(* *)
(**************************************************************************)
+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".
-(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
(* Basic_1: includes: drop_skip_bind drop1_skip_bind *)
(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
drops c (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
+interpretation "uniform slicing (local environment)"
+ 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
+
interpretation "general slicing (local environment)"
'RDropStar c f L1 L2 = (drops c f L1 L2).
∀f2. f ⊚ f1 ≡ f2 →
∃∃L2. R f2 L1 L2 & ⬇*[c, f] L2 ≡ K2 & L1 ≡[f] L2.
+(* Basic properties *********************************************************)
+
+lemma drops_eq_repl_back: ∀L1,L2,c. eq_repl_back … (λf. ⬇*[c, f] L1 ≡ L2).
+#L1 #L2 #c #f1 #H elim H -L1 -L2 -f1
+[ /4 width=3 by drops_atom, isid_eq_repl_back/
+| #I #L1 #L2 #V #f1 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
+ /3 width=3 by drops_drop/
+| #I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H elim (eq_inv_px … H) -H
+ /3 width=3 by drops_skip, lifts_eq_repl_back/
+]
+qed-.
+
+lemma drops_eq_repl_fwd: ∀L1,L2,c. eq_repl_fwd … (λf. ⬇*[c, f] L1 ≡ L2).
+#L1 #L2 #c @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
+qed-.
+
+lemma drops_inv_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≡ i2 →
+ ∀c, L1,L2. ⬇*[c,⫱*[i2]f] L1 ≡ L2 →
+ ⬇*[c,↑⫱*[⫯i2]f] L1 ≡ L2.
+/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
+
+(* Basic_2A1: includes: drop_FT *)
+lemma drops_TF: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
+#L1 #L2 #f #H elim H -L1 -L2 -f
+/3 width=1 by drops_atom, drops_drop, drops_skip/
+qed.
+
+(* Basic_2A1: includes: drop_gen *)
+lemma drops_gen: ∀L1,L2,c,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[c, f] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_TF/
+qed-.
+
+(* Basic_2A1: includes: drop_T *)
+lemma drops_F: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_TF/
+qed-.
+
+(* Basic_2A1: includes: drop_refl *)
+lemma drops_refl: ∀c,L,f. 𝐈⦃f⦄ → ⬇*[c, f] L ≡ L.
+#c #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/
+qed.
+
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[c, f1] L1 ≡ L & ⬇*[c, f2] L ≡ L2.
+#L1 #L2 #f #c #H elim H -L1 -L2 -f
+[ #f #Hc #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
+ #H lapply (Hc H) -c
+ #H elim (after_inv_isid3 … Hf H) -f //
+| #I #L1 #L2 #V #f #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/
+ ]
+| #I #L1 #L2 #V1 #V2 #f #_ #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: ∀L1,L,f1,c. ⬇*[c, f1] L1 ≡ L → ∀f2,f. f1 ⊚ f2 ≡ f → 𝐔⦃f2⦄ →
+ ∃∃L2. ⬇*[Ⓕ, f2] L ≡ L2 & ⬇*[Ⓕ, f] L1 ≡ L2.
+#L1 #L #f1 #c #H elim H -L1 -L -f1
+[ #f1 #Hc #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
+| #I #L1 #L #V #f1 #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/
+| #I #L1 #L #V1 #V #f1 #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-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_1: includes: drop_gen_refl *)
+(* Basic_2A1: includes: drop_inv_O2 *)
+lemma drops_fwd_isid: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → 𝐈⦃f⦄ → L1 = L2.
+#L1 #L2 #c #f #H elim H -L1 -L2 -f //
+[ #I #L1 #L2 #V #f #_ #_ #H elim (isid_inv_next … H) //
+| /5 width=5 by isid_inv_push, lifts_fwd_isid, eq_f3, sym_eq/
+]
+qed-.
+
+fact drops_fwd_drop2_aux: ∀X,Y,c,f2. ⬇*[c, f2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
+#X #Y #c #f2 #H elim H -X -Y -f2
+[ #f2 #Ht2 #J #K #W #H destruct
+| #I #L1 #L2 #V #f2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL
+ /3 width=7 by after_next, ex3_2_intro, drops_drop/
+| #I #L1 #L2 #V1 #V2 #f2 #HL #_ #_ #J #K #W #H destruct
+ lapply (isid_after_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/
+]
+qed-.
