-x /4 width=4 by llpx_sn_fwd_length, or_intror/
qed-.
-(* Properties on relocation *************************************************)
-
-lemma llpx_sn_lift_le: ∀R. d_liftable R →
- ∀K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
- ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 →
- ∀U. ⬆[l, k] T ≡ U → l0 ≤ l → llpx_sn R l0 U L1 L2.
-#R #HR #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
-[ #K1 #K2 #l0 #s #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
- * #Hli #H destruct
- [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
- /2 width=1 by llpx_sn_skip/
- | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -k -Hil0
- /3 width=3 by yle_trans, yle_inj/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
- * #Hli #H destruct [ -HK12 | -IHK12 ]
- [ elim (drop_trans_lt … HLK1 … HK11) // -K1
- elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2
- /3 width=18 by llpx_sn_lref/
- | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2
- /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
- ]
-| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
- * #Hil #H destruct
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, drop_fwd_be/
- | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -k
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
-| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
- ∀L1,L2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 →
- ∀U. ⬆[l, k] T ≡ U → l ≤ l0 → llpx_sn R (l0+k) U L1 L2.
-#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
-[ #K1 #K2 #l0 #s #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
- /2 width=1 by llpx_sn_sort/
-| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
- * #_ #H destruct
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
- [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
- | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
- ]
-| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
- * #Hil #H destruct
- [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -k -Hil0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
- lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2
- /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
- ]
-| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
- * #Hil #H destruct
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
- [ /3 width=7 by llpx_sn_free, drop_fwd_be/
- | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
- lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
- ]
-| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
- lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
- /2 width=1 by llpx_sn_gref/
-| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
- #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/
-| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
- #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R →
- ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
- ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 →
- ∀T. ⬆[l, k] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2.
-#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
-[ #L1 #L2 #l0 #s #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -k
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /2 width=1 by llpx_sn_skip/
- | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
- * #Hil #H destruct [ -HK12 | -IHK12 ]
- [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
- elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
- elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
- lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
- /3 width=10 by llpx_sn_lref/
- | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0
- elim (yle_inv_plus_inj2 … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
- lapply (drop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -k
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
-| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
- ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 →
- ∀T. ⬆[l, k] T ≡ U → l ≤ l0 → l0 ≤ l + k → llpx_sn R l T K1 K2.
-#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
-[ #L1 #L2 #l0 #s #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -k
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 #Hl0k elim (lift_inv_lref2 … H) -H
- * #Hil #H destruct
- [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/
- | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0
- /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 #Hl0k elim (lift_inv_lref2 … H) -H
- * #Hil #H destruct
- [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0k -Hil0
- /3 width=3 by ylt_yle_trans, ylt_inj/
- | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0k
- elim (yle_inv_plus_inj2 … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/
- ]
-| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 #Hl0k elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
- lapply (drop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -k
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 #Hl0k elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
-| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hl0 #Hl0k elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
-]
-qed-.
-
-lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
- ∀K1,K2,l,k. ⬇[Ⓕ, l, k] L1 ≡ K1 → ⬇[Ⓕ, l, k] L2 ≡ K2 →
- ∀T. ⬆[l, k] T ≡ U → l + k ≤ l0 → llpx_sn R (l0-k) T K1 K2.
-#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
-[ #L1 #L2 #l0 #s #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
- /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
- * #Hil #H destruct [ -Hil0 | -Hlml0 ]
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
- [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/
- | elim (yle_inv_plus_inj2 … Hil) -Hil
- /3 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx/
- ]
-| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
- * #Hil #H destruct
- [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0
- /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
- | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
- lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil
- /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
- ]
-| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
- [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
- lapply (drop_fwd_length_le4 … HLK2) -HLK2
- #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
- | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #k #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
- lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
- /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct
- @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
- <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
- @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
-| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #k #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,K2,k. ⬇[k] L1 ≡ K1 → ⬇[k] L2 ≡ K2 →
- ∀T. ⬆[0, k] T ≡ U → llpx_sn R 0 T K1 K2.
-/2 width=11 by llpx_sn_inv_lift_be/ qed-.
-
-lemma llpx_sn_drop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K1,k. ⬇[k] L1 ≡ K1 → ∀T. ⬆[0, k] T ≡ U →
- ∃∃K2. ⬇[k] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K1 #k #HLK1 #T #HTU elim (llpx_sn_fwd_drop_sn … HU … HLK1)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
-lemma llpx_sn_drop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
- ∀K2,k. ⬇[k] L2 ≡ K2 → ∀T. ⬆[0, k] T ≡ U →
- ∃∃K1. ⬇[k] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K2 #k #HLK2 #T #HTU elim (llpx_sn_fwd_drop_dx … HU … HLK2)
-/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
-qed-.
