(* *)
(**************************************************************************)
+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/substitution/drop.ma".
-include "basic_2/multiple/mr2_minus.ma".
-include "basic_2/multiple/lifts_vector.ma".
+include "basic_2/relocation/seq.ma".
+include "basic_2/relocation/lifts_bind.ma".
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-inductive drops (s:bool): list2 ynat nat → relation lenv ≝
-| drops_nil : ∀L. drops s (◊) L L
-| drops_cons: ∀L1,L,L2,cs,l,m.
- drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2
+(* Basic_1: includes: drop_skip_bind drop1_skip_bind *)
+(* 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: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → 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})
.
-interpretation "iterated slicing (local environment) abstract"
- 'RDropStar s cs T1 T2 = (drops s cs T1 T2).
-(*
-interpretation "iterated slicing (local environment) general"
- 'RDropStar des T1 T2 = (drops true des T1 T2).
-*)
+interpretation "uniform slicing (local environment)"
+ 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
-definition d_liftable1: relation2 lenv term → predicate bool ≝
- λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → R L U.
+interpretation "generic slicing (local environment)"
+ 'RDropStar b f L1 L2 = (drops b f L1 L2).
-definition d_liftables1: relation2 lenv term → predicate bool ≝
- λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
- ∀T,U. ⬆*[cs] 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_liftables1_all: relation2 lenv term → predicate bool ≝
- λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
- ∀Ts,Us. ⬆*[cs] Ts ≡ Us →
- all … (R K) Ts → all … (R L) Us.
+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.
-(* Basic inversion lemmas ***************************************************)
+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.
-fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2.
-#L1 #L2 #s #cs * -L1 -L2 -cs //
-#L1 #L #L2 #l #m #cs #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: drop1_gen_pnil *)
-lemma drops_inv_nil: ∀L1,L2,s. ⬇*[s, ◊] L1 ≡ L2 → L1 = L2.
-/2 width=4 by drops_inv_nil_aux/ qed-.
-
-fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
- ∀l,m,tl. cs = {l, m} @ tl →
- ∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
-#L1 #L2 #s #cs * -L1 -L2 -cs
-[ #L #l #m #tl #H destruct
-| #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
- /2 width=3 by ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
- ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
-/2 width=3 by drops_inv_cons_aux/ qed-.
-
-lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 →
- ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
- ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 &
- ⬇*[s, cs1] K1 ≡ K2 &
- ⬆*[cs1] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#I #s #cs #cs2 #i #H elim H -cs -cs2 -i
-[ #i #L1 #K2 #V2 #H
- >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (drops_inv_cons … H) -H #L #HL1 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ #K #V <yminus_succ2 #HK2 #HV2 #H destruct
- elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
- >pluss_SO2 >pluss_SO2
- >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ <yminus_succ >commutative_plus (**) (* <yminus_succ1_inj does not work *)
- /3 width=1 by minuss_lt, ylt_succ/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- /4 width=7 by minuss_ge, yle_succ, ex4_3_intro/
-]
-qed-.
+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.
+
+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 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 *********************************************************)
-(* Basic_1: was: drop1_skip_bind *)
-lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 →
- ∀I. ⬇*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
-#L1 #L2 #s #cs #H elim H -L1 -L2 -cs
-[ #L #V1 #V2 #HV12 #I
- >(lifts_inv_nil … HV12) -HV12 //
-| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I
- elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
-].
+lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆.
+#f @drops_atom #H destruct
qed.
-lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s.
-#R #s #HR #L #K #cs #H elim H -L -K -cs
-[ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=10 by/
+lemma drops_eq_repl_back: ∀b,L1,L2. 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/
+| #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
+ /3 width=3 by drops_skip, liftsb_eq_repl_back/
]
+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 *)
+qed-.
+
+(* Basic_2A1: includes: drop_FT *)
+lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+#f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
-lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s.
-#R #s #HR #L #K #cs #HLK #Ts #Us #H elim H -Ts -Us normalize //
-#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
+(* Basic_2A1: includes: drop_gen *)
+lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2.
+* /2 width=1 by drops_TF/
+qed-.
+
+(* Basic_2A1: includes: drop_T *)
+lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ 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: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ →
+ Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+#b #f #X #Y * -f -X -Y
+[ /3 width=1 by conj/
+| #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: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+/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 →
+ ⬇*[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)
+]
+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.
+/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}.
+#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
+ /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}.
+/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}.
+#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
+ /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}.
+/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} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & 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/
+| #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. ⬇*[b, f2] X ≘ K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & 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: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[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/
qed.
-(* Basic_1: removed theorems 1: drop1_getl_trans *)
+(* Forward lemmas test for identity *****************************************)
+
+(* Basic_1: includes: drop_gen_refl *)
+(* Basic_2A1: includes: drop_inv_O2 *)
+lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ 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: ∀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: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ 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: ∀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 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: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ (𝐈⦃f⦄ ∧ L1 = L2) ∨
+ ∃∃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=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_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 →
+ (𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
+ ∃∃g. 𝐔⦃g⦄ & ⬇*[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
+ /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_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
+ (𝐈⦃f⦄ ∧ L1 = K.ⓘ{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 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct /3 width=1 by or_introl, conj/
+ | /3 width=7 by ex4_3_intro, or_intror/
+ ]
+]
+qed-.
+
+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=4 by ex2_2_intro/
+]
+qed-.
+
+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 #H destruct
+| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
+ /4 width=3 by drops_drop, isuni_inv_next/
+| #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 // <(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. ⬇*[Ⓕ, 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}.
+* /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}.
+* /2 width=1 by drops_inv_TF/
+qed-.
+
+(* Forward lemmas with test for uniformity **********************************)
+
+(* Basic_1: was: drop_S *)
+(* Basic_2A1: was: drop_fwd_drop2 *)
+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: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ →
+ ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ 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
+ [ 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: ∀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=4 by ex2_2_intro/
+]
+qed-.
+
+(* Properties with uniform relocations **************************************)
+
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ 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/
+* /4 width=3 by drops_drop, ex1_2_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 #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, 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 #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: ∀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 #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 #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, 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/
+ ]
+]
+qed-.
+
+(* Properties with application **********************************************)
+
+lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≘ 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.
+#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 #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
+ drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
+ drop_fwd_length_minus2 drop_fwd_length_minus4
+*)
+(* Basic_1: removed theorems 53:
+ drop1_gen_pnil drop1_gen_pcons drop1_getl_trans
+ drop_ctail drop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ drop_clear drop_clear_O drop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+ getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+ drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+ getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
projects / helm.git / blobdiff