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/lifts.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: ∀f,I,L1,L2,V. drops b f L1 L2 → drops b (⫯f) (L1.ⓑ{I}V) L2
-| drops_skip: ∀f,I,L1,L2,V1,V2.
- drops b f L1 L2 → ⬆*[f] V2 ≡ V1 →
- drops b (â\86\91f) (L1.â\93\91{I}V1) (L2.â\93\91{I}V2)
+| 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.â\93\98{I1}) (L2.â\93\98{I2})
.
interpretation "uniform slicing (local environment)"
'RDropStar b f L1 L2 = (drops b f 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_liftable2: predicate (lenv → relation term) ≝
- λR. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≡ K →
- ∀U1. ⬆*[f] T1 ≡ U1 →
- ∃∃U2. ⬆*[f] T2 ≡ U2 & R L U1 U2.
-
-definition d_deliftable2_sn: predicate (lenv → relation term) ≝
- λR. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≡ K →
- ∀T1. ⬆*[f] T1 ≡ U1 →
- ∃∃T2. ⬆*[f] T2 ≡ U2 & R K T1 T2.
+ 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. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ ∀U. ⬆*[f] T ≘ U → R L U.
+
+definition d_deliftable1: predicate (relation2 lenv term) ≝
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ ∀T. ⬆*[f] T ≘ U → R K T.
+
+definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ ∀T. ⬆*[f] T ≘ U → R K T.
+
+definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → 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) ≝
+ 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) ≝
+ 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) ≝
+ 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. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
- ∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≡ f2 →
- ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≡ K2.
-
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀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¡ 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 #V #_ #IH #f2 #H elim (eq_inv_nx … H) -H
+| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
/3 width=3 by drops_drop/
-| #f1 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H elim (eq_inv_px … H) -H
- /3 width=3 by drops_skip, lifts_eq_repl_back/
+| #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H elim (eq_inv_px … H) -H
+ /3 width=3 by drops_skip, liftsb_eq_repl_back/
]
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-.
+lemma d_liftable2_sn_bi: ∀C,S. (∀f,c. is_mono … (S f c)) →
+ ∀R. d_liftable2_sn C S R → d_liftable2_bi C S R.
+#C #S #HS #R #HR #K #T1 #T2 #HT12 #b #f #L #HLK #U1 #HTU1 #U2 #HTU2
+elim (HR … HT12 … HLK … HTU1) -HR -K -T1 #X #HTX #HUX
+<(HS … HTX … HTU2) -T2 -U2 -b -f //
+qed-.
+
+lemma d_deliftable2_sn_bi: ∀C,S. (∀f. is_inj2 … (S f)) →
+ ∀R. d_deliftable2_sn C S R → d_deliftable2_bi C S R.
+#C #S #HS #R #HR #L #U1 #U2 #HU12 #b #f #K #HLK #T1 #HTU1 #T2 #HTU2
+elim (HR … HU12 … HLK … HTU1) -HR -L -U1 #X #HUX #HTX
+<(HS … HUX … HTU2) -U2 -T2 -b -f //
+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/
-| #f #I #L1 #L2 #V #_ #H destruct
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
+| #f #I #L1 #L2 #_ #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #H destruct
]
qed-.
(* 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,V. X = K.â\93\91{I}V 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 #W #H destruct
-| #f #I #L1 #L2 #V #HL #g #J #K #W #H1 #H2 <(injective_next … H2) -g destruct //
-| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K #W #_ #H2 elim (discr_push_next … H2)
+[ #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)
]
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,V. ⬇*[b, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[b, f] K ≡ Y.
-/2 width=7 by drops_inv_drop1_aux/ qed-.
+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: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y â\86\92 â\88\80g,I,K1,V1. X = K1.â\93\91{I}V1 â\86\92 f = â\86\91g →
- ∃∃K2,V2. ⬇*[b, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+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 →
+ ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
#b #f #X #Y * -f -X -Y
-[ #f #Hf #g #J #K1 #W1 #H destruct
-| #f #I #L1 #L2 #V #_ #g #J #K1 #W1 #_ #H2 elim (discr_next_push … H2)
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_push … H2) -g destruct
+[ #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
/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,I,K1,V1,Y. ⬇*[b, ↑f] K1.ⓑ{I}V1 ≡ Y →
- ∃∃K2,V2. ⬇*[b, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+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: â\88\80b,f,X,Y. â¬\87*[b, f] X â\89¡ Y â\86\92 â\88\80g,I,K2,V2. Y = K2.â\93\91{I}V2 â\86\92 f = â\86\91g →
- ∃∃K1,V1. ⬇*[b, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+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 →
+ ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
#b #f #X #Y * -f -X -Y
-[ #f #Hf #g #J #K2 #W2 #H destruct
-| #f #I #L1 #L2 #V #_ #g #J #K2 #W2 #_ #H2 elim (discr_next_push … H2)
-| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_push … H2) -g destruct
+[ #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
/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,I,X,K2,V2. ⬇*[b, ↑f] X ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[b, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+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: â\88\80b,f2,X,Y. â¬\87*[b, f2] X â\89¡ Y â\86\92 â\88\80I,K,V. Y = K.â\93\91{I}V →
- 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 â\86\92 â\88\80I,K. Y = K.â\93\98{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 #W #H destruct
-| #f2 #I #L1 #L2 #V #_ #IHL #J #K #W #H elim (IHL … H) -IHL
+[ #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/
-| #f2 #I #L1 #L2 #V1 #V2 #HL #_ #_ #J #K #W #H destruct
+| #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/
]
qed-.