-
(* Inversion lemmas on negated lazy pointwise extension *********************)
lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀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 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_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 *********************************************************)
/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 ********************************)
]
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 →
lemma ceq_inv_lift: d_deliftable2_sn ceq.
/2 width=3 by ex2_intro/ qed-.
-(* Note: cfull_inv_lift does not hold *)
+(* Note: d_deliftable2_sn cfull does not hold *)
lemma cfull_lift: d_liftable2 cfull.
#K #T1 #T2 #_ #b #f #L #_ #U1 #_ -K -T1 -b
elim (lifts_total T2 f) /2 width=3 by ex2_intro/
(* Properties with entrywise extension of context-sensitive relations *******)
-(* Basic_2A1: includes: lpx_sn_deliftable_dropable *)
-lemma lexs_deliftable_dropable: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
- dropable_sn (lexs RN RP).
-#RN #RP #HN #HP #b #f #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #X #f2 #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
+(* Basic_2A1: includes: lpx_sn_deliftable_dropable *) (**) (* changed after commit 13218 *)
+lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
+#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
/4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #f #I #L1 #K1 #V1 #_ #IH #X #f2 #H #f1 #Hf2 elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ]
- #g2 #Hg2 #H2 destruct elim (lexs_inv_push1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
- /3 width=3 by drops_drop, ex2_intro/
-| #f #I #L1 #K1 #V1 #W1 #HLK #HWV #IH #X #f2 #H #f1 #Hf2 elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ]
- #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H
- #L2 #V2 #HL12 #HV12 #H destruct elim (IH … HL12 … Hg2) -g2
- [ elim (HP … HV12 … HLK … HWV) | elim (HN … HV12 … HLK … HWV) ] -V1
- /3 width=5 by lexs_next, lexs_push, drops_skip, ex2_intro/
+| #f #I #L1 #K1 #V1 #_ #IH #Hf #f2 #X #H #f1 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct
+ elim (lexs_inv_push1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IH … HL12 … Hg2) -g2
+ /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/
+| #f #I #L1 #K1 #V1 #W1 #HLK #HWV #IH #Hf #f2 #X #H #f1 #Hf2
+ lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+ lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct
+ lapply (lifts_fwd_isid … HWV … Hf) -HWV #H0 destruct
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2
+ lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct
+ /4 width=3 by drops_refl, lexs_next, lexs_push, isid_push, ex2_intro/
]
qed-.
(* Basic_2A1: includes: lpx_sn_liftable_dedropable *)
-lemma lexs_liftable_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2 RN → d_liftable2 RP → dedropable_sn (lexs RN RP).
+lemma lexs_liftable_co_dedropable: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+ d_liftable2 RN → d_liftable2 RP → co_dedropable_sn (lexs RN RP).
#RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
[ #f #Hf #X #f1 #H #f2 #Hf2 >(lexs_inv_atom1 … H) -X
/4 width=4 by drops_atom, lexs_atom, ex3_intro/
]
qed-.
-fact lexs_dropable_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
- ∀f2,L1. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≡ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[RN, RP, f1] K2.
+fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
+ ∀f2,L1. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≡ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[RN, RP, f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
#H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/
qed-.
(* Basic_2A1: includes: lpx_sn_dropable *)
-lemma lexs_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
-/2 width=5 by lexs_dropable_aux/ qed-.
+lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP).
+/2 width=5 by lexs_dropable_dx_aux/ qed-.
-(* Basic_2A1: includes: lpx_sn_drop_conf *)
-lemma lexs_drops_conf_next: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
+(* Basic_2A1: includes: lpx_sn_drop_conf *) (**)
+lemma lexs_drops_conf_next: ∀RN,RP.
∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
- ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
+ ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≡ f2 →
∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
-#RN #RP #HRN #HRP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX
#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma lexs_drops_conf_push: ∀RN,RP. d_deliftable2_sn RN → d_deliftable2_sn RP →
+lemma lexs_drops_conf_push: ∀RN,RP.
∀f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
- ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 →
+ ∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≡ f2 →
∃∃K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
-#RN #RP #HRN #HRP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #f1 #Hf2
-elim (lexs_deliftable_dropable … HRN HRP … HLK1 … HL12 … Hf2) -L1 -f2 -HRN -HRP
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
+elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX
#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
∀f1. f ~⊚ ⫯f1 ≡ f2 →
∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
∀f1. f ~⊚ ↑f1 ≡ f2 →
∃∃K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_dropable … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
+elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
∀f2. f ~⊚ f1 ≡ ⫯f2 →
∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+elim (lexs_liftable_co_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX
#L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
∀f2. f ~⊚ f1 ≡ ↑f2 →
∃∃L2,V2. ⬇*[b,f] L2.ⓑ{I}V2 ≡ K2 & L1 ⦻*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+elim (lexs_liftable_co_dedropable … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX
#L2 #V2 #H2L12 #HV12 #H destruct /2 width=6 by ex4_2_intro/
qed-.
