include "ground_2/relocation/rtmap_sor.ma".
include "basic_2/notation/relations/freestar_3.ma".
-include "basic_2/grammar/lenv.ma".
+include "basic_2/syntax/lenv.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
inductive frees: relation3 lenv term rtmap ≝
-| frees_atom: ∀f,I. 𝐈⦃f⦄ → frees (⋆) (⓪{I}) f
-| frees_sort: ∀f,I,L,V,s. frees L (⋆s) f →
- frees (L.ⓑ{I}V) (⋆s) (↑f)
-| frees_zero: ∀f,I,L,V. frees L V f →
- frees (L.ⓑ{I}V) (#0) (⫯f)
-| frees_lref: ∀f,I,L,V,i. frees L (#i) f →
- frees (L.ⓑ{I}V) (#⫯i) (↑f)
-| frees_gref: ∀f,I,L,V,l. frees L (§l) f →
- frees (L.ⓑ{I}V) (§l) (↑f)
+| frees_sort: ∀f,L,s. 𝐈⦃f⦄ → frees L (⋆s) f
+| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (⫯*[i]↑f)
+| frees_pair: ∀f,I,L,V. frees L V f →
+ frees (L.ⓑ{I}V) (#0) (↑f)
+| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (↑f)
+| frees_lref: ∀f,I,L,i. frees L (#i) f →
+ frees (L.ⓘ{I}) (#↑i) (⫯f)
+| frees_gref: ∀f,L,l. 𝐈⦃f⦄ → frees L (§l) f
| frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
- f1 â\8b\93 ⫱f2 â\89¡ f → frees L (ⓑ{p,I}V.T) f
+ f1 â\8b\93 ⫱f2 â\89\98 f → frees L (ⓑ{p,I}V.T) f
| frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
- f1 â\8b\93 f2 â\89¡ f → frees L (ⓕ{I}V.T) f
+ f1 â\8b\93 f2 â\89\98 f → frees L (ⓕ{I}V.T) f
.
interpretation
"context-sensitive free variables (term)"
- 'FreeStar L T t = (frees L T t).
+ 'FreeStar L T f = (frees L T f).
(* Basic inversion lemmas ***************************************************)
-fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀J. L = ⋆ → X = ⓪{J} → 𝐈⦃f⦄.
-#f #L #X #H elim H -f -L -X /3 width=3 by isid_push/
-[5,6: #f1 #f2 #f [ #p ] #I #L #V #T #_ #_ #_ #_ #_ #J #_ #H destruct
-|*: #f #I #L #V [1,3,4: #x ] #_ #_ #J #H destruct
+fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+#L #X #f #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
]
qed-.
-lemma frees_inv_atom: ∀f,I. ⋆ ⊢ 𝐅*⦃⓪{I}⦄ ≡ f → 𝐈⦃f⦄.
-/2 width=6 by frees_inv_atom_aux/ qed-.
+lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_sort_aux/ qed-.
-fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
-#L #X #f #H elim H -f -L -X /3 width=3 by isid_push/
-[ #f #_ #L #V #_ #_ #x #H destruct
-| #f #_ #L #_ #i #_ #_ #x #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
+ ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+#f #L #X #H elim H -f -L -X
+[ #f #L #s #_ #j #_ #H destruct
+| #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #V #_ #_ #j #H destruct
+| #f #I #L #_ #j #H destruct
+| #f #I #L #i #_ #_ #j #H destruct
+| #f #L #l #_ #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
]
qed-.
-lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≡ f → 𝐈⦃f⦄.
-/2 width=5 by frees_inv_sort_aux/ qed-.
+lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+/2 width=5 by frees_inv_atom_aux/ qed-.
-fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀x. X = §x → 𝐈⦃f⦄.
-#f #L #X #H elim H -f -L -X /3 width=3 by isid_push/
-[ #f #_ #L #V #_ #_ #x #H destruct
-| #f #_ #L #_ #i #_ #_ #x #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
+ ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #X #_ #H destruct
+| #f #i #_ #Z #Y #X #H destruct
+| #f #I #L #V #Hf #Z #Y #X #H #_ destruct /2 width=3 by ex2_intro/
+| #f #I #L #_ #Z #Y #X #H destruct
+| #f #I #L #i #_ #Z #Y #X #_ #H destruct
+| #f #L #l #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
]
qed-.
-lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≡ f → 𝐈⦃f⦄.
-/2 width=5 by frees_inv_gref_aux/ qed-.
+lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+/2 width=6 by frees_inv_pair_aux/ qed-.
-fact frees_inv_zero_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → X = #0 →
- (L = ⋆ ∧ 𝐈⦃f⦄) ∨
- ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g & L = K.ⓑ{I}V & f = ⫯g.
+fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
+ ∃∃g. 𝐈⦃g⦄ & f = ↑g.