-lemma drops_fwd_drop2: ∀b,f2,I,X,K,V. ⬇*[b, f2] X ≡ K.ⓑ{I}V →
- 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.
-/2 width=5 by drops_fwd_drop2_aux/ 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 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 #V #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
-/3 width=1 by drops_skip, lifts_refl/
+#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
+/3 width=1 by drops_skip, liftsb_refl/
qed.
(* Forward lemmas test for identity *****************************************)
(* 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 #V #_ #_ #H elim (isid_inv_next … H) //
-| /5 width=5 by isid_inv_push, lifts_fwd_isid, eq_f3, sym_eq/
+[ #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,V. ⬇*[b, f2] X ≡ K.ⓑ{I}V →
- ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[b, f] X ≡ K.
-#b #f2 #I #X #K #V #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
+ ∀f1,f. 𝐈⦃f1⦄ → 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/
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¡ 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 * /2 width=2 by ex_intro/
-#L #I #V destruct
-elim (IH L) -IH /3 width=2 by drops_drop, 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¡ 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) ∨
- ∃∃g,I,K,V. ⬇*[Ⓣ, g] K ≡ L2 & 𝐔⦃g⦄ & L1 = K.ⓑ{I}V & f = ⫯g.
+ ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
#f #L1 #L2 * -f -L1 -L2
[ /4 width=1 by or_introl, conj/
-| /4 width=8 by isuni_inv_next, ex4_4_intro, or_intror/
-| /7 width=6 by drops_fwd_isid, lifts_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f3, sym_eq/
+| /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/
]
qed-.
(* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_pair1_isuni: ∀b,f,I,K,L2,V. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓑ{I}V ≡ L2 →
- (ð\9d\90\88â¦\83fâ¦\84 â\88§ L2 = K.â\93\91{I}V) ∨
- â\88\83â\88\83g. ð\9d\90\94â¦\83gâ¦\84 & â¬\87*[b, g] K â\89¡ L2 & f = ⫯g.
-#b #f #I #K #L2 #V #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
-[ lapply (drops_inv_skip1 … H) -H * #Y #X #HY #HX #H destruct
- <(drops_fwd_isid … HY Hg) -Y >(lifts_fwd_isid … HX Hg) -X
+lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{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 & 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
/4 width=3 by isid_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_pair2_isuni: ∀b,f,I,K,V,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≡ K.ⓑ{I}V →
- (ð\9d\90\88â¦\83fâ¦\84 â\88§ L1 = K.â\93\91{I}V) ∨
- ∃∃g,I1,K1,V1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & f = ⫯g.
-#b #f #I #K #V *
+lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
+ (ð\9d\90\88â¦\83fâ¦\84 â\88§ L1 = K.â\93\98{I}) ∨
+ ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
+#b #f #I #K *
[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
-| #L1 #I1 #V1 #Hf #H elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
[ #Hf #H destruct /3 width=1 by or_introl, conj/
- | /3 width=8 by ex4_4_intro, or_intror/
+ | /3 width=7 by ex4_3_intro, or_intror/
]
]
qed-.
-lemma drops_inv_pair2_isuni_next: ∀b,f,I,K,V,L1. 𝐔⦃f⦄ → ⬇*[b, ⫯f] L1 ≡ K.ⓑ{I}V →
- ∃∃I1,K1,V1. ⬇*[b, f] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1.
-#b #f #I #K #V #L1 #Hf #H elim (drops_inv_pair2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
+lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{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=5 by ex2_3_intro/
+| /2 width=4 by ex2_2_intro/
]
qed-.
-fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ →
- ∀I,K,V. L2 = K.ⓑ{I}V →
- ⬇*[Ⓣ, f] L1 ≡ K.ⓑ{I}V.
+fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}.
