drop_refl_atom_O2 drop_drop_lt drop_skip_lt
*)
inductive drops (b:bool): rtmap → relation lenv ≝
-| drops_atom: â\88\80f. (b = â\93\89 â\86\92 ð\9d\90\88â¦\83fâ¦\84) → drops b (f) (⋆) (⋆)
-| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ{I}) L2
+| drops_atom: â\88\80f. (b = â\93\89 â\86\92 ð\9d\90\88â\9dªfâ\9d«) → drops b (f) (⋆) (⋆)
+| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ[I]) L2
| drops_skip: ∀f,I1,I2,L1,L2.
drops b f L1 L2 → ⇧*[f] I2 ≘ I1 →
- drops b (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+ drops b (⫯f) (L1.ⓘ[I1]) (L2.ⓘ[I2])
.
interpretation "uniform slicing (local environment)"
∀U. ⇧*[f] T ≘ 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\98 K â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ 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\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀U. ⇧*[f] T ≘ U → R L U.
definition d_deliftable1: predicate (relation2 lenv term) ≝
∀T. ⇧*[f] T ≘ 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\98 K â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ 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\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀T. ⇧*[f] T ≘ U → R K T.
definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
∀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\98 K1 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ λR. â\88\80b,f,L1,K1. â\87©*[b,f] L1 â\89\98 K1 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
∃∃K2. R f1 K1 K2 & ⇩*[b,f] L2 ≘ 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\98 K2 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+ â\88\80b,f,K2. â\87©*[b,f] L2 â\89\98 K2 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
∀f1. f ~⊚ f1 ≘ f2 →
∃∃K1. ⇩*[b,f] L1 ≘ K1 & R f1 K1 K2.
(* Basic inversion lemmas ***************************************************)
fact drops_inv_atom1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → X = ⋆ →
- Y = â\8b\86 â\88§ (b = â\93\89 â\86\92 ð\9d\90\88â¦\83fâ¦\84).
+ Y = â\8b\86 â\88§ (b = â\93\89 â\86\92 ð\9d\90\88â\9dªfâ\9d«).
#b #f #X #Y * -f -X -Y
[ /3 width=1 by conj/
| #f #I #L1 #L2 #_ #H destruct
(* 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\98 Y â\86\92 Y = â\8b\86 â\88§ (b = â\93\89 â\86\92 ð\9d\90\88â¦\83fâ¦\84).
+lemma drops_inv_atom1: â\88\80b,f,Y. â\87©*[b,f] â\8b\86 â\89\98 Y â\86\92 Y = â\8b\86 â\88§ (b = â\93\89 â\86\92 ð\9d\90\88â\9dªfâ\9d«).
/2 width=3 by drops_inv_atom1_aux/ qed-.
-fact drops_inv_drop1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
+fact drops_inv_drop1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I,K. X = K.ⓘ[I] → f = ↑g →
⇩*[b,g] K ≘ Y.
#b #f #X #Y * -f -X -Y
[ #f #Hf #g #J #K #H destruct
(* Basic_1: includes: drop_gen_drop *)
(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀b,f,I,K,Y. ⇩*[b,↑f] K.ⓘ{I} ≘ Y → ⇩*[b,f] K ≘ Y.
+lemma drops_inv_drop1: ∀b,f,I,K,Y. ⇩*[b,↑f] K.ⓘ[I] ≘ Y → ⇩*[b,f] K ≘ Y.
/2 width=6 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_skip1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
- ∃∃I2,K2. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+fact drops_inv_skip1_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ[I1] → f = ⫯g →
+ ∃∃I2,K2. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ 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: ∀b,f,I1,K1,Y. ⇩*[b,⫯f] K1.ⓘ{I1} ≘ Y →
- ∃∃I2,K2. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⇩*[b,⫯f] K1.ⓘ[I1] ≘ Y →
+ ∃∃I2,K2. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & Y = K2.ⓘ[I2].
/2 width=5 by drops_inv_skip1_aux/ qed-.
-fact drops_inv_skip2_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
- ∃∃I1,K1. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
+fact drops_inv_skip2_aux: ∀b,f,X,Y. ⇩*[b,f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ[I2] → f = ⫯g →
+ ∃∃I1,K1. ⇩*[b,g] K1 ≘ K2 & ⇧*[g] I2 ≘ 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: ∀b,f,I2,X,K2. ⇩*[b,⫯f] X ≘ K2.ⓘ{I2} →
- ∃∃I1,K1. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
+lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⇩*[b,⫯f] X ≘ K2.ⓘ[I2] →
+ ∃∃I1,K1. ⇩*[b,f] K1 ≘ K2 & ⇧*[f] I2 ≘ I1 & X = K1.ⓘ[I1].
