(* *)
(**************************************************************************)
-include "ground_2/relocation/rtmap_coafter.ma".
-include "static_2/notation/relations/rdropstar_3.ma".
+include "ground/xoa/ex_1_2.ma".
+include "ground/xoa/ex_4_3.ma".
+include "ground/relocation/rtmap_coafter.ma".
include "static_2/notation/relations/rdropstar_4.ma".
+include "static_2/notation/relations/rdrop_3.ma".
include "static_2/relocation/seq.ma".
include "static_2/relocation/lifts_bind.ma".
(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
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
+inductive drops (b:bool): pr_map → relation lenv ≝
+| 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)"
- 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
-
interpretation "generic slicing (local environment)"
'RDropStar b f L1 L2 = (drops b f L1 L2).
+interpretation "uniform slicing (local environment)"
+ 'RDrop i L1 L2 = (drops true (pr_uni i) L1 L2).
+
definition d_liftable1: predicate (relation2 lenv term) ≝
λR. ∀K,T. R K T → ∀b,f,L. ⇩*[b,f] L ≘ K →
∀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.
- predicate (lenv → relation C) ≝
+definition d_liftable2_sn: ∀C:Type[0]. ∀S:?→relation C.
+ predicate (lenv→relation C) ≝
λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⇩*[b,f] L ≘ 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) ≝
+definition d_deliftable2_sn: ∀C:Type[0]. ∀S:?→relation C.
+ predicate (lenv→relation C) ≝
λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⇩*[b,f] L ≘ 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) ≝
+definition d_liftable2_bi: ∀C:Type[0]. ∀S:?→relation C.
+ predicate (lenv→relation C) ≝
λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⇩*[b,f] L ≘ 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) ≝
+definition d_deliftable2_bi: ∀C:Type[0]. ∀S:?→relation C.
+ predicate (lenv→relation C) ≝
λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⇩*[b,f] L ≘ 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\98 K1 â\86\92 ð\9d\90\94â¦\83fâ¦\84 →
+definition co_dropable_sn: predicate (?→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\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) ≝
+definition co_dropable_dx: predicate (?→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 →
- ∀f1. f ~⊚ f1 ≘ f2 →
+ 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.
-definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
+definition co_dedropable_sn: predicate (?→relation lenv) ≝
λR. ∀b,f,L1,K1. ⇩*[b,f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
∀f2. f ~⊚ f1 ≘ f2 →
∃∃L2. R f2 L1 L2 & ⇩*[b,f] L2 ≘ K2 & L1 ≡[f] L2.
#f @drops_atom #H destruct
qed.
-lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⇩*[b,f] L1 ≘ L2).
+lemma drops_eq_repl_back: ∀b,L1,L2. pr_eq_repl_back … (λf. ⇩*[b,f] L1 ≘ L2).
#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
-[ /4 width=3 by drops_atom, isid_eq_repl_back/
+[ /4 width=3 by drops_atom, pr_isi_eq_repl_back/
| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
/3 width=3 by drops_drop/
| #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H elim (eq_inv_px … H) -H
]
qed-.
-lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⇩*[b,f] L1 ≘ L2).
-#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
+lemma drops_eq_repl_fwd: ∀b,L1,L2. pr_eq_repl_fwd … (λf. ⇩*[b,f] L1 ≘ L2).
+#b #L1 #L2 @pr_eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
qed-.
(* Basic_2A1: includes: drop_FT *)
(* 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
-| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J #K #_ #H2 elim (discr_push_next … H2)
+| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(eq_inv_pr_next_bi … H2) -g destruct //
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J #K #_ #H2 elim (eq_inv_pr_push_next … H2)
]
qed-.
(* 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)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (eq_inv_pr_next_push … H2)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
(* 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)
-| #f #I1 #I2 #L1 #L2 #HL #HV #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
+| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (eq_inv_pr_next_push … H2)
+| #f #I1 #I2 #L1 #L2 #HL #HV #g #J2 #K2 #H1 #H2 <(eq_inv_pr_push_bi … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
(* 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
- /3 width=7 by after_next, ex3_2_intro, drops_drop/
+ /3 width=7 by pr_after_next, ex3_2_intro, drops_drop/
| #f2 #I1 #I2 #L1 #L2 #HL #_ #_ #J #K #H destruct
- lapply (after_isid_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/
+ lapply (pr_after_isi_dx 𝐢 … f2) /3 width=9 by pr_after_push, ex3_2_intro, drops_drop/
]
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
+#L #I #IHL #f #Hf elim (pr_isi_inv_gen … Hf) -Hf
/3 width=1 by drops_skip, liftsb_refl/
qed.
(* 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/
+[ #f #I #L1 #L2 #_ #_ #H elim (pr_isi_inv_next … H) //
+| /5 width=5 by pr_isi_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/
+#g1 #g #Hg1 #Hg #HK lapply (pr_after_mono_eq … Hg … Hf ??) -Hg -Hf
+/3 width=5 by drops_eq_repl_back, pr_isi_inv_eq_repl, pr_eq_next/
qed-.
(* 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/
+/3 width=1 by pr_isf_next, pr_isf_push, pr_isf_isi/
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/
+#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/
qed-.
(* 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/
-| /7 width=6 by drops_fwd_isid, liftsb_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f2, sym_eq/
+| /4 width=7 by pr_isu_inv_next, ex4_3_intro, or_intror/
+| /7 width=6 by drops_fwd_isid, liftsb_fwd_isid, pr_isu_inv_push, pr_isi_push, or_introl, conj, eq_f2, sym_eq/
]
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.
