(* *)
(**************************************************************************)
-include "ground_2/relocation/rtmap_sor.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)
+| 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 ⋓ ⫱f2 ≘ f → frees L (ⓑ{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 ⋓ ⫱f2 ≘ 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 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
+ f1 ⋓ f2 ≘ f → frees L (ⓕ[I]V.T) f
.
interpretation
(* Basic inversion lemmas ***************************************************)
-fact frees_inv_sort_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80x. X = â\8b\86x â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+fact frees_inv_sort_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80x. X = â\8b\86x â\86\92 ð\9d\90\88â\9dªfâ\9d«.
#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: â\88\80f,L,s. L â\8a¢ ð\9d\90\85+â¦\83â\8b\86sâ¦\84 â\89\98 f â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma frees_inv_sort: â\88\80f,L,s. L â\8a¢ ð\9d\90\85+â\9dªâ\8b\86sâ\9d« â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d«.
/2 width=5 by frees_inv_sort_aux/ qed-.
fact frees_inv_atom_aux:
- â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 ≘ f → ∀i. L = ⋆ → X = #i →
- â\88\83â\88\83g. ð\9d\90\88â¦\83gâ¦\84 & f = ⫯*[i]↑g.
+ â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« ≘ f → ∀i. L = ⋆ → X = #i →
+ â\88\83â\88\83g. ð\9d\90\88â\9dªgâ\9d« & 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: â\88\80f,i. â\8b\86 â\8a¢ ð\9d\90\85+â¦\83#iâ¦\84 â\89\98 f â\86\92 â\88\83â\88\83g. ð\9d\90\88â¦\83gâ¦\84 & f = ⫯*[i]↑g.
+lemma frees_inv_atom: â\88\80f,i. â\8b\86 â\8a¢ ð\9d\90\85+â\9dª#iâ\9d« â\89\98 f â\86\92 â\88\83â\88\83g. ð\9d\90\88â\9dªgâ\9d« & f = ⫯*[i]↑g.
/2 width=5 by frees_inv_atom_aux/ qed-.
fact frees_inv_pair_aux:
- â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80I,K,V. L = K.â\93\91{I}V → X = #0 →
- â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 ≘ g & f = ↑g.
+ â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80I,K,V. L = K.â\93\91[I]V → X = #0 →
+ â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â\9dªVâ\9d« ≘ 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:
- â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80I,K. L = K.â\93¤{I} → X = #0 →
- â\88\83â\88\83g. ð\9d\90\88â¦\83gâ¦\84 & f = ↑g.
+ â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80I,K. L = K.â\93¤[I] → X = #0 →
+ â\88\83â\88\83g. ð\9d\90\88â\9dªgâ\9d« & 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:
- â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80I,K,j. L = K.â\93\98{I} → X = #(↑j) →
- â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â¦\83#jâ¦\84 ≘ g & f = ⫯g.
+ â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80I,K,j. L = K.â\93\98[I] → X = #(↑j) →
+ â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â\9dª#jâ\9d« ≘ 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 →
- â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â¦\83#iâ¦\84 ≘ g & f = ⫯g.
+ ∀f,I,K,i. K.ⓘ[I] ⊢ 𝐅+❪#(↑i)❫ ≘ f →
+ â\88\83â\88\83g. K â\8a¢ ð\9d\90\85+â\9dª#iâ\9d« ≘ g & f = ⫯g.
/2 width=6 by frees_inv_lref_aux/ qed-.
-fact frees_inv_gref_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80x. X = §x â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+fact frees_inv_gref_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80x. X = §x â\86\92 ð\9d\90\88â\9dªfâ\9d«.
#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: â\88\80f,L,l. L â\8a¢ ð\9d\90\85+â¦\83§lâ¦\84 â\89\98 f â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma frees_inv_gref: â\88\80f,L,l. L â\8a¢ ð\9d\90\85+â\9dªÂ§lâ\9d« â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d«.
/2 width=5 by frees_inv_gref_aux/ qed-.
fact frees_inv_bind_aux:
- â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â¦\83Xâ¦\84 â\89\98 f â\86\92 â\88\80p,I,V,T. X = â\93\91{p,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\93\91{I}V â\8a¢ ð\9d\90\85+â¦\83Tâ¦\84 ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+ â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80p,I,V,T. X = â\93\91[p,I]V.T →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 f1 & L.â\93\91[I]V â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ 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:
- â\88\80f,p,I,L,V,T. L â\8a¢ ð\9d\90\85+â¦\83â\93\91{p,I}V.Tâ¦\84 ≘ 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 ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+ â\88\80f,p,I,L,V,T. L â\8a¢ ð\9d\90\85+â\9dªâ\93\91[p,I]V.Tâ\9d« ≘ f →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 f1 & L.â\93\91[I]V â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f2 & f1 ⋓ ⫱f2 ≘ 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\98 f â\86\92 â\88\80I,V,T. X = â\93\95{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 ≘ f2 & f1 ⋓ f2 ≘ f.
+fact frees_inv_flat_aux: â\88\80f,L,X. L â\8a¢ ð\9d\90\85+â\9dªXâ\9d« â\89\98 f â\86\92 â\88\80I,V,T. X = â\93\95[I]V.T →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 f1 & L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ 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:
- â\88\80f,I,L,V,T. L â\8a¢ ð\9d\90\85+â¦\83â\93\95{I}V.Tâ¦\84 ≘ 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 ≘ f2 & f1 ⋓ f2 ≘ f.
+ â\88\80f,I,L,V,T. L â\8a¢ ð\9d\90\85+â\9dªâ\93\95[I]V.Tâ\9d« ≘ f →
+ â\88\83â\88\83f1,f2. L â\8a¢ ð\9d\90\85+â\9dªVâ\9d« â\89\98 f1 & L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f2 & f1 ⋓ f2 ≘ f.
/2 width=4 by frees_inv_flat_aux/ 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 ≘ f).
+lemma frees_eq_repl_back: â\88\80L,T. eq_repl_back â\80¦ (λf. L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f).
#L #T #f1 #H elim H -f1 -L -T
[ /3 width=3 by frees_sort, isid_eq_repl_back/
| #f1 #i #Hf1 #g2 #H
]
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 ≘ f).
+lemma frees_eq_repl_fwd: â\88\80L,T. eq_repl_fwd â\80¦ (λf. L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« ≘ f).
#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
qed-.
-lemma frees_lref_push: â\88\80f,i. â\8b\86 â\8a¢ ð\9d\90\85+â¦\83#iâ¦\84 â\89\98 f â\86\92 â\8b\86 â\8a¢ ð\9d\90\85+â¦\83#â\86\91iâ¦\84 ≘ ⫯f.
+lemma frees_lref_push: â\88\80f,i. â\8b\86 â\8a¢ ð\9d\90\85+â\9dª#iâ\9d« â\89\98 f â\86\92 â\8b\86 â\8a¢ ð\9d\90\85+â\9dª#â\86\91iâ\9d« ≘ ⫯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: â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â¦\83Tâ¦\84 â\89\98 f â\86\92 ð\9d\90\85â¦\83fâ¦\84.
+lemma frees_fwd_isfin: â\88\80f,L,T. L â\8a¢ ð\9d\90\85+â\9dªTâ\9d« â\89\98 f â\86\92 ð\9d\90\85â\9dªfâ\9d«.
#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-.
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