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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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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 T U l = (ssta h g l L T U).
37 definition ssta_step: ∀h. sd h → lenv → relation term ≝ λh,g,L,T,U.
38 ∃l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄.
40 (* Basic inversion lemmas ************************************************)
42 fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
43 deg h g k0 l ∧ U = ⋆(next h k0).
44 #h #g #L #T #U #l * -L -T -U -l
45 [ #L #k #l #Hkl #k0 #H destruct /2 width=1/
46 | #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
47 | #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
48 | #a #I #L #V #T #U #l #_ #k0 #H destruct
49 | #L #V #T #U #l #_ #k0 #H destruct
50 | #L #W #T #U #l #_ #k0 #H destruct
53 (* Basic_1: was just: sty0_gen_sort *)
54 lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g] ⦃l, U⦄ →
55 deg h g k l ∧ U = ⋆(next h k).
58 fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀j. T = #j →
59 (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
62 (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
63 ⇧[0, j + 1] W ≡ U & l = l0 + 1
65 #h #g #L #T #U #l * -L -T -U -l
66 [ #L #k #l #_ #j #H destruct
67 | #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6/
68 | #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8/
69 | #a #I #L #V #T #U #l #_ #j #H destruct
70 | #L #V #T #U #l #_ #j #H destruct
71 | #L #W #T #U #l #_ #j #H destruct
75 (* Basic_1: was just: sty0_gen_lref *)
76 lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g] ⦃l, U⦄ →
77 (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
80 (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
81 ⇧[0, i + 1] W ≡ U & l = l0 + 1
85 fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
86 #h #g #L #T #U #l * -L -T -U -l
87 [ #L #k #l #_ #p0 #H destruct
88 | #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
89 | #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
90 | #a #I #L #V #T #U #l #_ #p0 #H destruct
91 | #L #V #T #U #l #_ #p0 #H destruct
92 | #L #W #T #U #l #_ #p0 #H destruct
95 lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g] ⦃l, U⦄ → ⊥.
98 fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
99 ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
100 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
101 #h #g #L #T #U #l * -L -T -U -l
102 [ #L #k #l #_ #a #I #X #Y #H destruct
103 | #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
104 | #L #K #W #V #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
105 | #b #J #L #V #T #U #l #HTU #a #I #X #Y #H destruct /2 width=3/
106 | #L #V #T #U #l #_ #a #I #X #Y #H destruct
107 | #L #W #T #U #l #_ #a #I #X #Y #H destruct
111 (* Basic_1: was just: sty0_gen_bind *)
112 lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g] ⦃l, U⦄ →
113 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
116 fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
117 ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
118 #h #g #L #T #U #l * -L -T -U -l
119 [ #L #k #l #_ #X #Y #H destruct
120 | #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
121 | #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
122 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
123 | #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3/
124 | #L #W #T #U #l #_ #X #Y #H destruct
128 (* Basic_1: was just: sty0_gen_appl *)
129 lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g] ⦃l, U⦄ →
130 ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
133 fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
134 ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
135 #h #g #L #T #U #l * -L -T -U -l
136 [ #L #k #l #_ #X #Y #H destruct
137 | #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
138 | #L #K #W #V #U #l #i #_ #_ #_ #X #Y #H destruct
139 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
140 | #L #V #T #U #l #_ #X #Y #H destruct
141 | #L #W #T #U #l #HTU #X #Y #H destruct //
145 (* Basic_1: was just: sty0_gen_cast *)
146 lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g] ⦃l, U⦄ →
147 ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
150 (* Advanced inversion lemmas ************************************************)
152 lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
153 #h #g #L #T #U #l #H elim H -L -T -U -l
155 elim (frsupp_inv_atom1_frsups … H)
156 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
157 elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
158 elim (lift_inv_lref2_be … H ? ?) -H //
159 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
160 elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
161 elim (lift_inv_lref2_be … H ? ?) -H //
162 | #a #I #L #V #T #U #l #_ #IHTU #H
163 elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
164 lapply (frsups_fwd_fw … H) -H normalize
165 <associative_plus <associative_plus #H
166 elim (le_plus_xySz_x_false … H)
167 | #L #V #T #U #l #_ #IHTU #H
168 elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
169 lapply (frsups_fwd_fw … H) -H normalize
170 <associative_plus <associative_plus #H
171 elim (le_plus_xySz_x_false … H)
176 fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → T = U → ⊥.
177 #h #g #L #T #U #l #H elim H -L -T -U -l
179 lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
180 <e1 -e1 #H elim (lt_refl_false … H)
181 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
182 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
183 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
184 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
185 | #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
186 | #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
187 | #L #W #T #U #l #HTU #_ #H destruct
188 elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
192 lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, T⦄ → ⊥.
193 /2 width=8 by ssta_inv_refl_aux/ qed-.
195 lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
196 #h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
197 [ * #_ #H destruct /2 width=6 by ssta_inv_refl/
198 | /2 width=8 by ssta_inv_frsupp/