(* *)
(**************************************************************************)
-include "ground_2/relocation/rtmap_sor.ma".
-include "static_2/notation/relations/freestar_3.ma".
+include "ground/relocation/rtmap_sor.ma".
+include "static_2/notation/relations/freeplus_3.ma".
include "static_2/syntax/lenv.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-inductive frees: relation3 lenv term rtmap ≝
-| frees_sort: â\88\80f,L,s. ð\9d\90\88â¦\83fâ¦\84 → frees L (⋆s) f
-| frees_atom: â\88\80f,i. ð\9d\90\88â¦\83fâ¦\84 → frees (⋆) (#i) (⫯*[i]↑f)
+inductive frees: relation3 lenv term pr_map ≝
+| frees_sort: â\88\80f,L,s. ð\9d\90\88â\9dªfâ\9d« → frees L (⋆s) f
+| frees_atom: â\88\80f,i. ð\9d\90\88â\9dªfâ\9d« → frees (⋆) (#i) (⫯*[i]↑f)
| frees_pair: ∀f,I,L,V. frees L V f →
- frees (L.ⓑ{I}V) (#0) (↑f)
-| frees_unit: â\88\80f,I,L. ð\9d\90\88â¦\83fâ¦\84 â\86\92 frees (L.â\93¤{I}) (#0) (↑f)
+ frees (L.ⓑ[I]V) (#0) (↑f)
+| frees_unit: â\88\80f,I,L. ð\9d\90\88â\9dªfâ\9d« â\86\92 frees (L.â\93¤[I]) (#0) (↑f)
| frees_lref: ∀f,I,L,i. frees L (#i) f →
- frees (L.ⓘ{I}) (#↑i) (⫯f)
-| frees_gref: â\88\80f,L,l. ð\9d\90\88â¦\83fâ¦\84 → 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\98 f â\86\92 frees L (â\93\91{p,I}V.T) f
+ frees (L.ⓘ[I]) (#↑i) (⫯f)
+| frees_gref: â\88\80f,L,l. ð\9d\90\88â\9dªfâ\9d« → 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 â«°f2 â\89\98 f â\86\92 frees L (â\93\91[p,I]V.T) f
| frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
- f1 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
+ f1 ⋓ f2 ≘ f → frees L (ⓕ[I]V.T) f
.
interpretation
"context-sensitive free variables (term)"
- 'FreeStar L T f = (frees L T f).
+ 'FreePlus L T f = (frees L T f).
(* Basic inversion lemmas ***************************************************)
-fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+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
]
qed-.
-lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+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_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
- ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+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/
]
qed-.
-lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+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_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.
+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
]
qed-.
-lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+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_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
- ∃∃g. 𝐈⦃g⦄ & 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
[ #f #L #s #_ #Z #Y #_ #H destruct
| #f #i #_ #Z #Y #H destruct
]
qed-.
-lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+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 → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
- ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & 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
[ #f #L #s #_ #Z #Y #j #_ #H destruct
| #f #i #_ #Z #Y #j #H destruct
]
qed-.
-lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
- ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+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_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃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
]
qed-.
-lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+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.
+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 #L #s #_ #q #J #W #U #H destruct
| #f #i #_ #q #J #W #U #H destruct
]
qed-.
-lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind:
+ ∀f,p,I,L,V,T. L ⊢ 𝐅+❪ⓑ[p,I]V.T❫ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+❪V❫ ≘ f1 & L.ⓑ[I]V ⊢ 𝐅+❪T❫ ≘ f2 & f1 ⋓ ⫰f2 ≘ f.
/2 width=4 by frees_inv_bind_aux/ qed-.
-fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅+❪X❫ ≘ f → ∀I,V,T. X = ⓕ[I]V.T →
+ ∃∃f1,f2. L ⊢ 𝐅+❪V❫ ≘ f1 & L ⊢ 𝐅+❪T❫ ≘ f2 & f1 ⋓ f2 ≘ f.
#f #L #X * -f -L -X
[ #f #L #s #_ #J #W #U #H destruct
| #f #i #_ #J #W #U #H destruct
]
qed-.
-lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+lemma frees_inv_flat:
+ ∀f,I,L,V,T. L ⊢ 𝐅+❪ⓕ[I]V.T❫ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+❪V❫ ≘ f1 & L ⊢ 𝐅+❪T❫ ≘ f2 & f1 ⋓ f2 ≘ f.
/2 width=4 by frees_inv_flat_aux/ qed-.
(* Basic properties ********************************************************)
-lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+lemma frees_eq_repl_back: ∀L,T. pr_eq_repl_back … (λf. L ⊢ 𝐅+❪T❫ ≘ f).
#L #T #f1 #H elim H -f1 -L -T
-[ /3 width=3 by frees_sort, isid_eq_repl_back/
+[ /3 width=3 by frees_sort, pr_isi_eq_repl_back/
| #f1 #i #Hf1 #g2 #H
- elim (eq_inv_pushs_sn … H) -H #g #Hg #H destruct
+ elim (pr_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/
+ /3 width=3 by frees_atom, pr_isi_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/
+ /3 width=3 by frees_unit, pr_isi_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/
+| /3 width=3 by frees_gref, pr_isi_eq_repl_back/
+| /3 width=7 by frees_bind, pr_sor_eq_repl_back/
+| /3 width=7 by frees_flat, pr_sor_eq_repl_back/
]
qed-.
-lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
-#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
+lemma frees_eq_repl_fwd: ∀L,T. pr_eq_repl_fwd … (λf. L ⊢ 𝐅+❪T❫ ≘ f).
+#L #T @pr_eq_repl_sym /2 width=3 by frees_eq_repl_back/
qed-.
-lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+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/
(* Forward lemmas with test for finite colength *****************************)
-lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+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/
+/4 width=5 by pr_sor_inv_isf_bi, pr_isf_isi, pr_isf_tl, pr_isf_pushs, pr_isf_push, pr_isf_next/
qed-.
(* Basic_2A1: removed theorems 30:
frees_inv_bind frees_inv_flat frees_inv_bind_O
frees_lref_eq frees_lref_be frees_weak
frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
- frees_lift_ge frees_inv_lift_be frees_inv_lift_ge
+ frees_lift_ge frees_inv_lift_be frees_inv_lift_ge
lreq_frees_trans frees_lreq_conf
llor_atom llor_skip llor_total
llor_tail_frees llor_tail_cofrees
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