include "ground_2/relocation/rtmap_coafter.ma".
include "basic_2/notation/relations/rdropstar_3.ma".
include "basic_2/notation/relations/rdropstar_4.ma".
-include "basic_2/relocation/lreq.ma".
+include "basic_2/relocation/seq.ma".
include "basic_2/relocation/lifts_bind.ma".
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
*)
inductive drops (b:bool): rtmap → relation lenv ≝
| drops_atom: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → drops b (f) (⋆) (⋆)
-| drops_drop: â\88\80f,I,L1,L2. drops b f L1 L2 â\86\92 drops b (⫯f) (L1.ⓘ{I}) L2
+| drops_drop: â\88\80f,I,L1,L2. drops b f L1 L2 â\86\92 drops b (â\86\91f) (L1.ⓘ{I}) L2
| drops_skip: ∀f,I1,I2,L1,L2.
- drops b f L1 L2 â\86\92 â¬\86*[f] I2 â\89¡ I1 →
- drops b (â\86\91f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ drops b f L1 L2 â\86\92 â¬\86*[f] I2 â\89\98 I1 →
+ drops b (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
.
interpretation "uniform slicing (local environment)"
'RDropStar b f L1 L2 = (drops b f L1 L2).
definition d_liftable1: predicate (relation2 lenv term) ≝
- λR. â\88\80K,T. R K T â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89¡ K →
- â\88\80U. â¬\86*[f] T â\89¡ U → R L U.
+ λR. â\88\80K,T. R K T â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89\98 K →
+ â\88\80U. â¬\86*[f] T â\89\98 U → R L U.
definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
- λR. â\88\80K,T. R K T â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89¡ K → 𝐔⦃f⦄ →
- â\88\80U. â¬\86*[f] T â\89¡ U → R L U.
+ λR. â\88\80K,T. R K T â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89\98 K → 𝐔⦃f⦄ →
+ â\88\80U. â¬\86*[f] T â\89\98 U → R L U.
definition d_deliftable1: predicate (relation2 lenv term) ≝
- λR. â\88\80L,U. R L U â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89¡ K →
- â\88\80T. â¬\86*[f] T â\89¡ U → R K T.
+ λR. â\88\80L,U. R L U â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89\98 K →
+ â\88\80T. â¬\86*[f] T â\89\98 U → R K T.
definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
- λR. â\88\80L,U. R L U â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89¡ K → 𝐔⦃f⦄ →
- â\88\80T. â¬\86*[f] T â\89¡ U → R K T.
+ λR. â\88\80L,U. R L U â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89\98 K → 𝐔⦃f⦄ →
+ â\88\80T. â¬\86*[f] T â\89\98 U → R K T.
definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. â\88\80K,T1,T2. R K T1 T2 â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89¡ K →
+ λC,S,R. â\88\80K,T1,T2. R K T1 T2 â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89\98 K →
∀U1. S f T1 U1 →
∃∃U2. S f T2 U2 & R L U1 U2.
definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. â\88\80L,U1,U2. R L U1 U2 â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89¡ K →
+ λC,S,R. â\88\80L,U1,U2. R L U1 U2 â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89\98 K →
∀T1. S f T1 U1 →
∃∃T2. S f T2 U2 & R K T1 T2.
definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. â\88\80K,T1,T2. R K T1 T2 â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89¡ K →
+ λC,S,R. â\88\80K,T1,T2. R K T1 T2 â\86\92 â\88\80b,f,L. â¬\87*[b, f] L â\89\98 K →
∀U1. S f T1 U1 →
∀U2. S f T2 U2 → R L U1 U2.
definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
predicate (lenv → relation C) ≝
- λC,S,R. â\88\80L,U1,U2. R L U1 U2 â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89¡ K →
+ λC,S,R. â\88\80L,U1,U2. R L U1 U2 â\86\92 â\88\80b,f,K. â¬\87*[b, f] L â\89\98 K →
∀T1. S f T1 U1 →
∀T2. S f T2 U2 → R K T1 T2.
definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
- λR. â\88\80b,f,L1,K1. â¬\87*[b, f] L1 â\89¡ K1 → 𝐔⦃f⦄ →
- â\88\80f2,L2. R f2 L1 L2 â\86\92 â\88\80f1. f ~â\8a\9a f1 â\89¡ f2 →
- â\88\83â\88\83K2. R f1 K1 K2 & â¬\87*[b, f] L2 â\89¡ K2.