#f #L #X * -f -L -X
-[ /3 width=1 by or_introl, conj/
-| #f #I #L #V #s #_ #H destruct
-| /3 width=7 by ex3_4_intro, or_intror/
-| #f #I #L #V #i #_ #H destruct
-| #f #I #L #V #l #_ #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #H destruct
+[ #f #L #s #_ #Z #Y #_ #H destruct
+| #f #i #_ #Z #Y #H destruct
+| #f #I #L #V #_ #Z #Y #H destruct
+| #f #I #L #Hf #Z #Y #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #i #_ #Z #Y #_ #H destruct
+| #f #L #l #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
]
qed-.
-lemma frees_inv_zero: ∀f,L. L ⊢ 𝐅*⦃#0⦄ ≡ f →
- (L = ⋆ ∧ 𝐈⦃f⦄) ∨
- ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≡ g & L = K.ⓑ{I}V & f = ⫯g.
-/2 width=3 by frees_inv_zero_aux/ qed-.
+lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+/2 width=7 by frees_inv_unit_aux/ qed-.
-fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀j. X = #(⫯j) →
- (L = ⋆ ∧ 𝐈⦃f⦄) ∨
- ∃∃g,I,K,V. K ⊢ 𝐅*⦃#j⦄ ≡ g & L = K.ⓑ{I}V & f = ↑g.
+fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
+ ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
#f #L #X * -f -L -X
-[ /3 width=1 by or_introl, conj/
-| #f #I #L #V #s #_ #j #H destruct
-| #f #I #L #V #_ #j #H destruct
-| #f #I #L #V #i #Hf #j #H destruct /3 width=7 by ex3_4_intro, or_intror/
-| #f #I #L #V #l #_ #j #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #j #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #j #H destruct
+[ #f #L #s #_ #Z #Y #j #_ #H destruct
+| #f #i #_ #Z #Y #j #H destruct
+| #f #I #L #V #_ #Z #Y #j #_ #H destruct
+| #f #I #L #_ #Z #Y #j #_ #H destruct
+| #f #I #L #i #Hf #Z #Y #j #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #f #L #l #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
]
qed-.
-lemma frees_inv_lref: ∀f,L,i. L ⊢ 𝐅*⦃#(⫯i)⦄ ≡ f →
- (L = ⋆ ∧ 𝐈⦃f⦄) ∨
- ∃∃g,I,K,V. K ⊢ 𝐅*⦃#i⦄ ≡ g & L = K.ⓑ{I}V & f = ↑g.
-/2 width=3 by frees_inv_lref_aux/ qed-.
+lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
+ ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+/2 width=6 by frees_inv_lref_aux/ qed-.
-fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≡ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≡ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≡ f2 & f1 ⋓ ⫱f2 ≡ f.
+fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
+#f #L #X #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+]
+qed-.
+
+lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_gref_aux/ qed-.
+
+fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
#f #L #X * -f -L -X
-[ #f #I #_ #q #J #W #U #H destruct
-| #f #I #L #V #s #_ #q #J #W #U #H destruct
+[ #f #L #s #_ #q #J #W #U #H destruct
+| #f #i #_ #q #J #W #U #H destruct
| #f #I #L #V #_ #q #J #W #U #H destruct
-| #f #I #L #V #i #_ #q #J #W #U #H destruct
-| #f #I #L #V #l #_ #q #J #W #U #H destruct
+| #f #I #L #_ #q #J #W #U #H destruct
+| #f #I #L #i #_ #q #J #W #U #H destruct
+| #f #L #l #_ #q #J #W #U #H destruct
| #f1 #f2 #f #p #I #L #V #T #HV #HT #Hf #q #J #W #U #H destruct /2 width=5 by ex3_2_intro/
| #f1 #f2 #f #I #L #V #T #_ #_ #_ #q #J #W #U #H destruct
]
qed-.
-lemma frees_inv_bind: â\88\80f,p,I,L,V,T. L â\8a¢ ð\9d\90\85*â¦\83â\93\91{p,I}V.Tâ¦\84 â\89¡ f →
- â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89¡ f1 & L.â\93\91{I}V â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f2 & f1 â\8b\93 ⫱f2 â\89¡ f.
+lemma frees_inv_bind: â\88\80f,p,I,L,V,T. L â\8a¢ ð\9d\90\85*â¦\83â\93\91{p,I}V.Tâ¦\84 â\89\98 f →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89\98 f1 & L.â\93\91{I}V â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f2 & f1 â\8b\93 ⫱f2 â\89\98 f.
/2 width=4 by frees_inv_bind_aux/ qed-.
-fact frees_inv_flat_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85*â¦\83Xâ¦\84 â\89¡ f → ∀I,V,T. X = ⓕ{I}V.T →
- â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89¡ f1 & L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f2 & f1 â\8b\93 f2 â\89¡ f.