-#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
+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 (pr_isu_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
- /4 width=3 by isid_push, or_introl, conj/
+ /4 width=3 by pr_isi_push, or_introl, conj/
| lapply (drops_inv_drop1 … H) -H /3 width=4 by ex3_intro, or_intror/
]
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}.
-#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 //
+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 pr_isu_next/ -Hf *
+[ #H elim (pr_isi_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
- /4 width=3 by drops_drop, isuni_inv_next/
+ /4 width=3 by drops_drop, pr_isu_inv_next/
| #f #I1 #I2 #L1 #L2 #HL12 #HI21 #_ #Hf #J #K #H destruct
- lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf
+ lapply (pr_isu_inv_push … Hf ??) -Hf [1,2: // ] #Hf
<(drops_fwd_isid … HL12) -K // <(liftsb_fwd_isid … HI21) -I1
- /3 width=3 by drops_refl, isid_push/
+ /3 width=3 by drops_refl, pr_isi_push/
]
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.
-/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
+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, pr_after_isu_isi_next/ 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
+[ /3 width=4 by drops_atom, pr_after_isi_sn, ex2_2_intro/
+| #L #I #IH #f #H elim (pr_map_split_tl f) * #g #H0 destruct
[ elim (drops_inv_skip1 … H) -H #J #K #_ #_ #H destruct
| lapply (drops_inv_drop1 … H) -H #HL
- elim (IH … HL) -IH -HL /3 width=8 by drops_drop, after_next, ex2_2_intro/
+ elim (IH … HL) -IH -HL /3 width=8 by drops_drop, pr_after_next, ex2_2_intro/
]
]
qed-.
-lemma drops_inv_succ: ∀L1,L2,i. ⇩*[↑i] L1 ≘ L2 →
- ∃∃I,K. ⇩*[i] K ≘ L2 & L1 = K.ⓘ{I}.
+lemma drops_inv_succ: ∀L1,L2,i. ⇩[↑i] L1 ≘ L2 →
+ ∃∃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 //
+[ #H elim (pr_isi_inv_next … H) -H //
| /2 width=4 by ex2_2_intro/
]
qed-.
(* 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
#H lapply (H0f H) -b
- #H elim (after_inv_isid3 … Hf H) -f //
-| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ #H elim (pr_after_inv_isi … Hf H) -f //
+| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (pr_after_inv_next … Hf) -Hf [1,3: * |*: // ]
[ #g1 #g2 #Hf #H1 #H2 destruct
- lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
+ lapply (pr_isu_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
elim (IHL12 … Hf) -f
- /4 width=5 by drops_drop, drops_skip, liftsb_refl, isuni_isid, ex2_intro/
+ /4 width=5 by drops_drop, drops_skip, liftsb_refl, pr_isu_isi, ex2_intro/
| #g1 #Hf #H destruct elim (IHL12 … Hf) -f
- /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
+ /3 width=5 by ex2_intro, drops_drop, pr_isu_inv_next/
]
-| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (pr_after_inv_push … Hf) -Hf [2,3: // ]
#g1 #g2 #Hf #H1 #H2 destruct elim (liftsb_split_trans … HI21 … Hf) -HI21
- elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
+ elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, pr_isu_fwd_push/
]
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
-| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (pr_after_inv_next_sn … Hf) -Hf [2,3: // ]
#g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/
| #f1 #I1 #I #L1 #L #HL1 #HI1 #IH #f2 #f #Hf #Hf2
- elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+ elim (pr_after_inv_push_sn … Hf) -Hf [1,3: * |*: // ]
#g2 #g #Hg #H2 #H0 destruct
- [ lapply (isuni_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH
- lapply (after_isid_inv_dx … Hg … Hg2) -Hg #Hg
- /5 width=7 by drops_eq_repl_back, drops_F, drops_refl, drops_skip, liftsb_eq_repl_back, isid_push, ex2_intro/
- | lapply (isuni_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HI1
+ [ lapply (pr_isu_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH
+ lapply (pr_after_isi_inv_dx … Hg … Hg2) -Hg #Hg
+ /5 width=7 by drops_eq_repl_back, drops_F, drops_refl, drops_skip, liftsb_eq_repl_back, pr_isi_push, ex2_intro/
+ | lapply (pr_isu_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HI1
elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
]
]
(* Properties with application **********************************************)
-lemma drops_tls_at: â\88\80f,i1,i2. @â¦\83i1,fâ¦\84 ≘ i2 →
- â\88\80b,L1,L2. â\87©*[b,⫱*[i2]f] L1 ≘ L2 →
- â\87©*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
-/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
+lemma drops_tls_at: â\88\80f,i1,i2. @â\9dªi1,fâ\9d« ≘ i2 →
+ â\88\80b,L1,L2. â\87©*[b,â«°*[i2]f] L1 ≘ L2 →
+ â\87©*[b,⫯⫰*[↑i2]f] L1 ≘ L2.
+/3 width=3 by drops_eq_repl_fwd, pr_pat_inv_succ_dx_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
+elim (drops_split_trans … H) -H [ |5: @(pr_after_nat_uni … Hf) |2,3: skip ] /2 width=1 by pr_after_isi_dx/ #Y #HLY #H
lapply (drops_tls_at … Hf … H) -H #H
elim (drops_inv_skip2 … H) -H #J #K #HK0 #HIJ #H destruct
/3 width=5 by drops_inv_gen, ex3_2_intro/
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