(* *)
(**************************************************************************)
+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".
interpretation "generic slicing (local environment)"
'RDropStar b f L1 L2 = (drops b f L1 L2).
-definition d_liftable1: relation2 lenv term → predicate bool ≝
- λR,b. ∀f,L,K. ⬇*[b, f] L ≡ K →
- ∀T,U. ⬆*[f] T ≡ U → R K T → R L U.
+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_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_liftable2_sn: 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.
-definition dropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀f2,L2. R f2 L1 L2 →
- ∀f1. f ⊚ f1 ≡ f2 →
- ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≡ K2.
+definition d_liftable2_bi: 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_bi: 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.
+
+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.
-definition dropable_dx: predicate (rtmap → relation lenv) ≝
- λR. ∀f2,L1,L2. R f2 L1 L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
- ∀f1. f ⊚ f1 ≡ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≡ K1 & R f1 K1 K2.
+definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
+ λR. ∀f2,L1,L2. R f2 L1 L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ f1 ≡ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≡ K1 & R f1 K1 K2.
-definition dedropable_sn: predicate (rtmap → 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.
+definition co_dedropable_sn: predicate (rtmap → 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.
(* Basic properties *********************************************************)
/3 width=1 by isfin_next, isfin_push, isfin_isid/
qed-.
-(* Properties with uniform relocations **************************************)
-
-lemma drops_uni_ex: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ ∨ ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V.
-#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/
-#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-.
-
-(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ →
- ∃∃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 #V #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/
- | #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
- elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
-]
-qed-.
+(* Properties with test for uniformity **************************************)
-lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ →
- ∃∃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 #V #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
- 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
- elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
- ]
-]
+lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≡ 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/
qed-.
(* Inversion lemmas with test for uniformity ********************************)
[ #H elim (isid_inv_next … H) -H //
| /2 width=5 by ex2_3_intro/
]
-qed-.
+qed-.
fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ →
∀I,K,V. L2 = K.ⓑ{I}V →
]
qed-.
+(* Properties with uniform relocations **************************************)
+
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ ∨ ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V.
+#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/
+#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-.
+
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ →
+ ∃∃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 #V #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/
+ | #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
+ elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
+]
+qed-.
+
+lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ →
+ ∃∃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 #V #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
+ 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
+ elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
(* Properties with application **********************************************)
lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≡ i2 →
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