+ λR. â\88\80b,f,L1,K1. â¬\87*[b, f] L1 â\89\98 K1 → 𝐔⦃f⦄ →
+ â\88\80f2,L2. R f2 L1 L2 â\86\92 â\88\80f1. f ~â\8a\9a f1 â\89\98 f2 →
+ â\88\83â\88\83K2. R f1 K1 K2 & â¬\87*[b, f] L2 â\89\98 K2.
definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
λR. ∀f2,L1,L2. R f2 L1 L2 →
- â\88\80b,f,K2. â¬\87*[b, f] L2 â\89¡ K2 → 𝐔⦃f⦄ →
- â\88\80f1. f ~â\8a\9a f1 â\89¡ f2 →
- â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & R f1 K1 K2.
+ â\88\80b,f,K2. â¬\87*[b, f] L2 â\89\98 K2 → 𝐔⦃f⦄ →
+ â\88\80f1. f ~â\8a\9a f1 â\89\98 f2 →
+ â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89\98 K1 & R f1 K1 K2.
definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
- λR. â\88\80b,f,L1,K1. â¬\87*[b, f] L1 â\89¡ K1 → ∀f1,K2. R f1 K1 K2 →
- â\88\80f2. f ~â\8a\9a f1 â\89¡ f2 →
- â\88\83â\88\83L2. R f2 L1 L2 & â¬\87*[b, f] L2 â\89¡ K2 & L1 ≡[f] L2.
+ λR. â\88\80b,f,L1,K1. â¬\87*[b, f] L1 â\89\98 K1 → ∀f1,K2. R f1 K1 K2 →
+ â\88\80f2. f ~â\8a\9a f1 â\89\98 f2 →
+ â\88\83â\88\83L2. R f2 L1 L2 & â¬\87*[b, f] L2 â\89\98 K2 & L1 ≡[f] L2.
(* Basic properties *********************************************************)
-lemma drops_atom_F: â\88\80f. â¬\87*[â\92», f] â\8b\86 â\89¡ ⋆.
+lemma drops_atom_F: â\88\80f. â¬\87*[â\92», f] â\8b\86 â\89\98 ⋆.
#f @drops_atom #H destruct
qed.
-lemma drops_eq_repl_back: â\88\80b,L1,L2. eq_repl_back â\80¦ (λf. â¬\87*[b, f] L1 â\89¡ L2).
+lemma drops_eq_repl_back: â\88\80b,L1,L2. eq_repl_back â\80¦ (λf. â¬\87*[b, f] L1 â\89\98 L2).
#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
[ /4 width=3 by drops_atom, isid_eq_repl_back/
| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
]
qed-.
-lemma drops_eq_repl_fwd: â\88\80b,L1,L2. eq_repl_fwd â\80¦ (λf. â¬\87*[b, f] L1 â\89¡ L2).
+lemma drops_eq_repl_fwd: â\88\80b,L1,L2. eq_repl_fwd â\80¦ (λf. â¬\87*[b, f] L1 â\89\98 L2).
#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_2A1: includes: drop_FT *)
-lemma drops_TF: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89¡ L2 â\86\92 â¬\87*[â\92», f] L1 â\89¡ L2.
+lemma drops_TF: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89\98 L2 â\86\92 â¬\87*[â\92», f] L1 â\89\98 L2.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: â\88\80b,f,L1,L2. â¬\87*[â\93\89, f] L1 â\89¡ L2 â\86\92 â¬\87*[b, f] L1 â\89¡ L2.
+lemma drops_gen: â\88\80b,f,L1,L2. â¬\87*[â\93\89, f] L1 â\89\98 L2 â\86\92 â¬\87*[b, f] L1 â\89\98 L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic_2A1: includes: drop_T *)
-lemma drops_F: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89¡ L2 â\86\92 â¬\87*[â\92», f] L1 â\89¡ L2.
