(**************************************************************************)
include "basic_2/relocation/lifts_lifts.ma".
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_weight.ma".
-(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Main properties on generic relocation ************************************)
+
+lemma d_liftable2_sn_bi: ∀R. d_liftable2_sn R → d_liftable2_bi R.
+#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
+<(lifts_mono … HTX … HTU2) -T2 -U2 -b -f //
+qed-.
+
+lemma d_deliftable2_sn_bi: ∀R. d_deliftable2_sn R → d_deliftable2_bi R.
+#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
+<(lifts_inj … HUX … HTU2) -U2 -T2 -b -f //
+qed-.
(* Main properties **********************************************************)
(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
- ∀L2,s2,t. ⬇*[s2, t] L1 ≡ L2 →
- ∀t2. t1 ⊚ t2 ≡ t → ⬇*[s2, t2] L ≡ L2.
-#L1 #L #s1 #t1 #H elim H -L1 -L -t1
-[ #t1 #_ #L2 #s2 #t #H #t2 #Ht12 elim (drops_inv_atom1 … H) -s1 -H
- #H #Ht destruct @drops_atom
- #H elim (after_inv_isid3 … Ht12) -Ht12 /2 width=1 by/
-| #I #K1 #K #V1 #t1 #_ #IH #L2 #s2 #t #H12 #t2 #Ht elim (after_inv_false1 … Ht) -Ht
- #u #H #Hu destruct /3 width=3 by drops_inv_drop1/
-| #I #K1 #K #V1 #V #t1 #_ #HV1 #IH #L2 #s2 #t #H #t2 #Ht elim (after_inv_true1 … Ht) -Ht
- #u2 #u * #H1 #H2 #Hu destruct
- [ elim (drops_inv_skip1 … H) -H /3 width=6 by drops_skip, lifts_div/
+theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≡ L →
+ ∀b2,f,L2. ⬇*[b2, f] L1 ≡ L2 →
+ ∀f2. f1 ⊚ f2 ≡ f → ⬇*[b2, f2] L ≡ L2.
+#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
+ #H #Hf destruct @drops_atom
+ #H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
+| #f1 #I #K1 #K #V1 #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+ #g #Hg #H destruct /3 width=3 by drops_inv_drop1/
+| #f1 #I #K1 #K #V1 #V #_ #HV1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
+ #g2 #g #Hf #H1 #H2 destruct
+ [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_div3/
| /4 width=3 by drops_inv_drop1, drops_drop/
]
]
(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
drops_drop_trans
*)
-theorem drops_trans: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
- ∀L2,s2,t2. ⬇*[s2, t2] L ≡ L2 →
- ∀t. t1 ⊚ t2 ≡ t → ⬇*[s1∨s2, t] L1 ≡ L2.
-#L1 #L #s1 #t1 #H elim H -L1 -L -t1
-[ #t1 #Ht1 #L2 #s2 #t2 #H #t #Ht elim (drops_inv_atom1 … H) -H
- #H #Ht2 destruct @drops_atom #H elim (orb_false_r … H) -H
- #H1 #H2 >(after_isid_inv_sn … Ht) -Ht /2 width=1 by/
-| #I #K1 #K #V1 #t1 #_ #IH #L #s2 #t2 #HKL #t #Ht elim (after_inv_false1 … Ht) -Ht
+theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≡ L →
+ ∀b2,f2,L2. ⬇*[b2, f2] L ≡ L2 →
+ ∀f. f1 ⊚ f2 ≡ f → ⬇*[b1∧b2, f] L1 ≡ L2.
+#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
+ #H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
+ #H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
+ /3 width=3 by isid_eq_repl_back/
+| #f1 #I #K1 #K #V1 #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
/3 width=3 by drops_drop/
-| #I #K1 #K #V1 #V #t1 #_ #HV1 #IH #L #s2 #t2 #H #t #Ht elim (after_inv_true1 … Ht) -Ht
- #u2 #u * #H1 #H2 #Hu destruct
- [ elim (drops_inv_skip1 … H) -H /3 width=6 by drops_skip, lifts_trans/
+| #f1 #I #K1 #K #V1 #V #_ #HV1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+ #g2 #g #Hg #H1 #H2 destruct
+ [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_trans/
| /4 width=3 by drops_inv_drop1, drops_drop/
]
]
qed-.