+
+lemma drops_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
+/2 width=5 by drops_fwd_drop2_aux/ qed-.
+
+lemma drops_after_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
+ ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[c, f] X ≡ K.
+#I #X #K #V #c #f2 #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-.
+
+(* Basic_1: was: drop_S *)
+(* Basic_2A1: was: drop_fwd_drop2 *)
+lemma drops_isuni_fwd_drop2: ∀I,X,K,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] X ≡ K.ⓑ{I}V → ⬇*[c, ⫯f] X ≡ K.
+/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
+
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma drops_fwd_isfin: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐅⦃f⦄.
+#L1 #L2 #f #H elim H -L1 -L2 -f
+/3 width=1 by isfin_next, isfin_push, isfin_isid/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ →
∃∃K1,V1. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1.
/2 width=5 by drops_inv_skip2_aux/ qed-.
-(* Basic properties *********************************************************)
-
-lemma drops_eq_repl_back: ∀L1,L2,c. eq_repl_back … (λf. ⬇*[c, f] L1 ≡ L2).
-#L1 #L2 #c #f1 #H elim H -L1 -L2 -f1
-[ /4 width=3 by drops_atom, isid_eq_repl_back/
-| #I #L1 #L2 #V #f1 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
- /3 width=3 by drops_drop/
-| #I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H elim (eq_inv_px … H) -H
- /3 width=3 by drops_skip, lifts_eq_repl_back/
+fact drops_inv_TF_aux: ∀L1,L2,f. ⬇*[Ⓕ, f] L1 ≡ L2 → 𝐔⦃f⦄ →
+ ∀I,K,V. L2 = K.ⓑ{I}V →
+ ⬇*[Ⓣ, f] L1 ≡ K.ⓑ{I}V.
+#L1 #L2 #f #H elim H -L1 -L2 -f
+[ #f #_ #_ #J #K #W #H destruct
+| #I #L1 #L2 #V #f #_ #IH #Hf #J #K #W #H destruct
+ /4 width=3 by drops_drop, isuni_inv_next/
+| #I #L1 #L2 #V1 #V2 #f #HL12 #HV21 #_ #Hf #J #K #W #H destruct
+ lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf
+ <(drops_fwd_isid … HL12) -K // <(lifts_fwd_isid … HV21) -V1
+ /3 width=3 by drops_refl, isid_push/
]
qed-.
-lemma drops_eq_repl_fwd: ∀L1,L2,c. eq_repl_fwd … (λf. ⬇*[c, f] L1 ≡ L2).
-#L1 #L2 #c @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
-qed-.
+(* Basic_2A1: includes: drop_inv_FT *)
+lemma drops_inv_TF: ∀I,L,K,V,f. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
+ ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V.
+/2 width=3 by drops_inv_TF_aux/ qed-.
-(* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: ∀c,L,f. 𝐈⦃f⦄ → ⬇*[c, f] L ≡ L.
-#c #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/
-qed.
+(* Advanced inversion lemmas ************************************************)
-(* Basic_2A1: includes: drop_FT *)
-lemma drops_TF: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
-#L1 #L2 #f #H elim H -L1 -L2 -f
-/3 width=1 by drops_atom, drops_drop, drops_skip/
-qed.
-
-(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: ∀L1,L2,c,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[c, f] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drops_TF/
+(* Basic_2A1: includes: drop_inv_gen *)
+lemma drops_inv_gen: ∀I,L,K,V,c,f. ⬇*[c, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
+ ⬇*[Ⓣ, f] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drops_inv_TF/
qed-.
-(* Basic_2A1: includes: drop_T *)
-lemma drops_F: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drops_TF/
+(* Basic_2A1: includes: drop_inv_T *)
+lemma drops_inv_F: ∀I,L,K,V,c,f. ⬇*[Ⓕ, f] L ≡ K.ⓑ{I}V → 𝐔⦃f⦄ →
+ ⬇*[c, f] L ≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drops_inv_TF/
qed-.
-(* Basic forward lemmas *****************************************************)
+(* Inversion lemmas with test for uniformity ********************************)
-(* Basic_1: includes: drop_gen_refl *)
-(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀L1,L2,c,f. ⬇*[c, f] L1 ≡ L2 → 𝐈⦃f⦄ → L1 = L2.
