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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 "ground_2/relocation/rtmap_sor.ma".
16 include "basic_2/notation/relations/freestar_3.ma".
17 include "basic_2/syntax/lenv.ma".
19 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
21 inductive frees: relation3 lenv term rtmap ≝
22 | frees_atom: ∀f,I. 𝐈⦃f⦄ → frees (⋆) (⓪{I}) f
23 | frees_sort: ∀f,I,L,V,s. frees L (⋆s) f →
24 frees (L.ⓑ{I}V) (⋆s) (↑f)
25 | frees_zero: ∀f,I,L,V. frees L V f →
26 frees (L.ⓑ{I}V) (#0) (⫯f)
27 | frees_lref: ∀f,I,L,V,i. frees L (#i) f →
28 frees (L.ⓑ{I}V) (#⫯i) (↑f)
29 | frees_gref: ∀f,I,L,V,l. frees L (§l) f →
30 frees (L.ⓑ{I}V) (§l) (↑f)
31 | frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
32 f1 ⋓ ⫱f2 ≡ f → frees L (ⓑ{p,I}V.T) f
33 | frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
34 f1 ⋓ f2 ≡ f → frees L (ⓕ{I}V.T) f
38 "context-sensitive free variables (term)"
39 'FreeStar L T t = (frees L T t).
41 (* Basic inversion lemmas ***************************************************)
43 fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀J. L = ⋆ → X = ⓪{J} → 𝐈⦃f⦄.
44 #f #L #X #H elim H -f -L -X /3 width=3 by isid_push/
45 [5,6: #f1 #f2 #f [ #p ] #I #L #V #T #_ #_ #_ #_ #_ #J #_ #H destruct
46 |*: #f #I #L #V [1,3,4: #x ] #_ #_ #J #H destruct
50 lemma frees_inv_atom: ∀f,I. ⋆ ⊢ 𝐅*⦃⓪{I}⦄ ≡ f → 𝐈⦃f⦄.
51 /2 width=6 by frees_inv_atom_aux/ qed-.
53 fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
54 #L #X #f #H elim H -f -L -X /3 width=3 by isid_push/
55 [ #f #_ #L #V #_ #_ #x #H destruct
56 | #f #_ #L #_ #i #_ #_ #x #H destruct
57 | #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
58 | #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
62 lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≡ f → 𝐈⦃f⦄.
63 /2 width=5 by frees_inv_sort_aux/ qed-.
65 fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀x. X = §x → 𝐈⦃f⦄.
66 #f #L #X #H elim H -f -L -X /3 width=3 by isid_push/
67 [ #f #_ #L #V #_ #_ #x #H destruct
68 | #f #_ #L #_ #i #_ #_ #x #H destruct
69 | #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
70 | #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
74 lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≡ f → 𝐈⦃f⦄.
75 /2 width=5 by frees_inv_gref_aux/ qed-.
77 fact frees_inv_zero_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → X = #0 →
79 ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g & L = K.ⓑ{I}V & f = ⫯g.
81 [ /3 width=1 by or_introl, conj/
82 | #f #I #L #V #s #_ #H destruct
83 | /3 width=7 by ex3_4_intro, or_intror/
84 | #f #I #L #V #i #_ #H destruct
85 | #f #I #L #V #l #_ #H destruct
86 | #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #H destruct
87 | #f1 #f2 #f #I #L #V #T #_ #_ #_ #H destruct
91 lemma frees_inv_zero: ∀f,L. L ⊢ 𝐅*⦃#0⦄ ≡ f →
93 ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g & L = K.ⓑ{I}V & f = ⫯g.
94 /2 width=3 by frees_inv_zero_aux/ qed-.
96 fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀j. X = #(⫯j) →
98 ∃∃g,I,K,V. K ⊢ 𝐅*⦃#j⦄ ≡ g & L = K.ⓑ{I}V & f = ↑g.
100 [ /3 width=1 by or_introl, conj/
101 | #f #I #L #V #s #_ #j #H destruct
102 | #f #I #L #V #_ #j #H destruct
103 | #f #I #L #V #i #Hf #j #H destruct /3 width=7 by ex3_4_intro, or_intror/
104 | #f #I #L #V #l #_ #j #H destruct
105 | #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #j #H destruct
106 | #f1 #f2 #f #I #L #V #T #_ #_ #_ #j #H destruct
110 lemma frees_inv_lref: ∀f,L,i. L ⊢ 𝐅*⦃#(⫯i)⦄ ≡ f →
112 ∃∃g,I,K,V. K ⊢ 𝐅*⦃#i⦄ ≡ g & L = K.ⓑ{I}V & f = ↑g.
