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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/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,V,W,T,U,l. ssta h g (l - 1) L V W → ssta h g l L T U →
31 ssta h g l L (ⓝV. T) (ⓝW. 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 #V #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 #V #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_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U →
83 ∀a,I,X,Y. T = ⓑ{a,I}Y.X →
84 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
85 #h #g #L #T #U #l * -L -T -U -l
86 [ #L #k #l #_ #a #I #X #Y #H destruct
87 | #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
88 | #L #K #W #V #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
89 | #b #J #L #V #T #U #l #HTU #a #I #X #Y #H destruct /2 width=3/
90 | #L #V #T #U #l #_ #a #I #X #Y #H destruct
91 | #L #V #W #T #U #l #_ #_ #a #I #X #Y #H destruct
95 (* Basic_1: was just: sty0_gen_bind *)
96 lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g, l] U →
97 ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
100 fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀X,Y. T = ⓐY.X →
101 ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
102 #h #g #L #T #U #l * -L -T -U -l
103 [ #L #k #l #_ #X #Y #H destruct
104 | #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
105 | #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
106 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
107 | #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3/
108 | #L #V #W #T #U #l #_ #_ #X #Y #H destruct
112 (* Basic_1: was just: sty0_gen_appl *)
113 lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g, l] U →
114 ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
117 fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀X,Y. T = ⓝY.X →
118 ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y •[g, l-1] Z1 & ⦃h, L⦄ ⊢ X •[g, l] Z2 &
120 #h #g #L #T #U #l * -L -T -U -l
121 [ #L #k #l #_ #X #Y #H destruct
122 | #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
123 | #L #K #W #V #U #l #i #_ #_ #_ #X #Y #H destruct
124 | #a #I #L #V #T #U #l #_ #X #Y #H destruct
125 | #L #V #T #U #l #_ #X #Y #H destruct
126 | #L #V #W #T #U #l #HVW #HTU #X #Y #H destruct /2 width=5/
130 (* Basic_1: was just: sty0_gen_cast *)
131 lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g, l] U →
132 ∃∃Z1,Z2. ⦃h, L⦄ ⊢ Y •[g, l-1] Z1 & ⦃h, L⦄ ⊢ X •[g, l] Z2 &
136 (* Advanced inversion lemmas ************************************************)
138 fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → T = U → ⊥.
139 #h #g #L #T #U #l #H elim H -L -T -U -l
141 lapply (next_lt h k) destruct -H -e0 (**) (* these premises are not erased *)
142 <e1 -e1 #H elim (lt_refl_false … H)
143 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
144 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
145 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
146 elim (lift_inv_lref2_be … HWU ? ?) -HWU //
147 | #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
148 | #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
149 | #L #V #W #T #U #l #_ #_ #_ #IHTU #H destruct /2 width=1/
153 lemma ssta_inv_refl: ∀h,g,L,T,l. ⦃h, L⦄ ⊢ T •[g, l] T → ⊥.