+lemma drops_F: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89\98 L2 â\86\92 â¬\87*[â\92», f] L1 â\89\98 L2.
* /2 width=1 by drops_TF/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact drops_inv_atom1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y → X = ⋆ →
+fact drops_inv_atom1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89\98 Y → X = ⋆ →
Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
#b #f #X #Y * -f -X -Y
[ /3 width=1 by conj/
(* Basic_1: includes: drop_gen_sort *)
(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: â\88\80b,f,Y. â¬\87*[b, f] â\8b\86 â\89¡ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+lemma drops_inv_atom1: â\88\80b,f,Y. â¬\87*[b, f] â\8b\86 â\89\98 Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
/2 width=3 by drops_inv_atom1_aux/ qed-.
-fact drops_inv_drop1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y â\86\92 â\88\80g,I,K. X = K.â\93\98{I} â\86\92 f = ⫯g →
- â¬\87*[b, g] K â\89¡ Y.
+fact drops_inv_drop1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89\98 Y â\86\92 â\88\80g,I,K. X = K.â\93\98{I} â\86\92 f = â\86\91g →
+ â¬\87*[b, g] K â\89\98 Y.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J #K #H destruct
| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
(* Basic_1: includes: drop_gen_drop *)
(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: â\88\80b,f,I,K,Y. â¬\87*[b, ⫯f] K.â\93\98{I} â\89¡ Y â\86\92 â¬\87*[b, f] K â\89¡ Y.
+lemma drops_inv_drop1: â\88\80b,f,I,K,Y. â¬\87*[b, â\86\91f] K.â\93\98{I} â\89\98 Y â\86\92 â¬\87*[b, f] K â\89\98 Y.
/2 width=6 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_skip1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y â\86\92 â\88\80g,I1,K1. X = K1.â\93\98{I1} â\86\92 f = â\86\91g →
- â\88\83â\88\83I2,K2. â¬\87*[b, g] K1 â\89¡ K2 & â¬\86*[g] I2 â\89¡ I1 & Y = K2.ⓘ{I2}.
+fact drops_inv_skip1_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89\98 Y â\86\92 â\88\80g,I1,K1. X = K1.â\93\98{I1} â\86\92 f = ⫯g →
+ â\88\83â\88\83I2,K2. â¬\87*[b, g] K1 â\89\98 K2 & â¬\86*[g] I2 â\89\98 I1 & Y = K2.ⓘ{I2}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J1 #K1 #H destruct
| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_l *)
(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: â\88\80b,f,I1,K1,Y. â¬\87*[b, â\86\91f] K1.â\93\98{I1} â\89¡ Y →
- â\88\83â\88\83I2,K2. â¬\87*[b, f] K1 â\89¡ K2 & â¬\86*[f] I2 â\89¡ I1 & Y = K2.ⓘ{I2}.
+lemma drops_inv_skip1: â\88\80b,f,I1,K1,Y. â¬\87*[b, ⫯f] K1.â\93\98{I1} â\89\98 Y →
+ â\88\83â\88\83I2,K2. â¬\87*[b, f] K1 â\89\98 K2 & â¬\86*[f] I2 â\89\98 I1 & Y = K2.ⓘ{I2}.
/2 width=5 by drops_inv_skip1_aux/ qed-.
-fact drops_inv_skip2_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y â\86\92 â\88\80g,I2,K2. Y = K2.â\93\98{I2} â\86\92 f = â\86\91g →
- â\88\83â\88\83I1,K1. â¬\87*[b, g] K1 â\89¡ K2 & â¬\86*[g] I2 â\89¡ I1 & X = K1.ⓘ{I1}.
+fact drops_inv_skip2_aux: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89\98 Y â\86\92 â\88\80g,I2,K2. Y = K2.â\93\98{I2} â\86\92 f = ⫯g →
+ â\88\83â\88\83I1,K1. â¬\87*[b, g] K1 â\89\98 K2 & â¬\86*[g] I2 â\89\98 I1 & X = K1.ⓘ{I1}.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J2 #K2 #H destruct
| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
(* Basic_1: includes: drop_gen_skip_r *)
(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: â\88\80b,f,I2,X,K2. â¬\87*[b, â\86\91f] X â\89¡ K2.ⓘ{I2} →
- â\88\83â\88\83I1,K1. â¬\87*[b, f] K1 â\89¡ K2 & â¬\86*[f] I2 â\89¡ I1 & X = K1.ⓘ{I1}.