/2 width=5 by drops_inv_skip2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⇩*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
- â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
+fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⇩*[b,f2] X ≘ Y → ∀I,K. Y = K.ⓘ[I] →
+ â\88\83â\88\83f1,f. ð\9d\90\88â\9dªf1â\9d« & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ 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: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ{I} →
- â\88\83â\88\83f1,f. ð\9d\90\88â¦\83f1â¦\84 & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ K.
+lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ[I] →
+ â\88\83â\88\83f1,f. ð\9d\90\88â\9dªf1â\9d« & f2 ⊚ ↑f1 ≘ f & ⇩*[b,f] X ≘ 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 → ⇩*[b,f] L ≘ L.
+lemma drops_refl: â\88\80b,L,f. ð\9d\90\88â\9dªfâ\9d« → ⇩*[b,f] L ≘ 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\98 L2 â\86\92 ð\9d\90\88â¦\83fâ¦\84 → L1 = L2.
+lemma drops_fwd_isid: â\88\80b,f,L1,L2. â\87©*[b,f] L1 â\89\98 L2 â\86\92 ð\9d\90\88â\9dªfâ\9d« → 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: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ{I} →
- â\88\80f1,f. ð\9d\90\88â¦\83f1â¦\84 → f2 ⊚ ↑f1 ≘ f → ⇩*[b,f] X ≘ K.
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⇩*[b,f2] X ≘ K.ⓘ[I] →
+ â\88\80f1,f. ð\9d\90\88â\9dªf1â\9d« → f2 ⊚ ↑f1 ≘ f → ⇩*[b,f] X ≘ 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\98 L2 â\86\92 ð\9d\90\85â¦\83fâ¦\84.
+lemma drops_fwd_isfin: â\88\80f,L1,L2. â\87©*[â\93\89,f] L1 â\89\98 L2 â\86\92 ð\9d\90\85â\9dªfâ\9d«.
#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 → ∀L. ∃K. ⇩*[Ⓕ,f] L ≘ K.
+lemma drops_isuni_ex: â\88\80f. ð\9d\90\94â\9dªfâ\9d« → ∀L. ∃K. ⇩*[Ⓕ,f] L ≘ 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\98 L2 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
- (ð\9d\90\88â¦\83fâ¦\84 ∧ L1 = L2) ∨
- 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 = ↑g.
+lemma drops_inv_isuni: â\88\80f,L1,L2. â\87©*[â\93\89,f] L1 â\89\98 L2 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
+ (ð\9d\90\88â\9dªfâ\9d« ∧ L1 = L2) ∨
+ â\88\83â\88\83g,I,K. â\87©*[â\93\89,g] K â\89\98 L2 & ð\9d\90\94â\9dªgâ\9d« & L1 = K.â\93\98[I] & f = ↑g.
#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} ≘ L2 →
- (ð\9d\90\88â¦\83fâ¦\84 â\88§ L2 = K.â\93\98{I}) ∨
- â\88\83â\88\83g. ð\9d\90\94â¦\83gâ¦\84 & ⇩*[b,g] K ≘ L2 & f = ↑g.
+lemma drops_inv_bind1_isuni: â\88\80b,f,I,K,L2. ð\9d\90\94â\9dªfâ\9d« â\86\92 â\87©*[b,f] K.â\93\98[I] ≘ L2 →
+ (ð\9d\90\88â\9dªfâ\9d« â\88§ L2 = K.â\93\98[I]) ∨
+ â\88\83â\88\83g. ð\9d\90\94â\9dªgâ\9d« & ⇩*[b,g] K ≘ L2 & f = ↑g.
#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\98 K.â\93\98{I} →
- (ð\9d\90\88â¦\83fâ¦\84 â\88§ L1 = K.â\93\98{I}) ∨
- 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 = ↑g.
+lemma drops_inv_bind2_isuni: â\88\80b,f,I,K,L1. ð\9d\90\94â\9dªfâ\9d« â\86\92 â\87©*[b,f] L1 â\89\98 K.â\93\98[I] →
+ (ð\9d\90\88â\9dªfâ\9d« â\88§ L1 = K.â\93\98[I]) ∨
+ â\88\83â\88\83g,I1,K1. ð\9d\90\94â\9dªgâ\9d« & â\87©*[b,g] K1 â\89\98 K.â\93\98[I] & L1 = K1.â\93\98[I1] & f = ↑g.
#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,â\86\91f] L1 â\89\98 K.â\93\98{I} →
- ∃∃I1,K1. ⇩*[b,f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+lemma drops_inv_bind2_isuni_next: â\88\80b,f,I,K,L1. ð\9d\90\94â\9dªfâ\9d« â\86\92 â\87©*[b,â\86\91f] L1 â\89\98 K.â\93\98[I] →
+ ∃∃I1,K1. ⇩*[b,f] K1 ≘ 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\98 L2 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
- ∀I,K. L2 = K.ⓘ{I} → ⇩*[Ⓣ,f] L1 ≘ K.ⓘ{I}.
