(* *)
(**************************************************************************)
-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/lreq.ma".
+include "basic_2/relocation/lifts.ma".
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+(* GENERAL 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 (c:bool): rtmap → relation lenv ≝
+| drops_atom: ∀f. (c = Ⓣ → 𝐈⦃f⦄) → drops c (f) (⋆) (⋆)
+| drops_drop: ∀I,L1,L2,V,f. drops c f L1 L2 → drops c (⫯f) (L1.ⓑ{I}V) L2
+| drops_skip: ∀I,L1,L2,V1,V2,f.
+ drops c f L1 L2 → ⬆*[f] V2 ≡ V1 →
+ drops c (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
-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 "general slicing (local environment)"
+ 'RDropStar c f L1 L2 = (drops c f 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.
-
-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_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.
+ λR,c. ∀L,K,f. ⬇*[c, f] L ≡ K →
+ ∀T,U. ⬆*[f] T ≡ U → R K T → R L U.
+
+definition d_liftable2: predicate (lenv → relation term) ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,c,f. ⬇*[c, 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 → ∀K,c,f. ⬇*[c, f] L ≡ K →
+ ∀T1. ⬆*[f] T1 ≡ U1 →
+ ∃∃T2. ⬆*[f] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: predicate (rtmap → relation lenv) ≝
+ λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀L2,f2. R f2 L1 L2 →
+ ∀f1. f ⊚ f1 ≡ f2 →
+ ∃∃K2. R f1 K1 K2 & ⬇*[c, f] L2 ≡ K2.
+
+definition dropable_dx: predicate (rtmap → relation lenv) ≝
+ λR. ∀L1,L2,f2. R f2 L1 L2 →
+ ∀K2,c,f. ⬇*[c, f] L2 ≡ K2 → 𝐔⦃f⦄ →
+ ∀f1. f ⊚ f1 ≡ f2 →
+ ∃∃K1. ⬇*[c, f] L1 ≡ K1 & R f1 K1 K2.
+
+definition dedropable_sn: predicate (rtmap → relation lenv) ≝
+ λR. ∀L1,K1,c,f. ⬇*[c, f] L1 ≡ K1 → ∀K2,f1. R f1 K1 K2 →
+ ∀f2. f ⊚ f1 ≡ f2 →
+ ∃∃L2. R f2 L1 L2 & ⬇*[c, f] L2 ≡ K2 & L1 ≡[f] L2.
(* Basic inversion lemmas ***************************************************)
-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
+fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ →
+ Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
+#X #Y #c #f * -X -Y -f
+[ /3 width=1 by conj/
+| #I #L1 #L2 #V #f #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #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/
+(* Basic_1: includes: drop_gen_sort *)
+(* Basic_2A1: includes: drop_inv_atom1 *)
+lemma drops_inv_atom1: ∀Y,c,f. ⬇*[c, f] ⋆ ≡ Y → Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
+/2 width=3 by drops_inv_atom1_aux/ qed-.
+
+fact drops_inv_drop1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K,V,g. X = K.ⓑ{I}V → f = ⫯g →
+ ⬇*[c, g] K ≡ Y.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K #W #g #H destruct
+| #I #L1 #L2 #V #f #HL #J #K #W #g #H1 #H2 <(injective_next … H2) -g destruct //
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K #W #g #_ #H2 elim (discr_push_next … H2)
]
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/
+(* Basic_1: includes: drop_gen_drop *)
+(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
+lemma drops_inv_drop1: ∀I,K,Y,V,c,f. ⬇*[c, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[c, f] K ≡ Y.
+/2 width=7 by drops_inv_drop1_aux/ qed-.
+
+
+fact drops_inv_skip1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K1,V1,g. X = K1.ⓑ{I}V1 → f = ↑g →
+ ∃∃K2,V2. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K1 #W1 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K1 #W1 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
]
qed-.
-(* Basic properties *********************************************************)
+(* Basic_1: includes: drop_gen_skip_l *)
+(* Basic_2A1: includes: drop_inv_skip1 *)
+lemma drops_inv_skip1: ∀I,K1,V1,Y,c,f. ⬇*[c, ↑f] K1.ⓑ{I}V1 ≡ Y →
+ ∃∃K2,V2. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+/2 width=5 by drops_inv_skip1_aux/ qed-.
+
+fact drops_inv_skip2_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K2,V2,g. Y = K2.ⓑ{I}V2 → f = ↑g →
+ ∃∃K1,V1. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K2 #W2 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K2 #W2 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
-(* 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/
-].
-qed.
+(* Basic_1: includes: drop_gen_skip_r *)
+(* Basic_2A1: includes: drop_inv_skip2 *)
+lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 →
+ ∃∃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 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: ∀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-.
+
+(* 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.
-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_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_1: removed theorems 1: drop1_getl_trans *)
+(* 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 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-.
+
+(* Basic_2A1: removed theorems 14:
+ 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_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