+lemma drops_inv_skip2: â\88\80b,f,I2,X,K2. â¬\87*[b, ⫯f] X â\89\98 K2.ⓘ{I2} →
+ â\88\83â\88\83I1,K1. â¬\87*[b, f] K1 â\89\98 K2 & â¬\86*[f] I2 â\89\98 I1 & X = K1.ⓘ{I1}.
/2 width=5 by drops_inv_skip2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-fact drops_fwd_drop2_aux: â\88\80b,f2,X,Y. â¬\87*[b, f2] X â\89¡ Y → ∀I,K. Y = K.ⓘ{I} →
- â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 â\8a\9a ⫯f1 â\89¡ f & â¬\87*[b, f] X â\89¡ K.
+fact drops_fwd_drop2_aux: â\88\80b,f2,X,Y. â¬\87*[b, f2] X â\89\98 Y → ∀I,K. Y = K.ⓘ{I} →
+ â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 â\8a\9a â\86\91f1 â\89\98 f & â¬\87*[b, f] X â\89\98 K.
#b #f2 #X #Y #H elim H -f2 -X -Y
[ #f2 #Hf2 #J #K #H destruct
| #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
]
qed-.
-lemma drops_fwd_drop2: â\88\80b,f2,I,X,K. â¬\87*[b, f2] X â\89¡ K.ⓘ{I} →
- â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 â\8a\9a ⫯f1 â\89¡ f & â¬\87*[b, f] X â\89¡ K.
+lemma drops_fwd_drop2: â\88\80b,f2,I,X,K. â¬\87*[b, f2] X â\89\98 K.ⓘ{I} →
+ â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 â\8a\9a â\86\91f1 â\89\98 f & â¬\87*[b, f] X â\89\98 K.
/2 width=4 by drops_fwd_drop2_aux/ qed-.
(* Properties with test for identity ****************************************)
(* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: â\88\80b,L,f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L â\89¡ L.
+lemma drops_refl: â\88\80b,L,f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L â\89\98 L.
#b #L elim L -L /2 width=1 by drops_atom/
#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
/3 width=1 by drops_skip, liftsb_refl/
(* Basic_1: includes: drop_gen_refl *)
(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89¡ L2 → 𝐈⦃f⦄ → L1 = L2.
+lemma drops_fwd_isid: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89\98 L2 → 𝐈⦃f⦄ → L1 = L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2 //
[ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
| /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
]
qed-.
-lemma drops_after_fwd_drop2: â\88\80b,f2,I,X,K. â¬\87*[b, f2] X â\89¡ K.ⓘ{I} →
- â\88\80f1,f. ð\9d\90\88â¦\83f1â¦\84 â\86\92 f2 â\8a\9a ⫯f1 â\89¡ f â\86\92 â¬\87*[b, f] X â\89¡ K.
+lemma drops_after_fwd_drop2: â\88\80b,f2,I,X,K. â¬\87*[b, f2] X â\89\98 K.ⓘ{I} →
+ â\88\80f1,f. ð\9d\90\88â¦\83f1â¦\84 â\86\92 f2 â\8a\9a â\86\91f1 â\89\98 f â\86\92 â¬\87*[b, f] X â\89\98 K.
#b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
#g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
/3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
(* Forward lemmas with test for finite colength *****************************)
-lemma drops_fwd_isfin: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89¡ L2 → 𝐅⦃f⦄.
+lemma drops_fwd_isfin: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89\98 L2 → 𝐅⦃f⦄.
#f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by isfin_next, isfin_push, isfin_isid/
qed-.
(* Properties with test for uniformity **************************************)
-lemma drops_isuni_ex: â\88\80f. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â\88\80L. â\88\83K. â¬\87*[â\92», f] L â\89¡ K.
+lemma drops_isuni_ex: â\88\80f. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â\88\80L. â\88\83K. â¬\87*[â\92», f] L â\89\98 K.