+fact frees_inv_flat_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85*â¦\83Xâ¦\84 â\89\98 f → ∀I,V,T. X = ⓕ{I}V.T →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89\98 f1 & L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f2 & f1 â\8b\93 f2 â\89\98 f.
#f #L #X * -f -L -X
-[ #f #I #_ #J #W #U #H destruct
-| #f #I #L #V #s #_ #J #W #U #H destruct
+[ #f #L #s #_ #J #W #U #H destruct
+| #f #i #_ #J #W #U #H destruct
| #f #I #L #V #_ #J #W #U #H destruct
-| #f #I #L #V #i #_ #J #W #U #H destruct
-| #f #I #L #V #l #_ #J #W #U #H destruct
+| #f #I #L #_ #J #W #U #H destruct
+| #f #I #L #i #_ #J #W #U #H destruct
+| #f #L #l #_ #J #W #U #H destruct
| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #J #W #U #H destruct
| #f1 #f2 #f #I #L #V #T #HV #HT #Hf #J #W #U #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma frees_inv_flat: â\88\80f,I,L,V,T. L â\8a¢ ð\9d\90\85*â¦\83â\93\95{I}V.Tâ¦\84 â\89¡ f →
- â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89¡ f1 & L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f2 & f1 â\8b\93 f2 â\89¡ f.
+lemma frees_inv_flat: â\88\80f,I,L,V,T. L â\8a¢ ð\9d\90\85*â¦\83â\93\95{I}V.Tâ¦\84 â\89\98 f →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85*â¦\83Vâ¦\84 â\89\98 f1 & L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f2 & f1 â\8b\93 f2 â\89\98 f.
/2 width=4 by frees_inv_flat_aux/ qed-.
-(* Basic forward lemmas ****************************************************)
-
-lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≡ f → 𝐅⦃f⦄.
-#f #L #T #H elim H -f -L -T
-/3 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_push, isfin_next/
-qed-.
-
(* Basic properties ********************************************************)
-lemma frees_eq_repl_back: â\88\80L,T. eq_repl_back â\80¦ (λf. L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f).
+lemma frees_eq_repl_back: â\88\80L,T. eq_repl_back â\80¦ (λf. L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f).
#L #T #f1 #H elim H -f1 -L -T
-[ /3 width=3 by frees_atom, isid_eq_repl_back/
-| #f1 #I #L #V #s #_ #IH #f2 #Hf12
- elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_sort/
-| #f1 #I #L #V #_ #IH #f2 #Hf12
- elim (eq_inv_nx … Hf12) -Hf12 /3 width=3 by frees_zero/
-| #f1 #I #L #V #i #_ #IH #f2 #Hf12
- elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_lref/
-| #f1 #I #L #V #l #_ #IH #f2 #Hf12
- elim (eq_inv_px … Hf12) -Hf12 /3 width=3 by frees_gref/
+[ /3 width=3 by frees_sort, isid_eq_repl_back/
+| #f1 #i #Hf1 #g2 #H
+ elim (eq_inv_pushs_sn … H) -H #g #Hg #H destruct
+ elim (eq_inv_nx … Hg) -Hg
+ /3 width=3 by frees_atom, isid_eq_repl_back/
+| #f1 #I #L #V #_ #IH #g2 #H
+ elim (eq_inv_nx … H) -H
+ /3 width=3 by frees_pair/
+| #f1 #I #L #Hf1 #g2 #H
+ elim (eq_inv_nx … H) -H
+ /3 width=3 by frees_unit, isid_eq_repl_back/
+| #f1 #I #L #i #_ #IH #g2 #H
+ elim (eq_inv_px … H) -H /3 width=3 by frees_lref/
+| /3 width=3 by frees_gref, isid_eq_repl_back/
| /3 width=7 by frees_bind, sor_eq_repl_back3/
| /3 width=7 by frees_flat, sor_eq_repl_back3/
]
qed-.
-lemma frees_eq_repl_fwd: â\88\80L,T. eq_repl_fwd â\80¦ (λf. L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f).
+lemma frees_eq_repl_fwd: â\88\80L,T. eq_repl_fwd â\80¦ (λf. L â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 f).
#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
qed-.
-lemma frees_sort_gen: ∀f,L,s. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃⋆s⦄ ≡ f.
-#f #L elim L -L
-/4 width=3 by frees_eq_repl_back, frees_sort, frees_atom, eq_push_inv_isid/
+lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+#f #i #H
+elim (frees_inv_atom … H) -H #g #Hg #H destruct
+/2 width=1 by frees_atom/
qed.
-lemma frees_gref_gen: ∀f,L,p. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃§p⦄ ≡ f.
-#f #L elim L -L
-/4 width=3 by frees_eq_repl_back, frees_gref, frees_atom, eq_push_inv_isid/
-qed.
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+#f #L #T #H elim H -f -L -T
+/4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
+qed-.
(* Basic_2A1: removed theorems 30:
frees_eq frees_be frees_inv
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