#f #L1 #L2 #H elim H -f -L1 -L2
-[ #f #_ #_ #J #K #W #H destruct
-| #f #I #L1 #L2 #V #_ #IH #Hf #J #K #W #H destruct
+[ #f #_ #_ #J #K #H destruct
+| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
/4 width=3 by drops_drop, isuni_inv_next/
-| #f #I #L1 #L2 #V1 #V2 #HL12 #HV21 #_ #Hf #J #K #W #H destruct
+| #f #I1 #I2 #L1 #L2 #HL12 #HI21 #_ #Hf #J #K #H destruct
lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf
- <(drops_fwd_isid … HL12) -K // <(lifts_fwd_isid … HV21) -V1
+ <(drops_fwd_isid … HL12) -K // <(liftsb_fwd_isid … HI21) -I1
/3 width=3 by drops_refl, isid_push/
]
qed-.
(* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: ∀f,I,L,K,V. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
- ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V.
+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,V. ⬇*[b, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
- ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V.
+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,V. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
- ⬇*[b, f] L ≡ K.ⓑ{I}V.
+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: ∀b,f,I,X,K,V. 𝐔⦃f⦄ → ⬇*[b, f] X ≡ K.ⓑ{I}V → ⬇*[b, ⫯f] X ≡ K.
+lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{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: 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 #V #IH #f #H elim (pn_split f) * #g #H0 destruct
- [ elim (drops_inv_skip1 … H) -H #K #W #_ #_ #H destruct
+| #L #I #IH #f #H elim (pn_split 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/
]
]
qed-.
-lemma drops_inv_succ: ∀l,L1,L2. ⬇*[⫯l] L1 ≡ L2 →
- ∃∃I,K,V. ⬇*[l] K ≡ L2 & L1 = K.ⓑ{I}V.
-#l #L1 #L2 #H elim (drops_inv_isuni … H) -H // *
+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 //
-| /2 width=5 by ex2_3_intro/
+| /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¡ â\8b\86 â\88¨ â\88\83â\88\83I,K,V. â¬\87*[i] L â\89¡ K.â\93\91{I}V.
+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}.
#L elim L -L /2 width=1 by or_introl/
-#L #I #V #IH * /4 width=4 by drops_refl, ex1_3_intro, or_intror/
+#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/
-* /4 width=4 by drops_drop, ex1_3_intro, or_intror/
-qed-.
+* /4 width=3 by drops_drop, ex1_2_intro, or_intror/
+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
#H elim (after_inv_isid3 … Hf H) -f //
-| #f #I #L1 #L2 #V #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
[ #g1 #g2 #Hf #H1 #H2 destruct
lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
elim (IHL12 … Hf) -f
- /4 width=5 by drops_drop, drops_skip, lifts_refl, isuni_isid, ex2_intro/
+ /4 width=5 by drops_drop, drops_skip, liftsb_refl, isuni_isid, ex2_intro/
| #g1 #Hf #H destruct elim (IHL12 … Hf) -f
/3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
]
-| #f #I #L1 #L2 #V1 #V2 #_ #HV21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
- #g1 #g2 #Hf #H1 #H2 destruct elim (lifts_split_trans … HV21 … Hf) -HV21
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … 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/
]
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 #V #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 (after_inv_nxx … Hf) -Hf [2,3: // ]
#g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/
-| #f1 #I #L1 #L #V1 #V #HL1 #HV1 #IH #f2 #f #Hf #Hf2
+| #f1 #I1 #I #L1 #L #HL1 #HI1 #IH #f2 #f #Hf #Hf2
elim (after_inv_pxx … 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, lifts_eq_repl_back, isid_push, ex2_intro/
- | lapply (isuni_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HV1
+ /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
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 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_pair2: ∀b,f,I,L,K0,V. ⬇*[b, f] L ≡ K0.ⓑ{I}V → ∀n. @⦃O, f⦄ ≡ n →
- ∃∃K,W. ⬇*[n]L ≡ K.ⓑ{I}W & ⬇*[b, ⫱*[⫯n]f] K ≡ K0 & ⬆*[⫱*[⫯n]f] V ≡ W.
-#b #f #I #L #K0 #V #H #n #Hf
+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
-elim (drops_inv_skip2 … H) -H #K #W #HK0 #HVW #H destruct
+elim (drops_inv_skip2 … H) -H #J #K #HK0 #HIJ #H destruct
/3 width=5 by drops_inv_gen, ex3_2_intro/
qed-.
+(* Properties with context-sensitive equivalence for terms ******************)
+
+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_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 *)
+lemma cfull_lift_sn: d_liftable2_sn … liftsb cfull.
+#K #I1 #I2 #_ #b #f #L #_ #J1 #_ -K -I1 -b
+elim (liftsb_total I2 f) /2 width=3 by ex2_intro/
+qed-.
+
(* Basic_2A1: removed theorems 12:
drops_inv_nil drops_inv_cons d1_liftable_liftables
drop_refl_atom_O2 drop_inv_pair1
projects / helm.git / blobdiff