113 /2 width=3 by frees_inv_lref_aux/ qed-.
115 fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
116 ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ ⫱f2 ≡ f.
118 [ #f #I #_ #q #J #W #U #H destruct
119 | #f #I #L #V #s #_ #q #J #W #U #H destruct
120 | #f #I #L #V #_ #q #J #W #U #H destruct
121 | #f #I #L #V #i #_ #q #J #W #U #H destruct
122 | #f #I #L #V #l #_ #q #J #W #U #H destruct
123 | #f1 #f2 #f #p #I #L #V #T #HV #HT #Hf #q #J #W #U #H destruct /2 width=5 by ex3_2_intro/
124 | #f1 #f2 #f #I #L #V #T #_ #_ #_ #q #J #W #U #H destruct
128 lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≡ f →
129 ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ ⫱f2 ≡ f.
130 /2 width=4 by frees_inv_bind_aux/ qed-.
132 fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀I,V,T. X = ⓕ{I}V.T →
133 ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ f2 ≡ f.
135 [ #f #I #_ #J #W #U #H destruct
136 | #f #I #L #V #s #_ #J #W #U #H destruct
137 | #f #I #L #V #_ #J #W #U #H destruct
138 | #f #I #L #V #i #_ #J #W #U #H destruct
139 | #f #I #L #V #l #_ #J #W #U #H destruct
140 | #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #J #W #U #H destruct
141 | #f1 #f2 #f #I #L #V #T #HV #HT #Hf #J #W #U #H destruct /2 width=5 by ex3_2_intro/
145 lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≡ f →
146 ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ f2 ≡ f.
147 /2 width=4 by frees_inv_flat_aux/ qed-.
149 (* Advanced inversion lemmas ***********************************************)
151 lemma frees_inv_zero_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≡ f →
152 ∃∃g. K ⊢ 𝐅*⦃V⦄ ≡ g & f = ⫯g.
153 #f #I #K #V #H elim (frees_inv_zero … H) -H *
155 | #g #Z #Y #X #Hg #H1 #H2 destruct /3 width=3 by ex2_intro/
159 lemma frees_inv_lref_pair: ∀f,I,K,V,i. K.ⓑ{I}V ⊢ 𝐅*⦃#(⫯i)⦄ ≡ f →
160 ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≡ g & f = ↑g.
161 #f #I #K #V #i #H elim (frees_inv_lref … H) -H *
163 | #g #Z #Y #X #Hg #H1 #H2 destruct /3 width=3 by ex2_intro/
167 (* Basic forward lemmas ****************************************************)
169 lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≡ f → 𝐅⦃f⦄.
170 #f #L #T #H elim H -f -L -T
171 /3 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_push, isfin_next/
174 (* Basic properties ********************************************************)
176 lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≡ f).
177 #L #T #f1 #H elim H -f1 -L -T
178 [ /3 width=3 by frees_atom, isid_eq_repl_back/
179 | #f1 #I #L #V #s #_ #IH #f2 #Hf12
180 elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_sort/
181 | #f1 #I #L #V #_ #IH #f2 #Hf12
182 elim (eq_inv_nx … Hf12) -Hf12 /3 width=3 by frees_zero/
183 | #f1 #I #L #V #i #_ #IH #f2 #Hf12
184 elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_lref/
185 | #f1 #I #L #V #l #_ #IH #f2 #Hf12
186 elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_gref/
187 | /3 width=7 by frees_bind, sor_eq_repl_back3/
188 | /3 width=7 by frees_flat, sor_eq_repl_back3/
192 lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≡ f).
193 #L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
196 lemma frees_sort_gen: ∀f,L,s. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃⋆s⦄ ≡ f.
198 /4 width=3 by frees_eq_repl_back, frees_sort, frees_atom, eq_push_inv_isid/
201 lemma frees_gref_gen: ∀f,L,p. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃§p⦄ ≡ f.
203 /4 width=3 by frees_eq_repl_back, frees_gref, frees_atom, eq_push_inv_isid/
206 (* Basic_2A1: removed theorems 30:
207 frees_eq frees_be frees_inv
208 frees_inv_sort frees_inv_gref frees_inv_lref frees_inv_lref_free
209 frees_inv_lref_skip frees_inv_lref_ge frees_inv_lref_lt
210 frees_inv_bind frees_inv_flat frees_inv_bind_O
211 frees_lref_eq frees_lref_be frees_weak
212 frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
213 frees_lift_ge frees_inv_lift_be frees_inv_lift_ge
214 lreq_frees_trans frees_lreq_conf
215 llor_atom llor_skip llor_total
216 llor_tail_frees llor_tail_cofrees