#f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
#f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
#L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
(* Inversion lemmas with test for uniformity ********************************)
-lemma drops_inv_isuni: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89¡ L2 → 𝐔⦃f⦄ →
+lemma drops_inv_isuni: â\88\80f,L1,L2. â¬\87*[â\93\89, f] L1 â\89\98 L2 → 𝐔⦃f⦄ →
(𝐈⦃f⦄ ∧ L1 = L2) ∨
- â\88\83â\88\83g,I,K. â¬\87*[â\93\89, g] K â\89¡ L2 & ð\9d\90\94â¦\83gâ¦\84 & L1 = K.â\93\98{I} & f = ⫯g.
+ â\88\83â\88\83g,I,K. â¬\87*[â\93\89, g] K â\89\98 L2 & ð\9d\90\94â¦\83gâ¦\84 & L1 = K.â\93\98{I} & f = â\86\91g.
#f #L1 #L2 * -f -L1 -L2
[ /4 width=1 by or_introl, conj/
| /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
qed-.
(* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_bind1_isuni: â\88\80b,f,I,K,L2. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] K.â\93\98{I} â\89¡ L2 →
+lemma drops_inv_bind1_isuni: â\88\80b,f,I,K,L2. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] K.â\93\98{I} â\89\98 L2 →
(𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
- â\88\83â\88\83g. ð\9d\90\94â¦\83gâ¦\84 & â¬\87*[b, g] K â\89¡ L2 & f = ⫯g.
+ â\88\83â\88\83g. ð\9d\90\94â¦\83gâ¦\84 & â¬\87*[b, g] K â\89\98 L2 & f = â\86\91g.
#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
[ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
<(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
qed-.
(* Basic_2A1: was: drop_inv_O1_pair2 *)
-lemma drops_inv_bind2_isuni: â\88\80b,f,I,K,L1. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L1 â\89¡ K.ⓘ{I} →
+lemma drops_inv_bind2_isuni: â\88\80b,f,I,K,L1. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L1 â\89\98 K.ⓘ{I} →
(𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
- â\88\83â\88\83g,I1,K1. ð\9d\90\94â¦\83gâ¦\84 & â¬\87*[b, g] K1 â\89¡ K.â\93\98{I} & L1 = K1.â\93\98{I1} & f = ⫯g.
+ â\88\83â\88\83g,I1,K1. ð\9d\90\94â¦\83gâ¦\84 & â¬\87*[b, g] K1 â\89\98 K.â\93\98{I} & L1 = K1.â\93\98{I1} & f = â\86\91g.
#b #f #I #K *
[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
]
qed-.
-lemma drops_inv_bind2_isuni_next: â\88\80b,f,I,K,L1. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, ⫯f] L1 â\89¡ K.ⓘ{I} →
- â\88\83â\88\83I1,K1. â¬\87*[b, f] K1 â\89¡ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+lemma drops_inv_bind2_isuni_next: â\88\80b,f,I,K,L1. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, â\86\91f] L1 â\89\98 K.ⓘ{I} →
+ â\88\83â\88\83I1,K1. â¬\87*[b, f] K1 â\89\98 K.ⓘ{I} & L1 = K1.ⓘ{I1}.
#b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
[ #H elim (isid_inv_next … H) -H //
| /2 width=4 by ex2_2_intro/
]
qed-.
-fact drops_inv_TF_aux: â\88\80f,L1,L2. â¬\87*[â\92», f] L1 â\89¡ L2 → 𝐔⦃f⦄ →
- â\88\80I,K. L2 = K.â\93\98{I} â\86\92 â¬\87*[â\93\89, f] L1 â\89¡ K.ⓘ{I}.
+fact drops_inv_TF_aux: â\88\80f,L1,L2. â¬\87*[â\92», f] L1 â\89\98 L2 → 𝐔⦃f⦄ →
+ â\88\80I,K. L2 = K.â\93\98{I} â\86\92 â¬\87*[â\93\89, f] L1 â\89\98 K.ⓘ{I}.
#f #L1 #L2 #H elim H -f -L1 -L2
[ #f #_ #_ #J #K #H destruct
| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
qed-.
(* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: â\88\80f,I,L,K. â¬\87*[â\92», f] L â\89¡ K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[â\93\89, f] L â\89¡ K.ⓘ{I}.
+lemma drops_inv_TF: â\88\80f,I,L,K. â¬\87*[â\92», f] L â\89\98 K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[â\93\89, f] L â\89\98 K.ⓘ{I}.
/2 width=3 by drops_inv_TF_aux/ qed-.
(* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: â\88\80b,f,I,L,K. â¬\87*[b, f] L â\89¡ K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[â\93\89, f] L â\89¡ K.ⓘ{I}.
+lemma drops_inv_gen: â\88\80b,f,I,L,K. â¬\87*[b, f] L â\89\98 K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[â\93\89, f] L â\89\98 K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: â\88\80b,f,I,L,K. â¬\87*[â\92», f] L â\89¡ K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L â\89¡ K.ⓘ{I}.
+lemma drops_inv_F: â\88\80b,f,I,L,K. â¬\87*[â\92», f] L â\89\98 K.â\93\98{I} â\86\92 ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] L â\89\98 K.ⓘ{I}.
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_1: was: drop_S *)
(* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: â\88\80b,f,I,X,K. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] X â\89¡ K.â\93\98{I} â\86\92 â¬\87*[b, ⫯f] X â\89¡ K.
+lemma drops_isuni_fwd_drop2: â\88\80b,f,I,X,K. ð\9d\90\94â¦\83fâ¦\84 â\86\92 â¬\87*[b, f] X â\89\98 K.â\93\98{I} â\86\92 â¬\87*[b, â\86\91f] X â\89\98 K.
/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
(* Inversion lemmas with uniform relocations ********************************)
-lemma drops_inv_atom2: â\88\80b,L,f. â¬\87*[b, f] L â\89¡ ⋆ →
- â\88\83â\88\83n,f1. â¬\87*[b, ð\9d\90\94â\9d´nâ\9dµ] L â\89¡ â\8b\86 & ð\9d\90\94â\9d´nâ\9dµ â\8a\9a f1 â\89¡ f.
+lemma drops_inv_atom2: â\88\80b,L,f. â¬\87*[b, f] L â\89\98 ⋆ →
+ â\88\83â\88\83n,f1. â¬\87*[b, ð\9d\90\94â\9d´nâ\9dµ] L â\89\98 â\8b\86 & ð\9d\90\94â\9d´nâ\9dµ â\8a\9a f1 â\89\98 f.
#b #L elim L -L
[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
]
qed-.
-lemma drops_inv_succ: â\88\80L1,L2,i. â¬\87*[⫯i] L1 â\89¡ L2 →
- â\88\83â\88\83I,K. â¬\87*[i] K â\89¡ L2 & L1 = K.ⓘ{I}.
+lemma drops_inv_succ: â\88\80L1,L2,i. â¬\87*[â\86\91i] L1 â\89\98 L2 →
+ â\88\83â\88\83I,K. â¬\87*[i] K â\89\98 L2 & L1 = K.ⓘ{I}.
#L1 #L2 #i #H elim (drops_inv_isuni … H) -H // *
[ #H elim (isid_inv_next … H) -H //
| /2 width=4 by ex2_2_intro/
(* Properties with uniform relocations **************************************)
-lemma drops_F_uni: â\88\80L,i. â¬\87*[â\92», ð\9d\90\94â\9d´iâ\9dµ] L â\89¡ â\8b\86 â\88¨ â\88\83â\88\83I,K. â¬\87*[i] L â\89¡ K.ⓘ{I}.
+lemma drops_F_uni: â\88\80L,i. â¬\87*[â\92», ð\9d\90\94â\9d´iâ\9dµ] L â\89\98 â\8b\86 â\88¨ â\88\83â\88\83I,K. â¬\87*[i] L â\89\98 K.ⓘ{I}.
#L elim L -L /2 width=1 by or_introl/
#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
qed-.
(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89¡ L2 â\86\92 â\88\80f1,f2. f1 â\8a\9a f2 â\89¡ f → 𝐔⦃f1⦄ →
- â\88\83â\88\83L. â¬\87*[b, f1] L1 â\89¡ L & â¬\87*[b, f2] L â\89¡ L2.