+theorem drops_conf_div: ∀f1,L,K. ⬇*[Ⓣ,f1] L ≡ K → ∀f2. ⬇*[Ⓣ,f2] L ≡ K →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≗ f2.
+#f1 #L #K #H elim H -f1 -L -K
+[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
+ /3 width=1 by isid_inv_eq_repl/
+| #f1 #I #L #K #V #Hf1 #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
+ lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
+ #H destruct elim (drops_inv_x_pair_xy … Hf1)
+ | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
+ ]
+| #f1 #I #L #K #V #W #Hf1 #_ #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
+ /4 width=5 by isuni_fwd_push, eq_push/
+ | lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
+ lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
+ #H destruct elim (drops_inv_x_pair_xy … Hg2)
+ ]
+]
+qed-.
+
(* Advanced properties ******************************************************)
(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: ∀L,L1,s1,t. ⬇*[s1, t] L ≡ L1 →
- ∀L2,s2. ⬇*[s2, t] L ≡ L2 → L1 = L2.
-#L #L1 #s1 #t elim (isid_after_dx t)
+lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≡ L1 →
+ ∀b2,L2. ⬇*[b2, f] L ≡ L2 → L1 = L2.
+#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
/3 width=8 by drops_conf, drops_fwd_isid/
qed-.
(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: ∀L,L2,s2,t. ⬇*[s2, t] L ≡ L2 →
- ∀I,K1,V1,s1,t1. ⬇*[s1, t1] L ≡ K1.ⓑ{I}V1 →
- ∀t2. t1 ⊚ Ⓣ@t2 ≡ t →
+lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≡ L2 →
+ ∀b1,f1,I,K1,V1. ⬇*[b1, f1] L ≡ K1.ⓑ{I}V1 →
+ ∀f2. f1 ⊚ ↑f2 ≡ f →
∃∃K2,V2. L2 = K2.ⓑ{I}V2 &
- ⬇*[s2, t2] K1 ≡ K2 & ⬆*[t2] V2 ≡ V1.
-#L #L2 #s2 #t #H2 #I #K1 #V1 #s1 #t1 #H1 #t2 #Ht lapply (drops_conf … H1 … H2 … Ht) -L -Ht
+ ⬇*[b2, f2] K1 ≡ K2 & ⬆*[f2] V2 ≡ V1.
+#b2 #f #L #L2 #H2 #b1 #f1 #I #K1 #V1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
qed-.
(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
- ∀I,K2,V2,s2,t2. ⬇*[s2, t2] L ≡ K2.ⓑ{I}V2 →
- ∀t. t1 ⊚ t2 ≡ Ⓣ@t →
+lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≡ L →
+ ∀b2,f2,I,K2,V2. ⬇*[b2, f2] L ≡ K2.ⓑ{I}V2 →
+ ∀f. f1 ⊚ f2 ≡ ↑f →
∃∃K1,V1. L1 = K1.ⓑ{I}V1 &
- ⬇*[s1∨s2, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1.
-#L1 #L #s1 #t1 #H1 #I #K2 #V2 #s2 #t2 #H2 #t #Ht
-lapply (drops_trans … H1 … H2 … Ht) -L -Ht
+ ⬇*[b1∧b2, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1.
+#b1 #f1 #L1 #L #H1 #b2 #f2 #I #K2 #V2 #H2 #f #Hf
+lapply (drops_trans … H1 … H2 … Hf) -L -Hf
#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
qed-.
+
+(* Basic_2A1: includes: drops_conf_div *)
+lemma drops_conf_div_pair: ∀f1,f2,I1,I2,L,K,V1,V2.
+ ⬇*[Ⓣ,f1] L ≡ K.ⓑ{I1}V1 → ⬇*[Ⓣ,f2] L ≡ K.ⓑ{I2}V2 →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → ∧∧ f1 ≗ f2 & I1 = I2 & V1 = V2.
+#f1 #f2 #I1 #I2 #L #K #V1 #V2 #Hf1 #Hf2 #HU1 #HU2
+lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
+lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
+lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
+lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
+lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
+lapply (drops_mono … H0 … Hf2) -L #H
+destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≡ ⋆ → ∀I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V → ⊥.
+#L #i #H1 #I #K #V #H2
+lapply (drops_F … H2) -H2 #H2
+lapply (drops_mono … H2 … H1) -L -i #H destruct
+qed-.
projects / helm.git / blobdiff