+fact drops_inv_TF_aux: â\88\80f,L1,L2. â\87©*[â\92»,f] L1 â\89\98 L2 â\86\92 ð\9d\90\94â\9dªfâ\9d« →
+ ∀I,K. L2 = K.ⓘ[I] → ⇩*[Ⓣ,f] L1 ≘ 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: ∀f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[Ⓣ,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_TF: ∀f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[Ⓣ,f] L ≘ K.ⓘ[I].
/2 width=3 by drops_inv_TF_aux/ qed-.
(* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: ∀b,f,I,L,K. ⇩*[b,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[Ⓣ,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_gen: ∀b,f,I,L,K. ⇩*[b,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[Ⓣ,f] L ≘ K.ⓘ[I].
* /2 width=1 by drops_inv_TF/
qed-.
(* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: ∀b,f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⇩*[b,f] L ≘ K.ⓘ{I}.
+lemma drops_inv_F: ∀b,f,I,L,K. ⇩*[Ⓕ,f] L ≘ K.ⓘ[I] → 𝐔❪f❫ → ⇩*[b,f] L ≘ 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\98 K.â\93\98{I} → ⇩*[b,↑f] X ≘ K.
+lemma drops_isuni_fwd_drop2: â\88\80b,f,I,X,K. ð\9d\90\94â\9dªfâ\9d« â\86\92 â\87©*[b,f] X â\89\98 K.â\93\98[I] → ⇩*[b,↑f] X ≘ K.
/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
(* Inversion lemmas with uniform relocations ********************************)
lemma drops_inv_atom2: ∀b,L,f. ⇩*[b,f] L ≘ ⋆ →
- 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 ⊚ f1 ≘ f.
+ â\88\83â\88\83n,f1. â\87©*[b,ð\9d\90\94â\9d¨nâ\9d©] L â\89\98 â\8b\86 & ð\9d\90\94â\9d¨nâ\9d© ⊚ f1 ≘ 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: ∀L1,L2,i. ⇩*[↑i] L1 ≘ L2 →
- ∃∃I,K. ⇩*[i] K ≘ L2 & L1 = K.ⓘ{I}.
+ ∃∃I,K. ⇩*[i] K ≘ 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\98 â\8b\86 â\88¨ â\88\83â\88\83I,K. â\87©*[i] L â\89\98 K.â\93\98{I}.
+lemma drops_F_uni: â\88\80L,i. â\87©*[â\92»,ð\9d\90\94â\9d¨iâ\9d©] L â\89\98 â\8b\86 â\88¨ â\88\83â\88\83I,K. â\87©*[i] L â\89\98 K.â\93\98[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\98 L2 â\86\92 â\88\80f1,f2. f1 â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\94â¦\83f1â¦\84 →
+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 â\86\92 ð\9d\90\94â\9dªf1â\9d« →
∃∃L. ⇩*[b,f1] L1 ≘ L & ⇩*[b,f2] L ≘ L2.
#b #f #L1 #L2 #H elim H -f -L1 -L2
[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
]
qed-.
-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 â\86\92 ð\9d\90\94â¦\83f2â¦\84 →
+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 â\86\92 ð\9d\90\94â\9dªf2â\9d« →
∃∃L2. ⇩*[Ⓕ,f2] L ≘ L2 & ⇩*[Ⓕ,f] L1 ≘ L2.
#b #f1 #L1 #L #H elim H -f1 -L1 -L
[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
(* Properties with application **********************************************)
-lemma drops_tls_at: â\88\80f,i1,i2. @â¦\83i1,fâ¦\84 ≘ i2 →
+lemma drops_tls_at: â\88\80f,i1,i2. @â\9dªi1,fâ\9d« ≘ i2 →
∀b,L1,L2. ⇩*[b,⫱*[i2]f] L1 ≘ L2 →
⇩*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
-lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⇩*[b,f] L ≘ K0.ⓘ{I} → ∀i. @⦃O,f⦄ ≘ i →
- ∃∃J,K. ⇩*[i]L ≘ K.ⓘ{J} & ⇩*[b,⫱*[↑i]f] K ≘ K0 & ⇧*[⫱*[↑i]f] I ≘ J.
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⇩*[b,f] L ≘ K0.ⓘ[I] → ∀i. @❪O,f❫ ≘ i →
+ ∃∃J,K. ⇩*[i]L ≘ K.ⓘ[J] & ⇩*[b,⫱*[↑i]f] K ≘ K0 & ⇧*[⫱*[↑i]f] I ≘ 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
projects / helm.git / blobdiff