -T -U -l
99 [ #L #k #l #_ #a #I #X #Y #H destruct
100 | #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
101 | #L #K #W #V #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
102 | #b #J #L #V #T #U #l #HTU #a #I #X #Y #H destruct /2 width=3/
103 | #L #V #T #U #l #_ #a #I #X #Y #H destruct
104 | #L #W #T #U #l #_ #a #I #X #Y #H destruct
108 (* Basic_1: was just: sty0_gen_bind *)
109 lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g, l] U →
110 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
113 fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀X,Y. T = ⓐY.X →
114 ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
115 #h #g #L #T #U #l * -L -T -U -l
116 [ #L #k #l #_ #X #Y #H destruct
117 | #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
118 | #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
119 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
120 | #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3/
121 | #L #W #T #U #l #_ #X #Y #H destruct
125 (* Basic_1: was just: sty0_gen_appl *)
126 lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g, l] U →
127 ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
130 fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U →
131 ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g, l] U.
132 #h #g #L #T #U #l * -L -T -U -l
133 [ #L #k #l #_ #X #Y #H destruct
134 | #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
135 | #L #K #W #V #U #l #i #_ #_ #_ #X #Y #H destruct
136 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
137 | #L #V #T #U #l #_ #X #Y #H destruct
138 | #L #W #T #U #l #HTU #X #Y #H destruct //
142 (* Basic_1: was just: sty0_gen_cast *)
143 lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g, l] U →
144 ⦃h, L⦄ ⊢ X •[g, l] U.
147 (* Advanced inversion lemmas ************************************************)
149 lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
150 #h #g #L #T #U #l #H elim H -L -T -U -l
152 elim (frsupp_inv_atom1_frsups … H)
153 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
154 elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
155 elim (lift_inv_lref2_be … H ? ?) -H //
156 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
157 elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
158 elim (lift_inv_lref2_be … H ? ?) -H //
159 | #a #I #L #V #T #U #l #_ #IHTU #H
160 elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
161 lapply (frsups_fwd_fw … H) -H normalize
162 <associative_plus <associative_plus #H
163 elim (le_plus_xySz_x_false … H)
164 | #L #V #T #U #l #_ #IHTU #H
165 elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
166 lapply (frsups_fwd_fw … H) -H normalize
167 <associative_plus <associative_plus #H
168 elim (le_plus_xySz_x_false … H)
173 fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → T = U → ⊥.
174 #h #g #L #T #U #l #H elim H -L -T -U -l
176 lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
177 <e1 -e1 #H elim (lt_refl_false … H)
178 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
179 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
180 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
181 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
182 | #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
183 | #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
184 | #L #W #T #U #l #HTU #_ #H destruct
185 elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
189 lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g, l] T → ⊥.
190 /2 width=8 by ssta_inv_refl_aux/ qed-.
192 lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
193 #h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
194 [ * #_ #H destruct /2 width=6 by ssta_inv_refl/
195 | /2 width=8 by ssta_inv_frsupp/