+lemma drops_split_trans: â\88\80b,f,L1,L2. â¬\87*[b, f] L1 â\89\98 L2 â\86\92 â\88\80f1,f2. f1 â\8a\9a f2 â\89\98 f → 𝐔⦃f1⦄ →
+ â\88\83â\88\83L. â¬\87*[b, f1] L1 â\89\98 L & â¬\87*[b, f2] L â\89\98 L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2
[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
#H lapply (H0f H) -b
]
qed-.
-lemma drops_split_div: â\88\80b,f1,L1,L. â¬\87*[b, f1] L1 â\89¡ L â\86\92 â\88\80f2,f. f1 â\8a\9a f2 â\89¡ f → 𝐔⦃f2⦄ →
- â\88\83â\88\83L2. â¬\87*[â\92», f2] L â\89¡ L2 & â¬\87*[â\92», f] L1 â\89¡ L2.
+lemma drops_split_div: â\88\80b,f1,L1,L. â¬\87*[b, f1] L1 â\89\98 L â\86\92 â\88\80f2,f. f1 â\8a\9a f2 â\89\98 f → 𝐔⦃f2⦄ →
+ â\88\83â\88\83L2. â¬\87*[â\92», f2] L â\89\98 L2 & â¬\87*[â\92», f] L1 â\89\98 L2.
#b #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
(* Properties with application **********************************************)
-lemma drops_tls_at: â\88\80f,i1,i2. @â¦\83i1,fâ¦\84 â\89¡ i2 →
- â\88\80b,L1,L2. â¬\87*[b,⫱*[i2]f] L1 â\89¡ L2 →
- â¬\87*[b,â\86\91⫱*[⫯i2]f] L1 â\89¡ L2.
+lemma drops_tls_at: â\88\80f,i1,i2. @â¦\83i1,fâ¦\84 â\89\98 i2 →
+ â\88\80b,L1,L2. â¬\87*[b,⫱*[i2]f] L1 â\89\98 L2 →
+ â¬\87*[b,⫯⫱*[â\86\91i2]f] L1 â\89\98 L2.
/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
-lemma drops_split_trans_bind2: â\88\80b,f,I,L,K0. â¬\87*[b, f] L â\89¡ K0.â\93\98{I} â\86\92 â\88\80i. @â¦\83O, fâ¦\84 â\89¡ i →
- â\88\83â\88\83J,K. â¬\87*[i]L â\89¡ K.â\93\98{J} & â¬\87*[b, ⫱*[⫯i]f] K â\89¡ K0 & â¬\86*[⫱*[⫯i]f] I â\89¡ J.
+lemma drops_split_trans_bind2: â\88\80b,f,I,L,K0. â¬\87*[b, f] L â\89\98 K0.â\93\98{I} â\86\92 â\88\80i. @â¦\83O, fâ¦\84 â\89\98 i →
+ â\88\83â\88\83J,K. â¬\87*[i]L â\89\98 K.â\93\98{J} & â¬\87*[b, ⫱*[â\86\91i]f] K â\89\98 K0 & â¬\86*[⫱*[â\86\91i]f] I â\89\98 J.
#b #f #I #L #K0 #H #i #Hf
elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
lapply (drops_tls_at … Hf … H) -H #H
(* Properties with context-sensitive equivalence for terms ******************)
-lemma ceq_lift_sn: d_liftable2_sn … liftsb ceq.
+lemma ceq_lift_sn: d_liftable2_sn … liftsb ceq_ext.
+#K #I1 #I2 #H <(ceq_ext_inv_eq … H) -I2
/2 width=3 by ex2_intro/ qed-.
-lemma ceq_inv_lift_sn: d_deliftable2_sn … liftsb ceq.
+lemma ceq_inv_lift_sn: d_deliftable2_sn … liftsb ceq_ext.
+#L #J1 #J2 #H <(ceq_ext_inv_eq … H) -J2
/2 width=3 by ex2_intro/ qed-.
(* Note: d_deliftable2_sn cfull does not hold *)
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