-#L1 #L2 #c #f #H elim H -L1 -L2 -f //
-[ #I #L1 #L2 #V #f #_ #_ #H elim (isid_inv_next … H) //
-| /5 width=5 by isid_inv_push, lifts_fwd_isid, eq_f3, sym_eq/
+lemma drops_inv_isuni: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐔⦃f⦄ →
+ (𝐈⦃f⦄ ∧ L1 = L2) ∨
+ ∃∃I,K,V,g. ⬇*[Ⓣ, g] K ≡ L2 & 𝐔⦃g⦄ & L1 = K.ⓑ{I}V & f = ⫯g.
+#L1 #L2 #f * -L1 -L2 -f
+[ /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/
]
qed-.
-fact drops_fwd_drop2_aux: ∀X,Y,c,f2. ⬇*[c, f2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
-#X #Y #c #f2 #H elim H -X -Y -f2
-[ #f2 #Ht2 #J #K #W #H destruct
-| #I #L1 #L2 #V #f2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL
- /3 width=7 by after_next, ex3_2_intro, drops_drop/
-| #I #L1 #L2 #V1 #V2 #f2 #HL #_ #_ #J #K #W #H destruct
- lapply (isid_after_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/
+(* Basic_2A1: was: drop_inv_O1_pair1 *)
+lemma drops_inv_pair1_isuni: ∀I,K,L2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] K.ⓑ{I}V ≡ L2 →
+ (𝐈⦃f⦄ ∧ L2 = K.ⓑ{I}V) ∨
+ ∃∃g. 𝐔⦃g⦄ & ⬇*[c, g] K ≡ L2 & f = ⫯g.
+#I #K #L2 #V #c #f #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
+ /4 width=3 by isid_push, or_introl, conj/
+| lapply (drops_inv_drop1 … H) -H /3 width=4 by ex3_intro, or_intror/
]
qed-.
-lemma drops_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ⫯f1 ≡ f & ⬇*[c, f] X ≡ K.
-/2 width=5 by drops_fwd_drop2_aux/ qed-.
-
-lemma drops_after_fwd_drop2: ∀I,X,K,V,c,f2. ⬇*[c, f2] X ≡ K.ⓑ{I}V →
- ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ⫯f1 ≡ f → ⬇*[c, f] X ≡ K.
-#I #X #K #V #c #f2 #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/
+(* Basic_2A1: was: drop_inv_O1_pair2 *)
+lemma drops_inv_pair2_isuni: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, f] L1 ≡ K.ⓑ{I}V →
+ (𝐈⦃f⦄ ∧ L1 = K.ⓑ{I}V) ∨
+ ∃∃I1,K1,V1,g. 𝐔⦃g⦄ & ⬇*[c, g] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & f = ⫯g.
+#I #K #V #c #f *
+[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #Hf #H elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct /3 width=1 by or_introl, conj/
+ | /3 width=8 by ex4_4_intro, or_intror/
+ ]
+]
qed-.
-(* Basic_1: was: drop_S *)
-(* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: ∀I,X,K,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] X ≡ K.ⓑ{I}V → ⬇*[c, ⫯f] X ≡ K.
-/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
+lemma drops_inv_pair2_isuni_next: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, ⫯f] L1 ≡ K.ⓑ{I}V →
+ ∃∃I1,K1,V1. ⬇*[c, f] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1.
+#I #K #V #c #f #L1 #Hf #H elim (drops_inv_pair2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma drops_inv_succ: ∀L1,L2,l. ⬇*[⫯l] L1 ≡ L2 →
+ ∃∃I,K,V. ⬇*[l] K ≡ L2 & L1 = K.ⓑ{I}V.
+#L1 #L2 #l #H elim (drops_inv_isuni … H) -H // *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=5 by ex2_3_intro/
+]
+qed-.
-(* Basic_2A1: removed theorems 14:
+(* Basic_2A1: removed theorems 12:
drops_inv_nil drops_inv_cons d1_liftable_liftables
- drop_refl_atom_O2
- drop_inv_O1_pair1 drop_inv_pair1 drop_inv_O1_pair2
+ 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
*)
projects / helm.git / blobdiff