(* Properties with ranged equivalence for local environments ****************)
-lemma lreq_dedropable: dedropable_sn lreq.
-@lexs_liftable_dedropable
+lemma lreq_co_dedropable: co_dedropable_sn lreq.
+@lexs_liftable_co_dedropable
/2 width=6 by cfull_lift, ceq_lift, cfull_refl, ceq_refl/
qed-.
-lemma lreq_dropable: ∀RN,RP. dropable_dx (lexs RN RP).
-@lexs_dropable qed-.
+lemma lreq_co_dropable_sn: co_dropable_sn lreq.
+@lexs_co_dropable_sn qed-.
+
+lemma lreq_co_dropable_dx: co_dropable_dx lreq.
+@lexs_co_dropable_dx qed-.
(* Basic_2A1: includes: lreq_drop_trans_be *)
lemma lreq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
lemma drops_lreq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
∀b,f,I,L1,V. ⬇*[b,f] L1.ⓑ{I}V ≡ K1 →
∀f2. f ~⊚ f1 ≡ ⫯f2 →
- ∃∃L2. ⬇*[b,f] L2.ⓑ{I}V ≡ K2 & L1 ≡[f2] L2 & L1.ⓑ{I}V≡[f]L2.ⓑ{I}V.
+ ∃∃L2. ⬇*[b,f] L2.ⓑ{I}V ≡ K2 & L1 ≡[f2] L2 & L1.ⓑ{I}V ≡[f] L2.ⓑ{I}V.
#f1 #K1 #K2 #HK12 #b #f #I #L1 #V #HLK1 #f2 #Hf2
elim (drops_lexs_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
/2 width=6 by cfull_lift, ceq_lift, cfull_refl, ceq_refl, ex3_intro/
]
qed-.
-lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (LTC … R).
-#R #HR #b #f #L1 #K1 #HLK1 #f2 #L2 #H elim H -L2
-[ #L2 #HL12 #f1 #H elim (HR … HLK1 … HL12 … H) -HR -f2 -L1
+lemma co_dropable_sn_TC: ∀R. co_dropable_sn R → co_dropable_sn (LTC … R).
+#R #HR #b #f #L1 #K1 #HLK1 #Hf #f2 #L2 #H elim H -L2
+[ #L2 #HL12 #f1 #H elim (HR … HLK1 … Hf … HL12 … H) -HR -Hf -f2 -L1
/3 width=3 by inj, ex2_intro/
| #L #L2 #_ #HL2 #IH #f1 #H elim (IH … H) -IH
#K #HK1 #HLK elim (HR … HLK … HL2 … H) -HR -f2 -L
]
qed-.
-lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (LTC … R).
+lemma co_dropable_dx_TC: ∀R. co_dropable_dx R → co_dropable_dx (LTC … R).
#R #HR #f2 #L1 #L2 #H elim H -L2
[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #f1 #Hf2 elim (HR … HL12 … HLK2 … Hf … Hf2) -HR -Hf -f2 -L2
/3 width=3 by inj, ex2_intro/
]
qed-.
-lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (LTC … R).
+lemma co_dedropable_sn_TC: ∀R. co_dedropable_sn R → co_dedropable_sn (LTC … R).
#R #HR #b #f #L1 #K1 #HLK1 #f1 #K2 #H elim H -K2
[ #K2 #HK12 #f2 #Hf elim (HR … HLK1 … HK12 … Hf) -HR -f1 -K1
/3 width=4 by inj, ex3_intro/
qed.
lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+ ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_bind, ex_intro/
qed.
lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
+ ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
#h #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_flat, ex_intro/
qed.
(* *)
(**************************************************************************)
-include "basic_2/substitution/lpx_sn_drop.ma".
+include "basic_2/static/lfxs_drops.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_transition/lfpx.ma".
(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
-(* Properties on local environment slicing ***********************************)
+(* Properties with generic slicing for local environments *******************)
-lemma lpx_drop_conf: ∀h,o,G. dropable_sn (lpx h o G).
-/3 width=6 by lpx_sn_deliftable_dropable, cpx_inv_lift1/ qed-.
+lemma drops_lfpx_trans: ∀h,G. dedropable_sn (cpx h G).
+/3 width=6 by lfxs_liftable_dedropable, cpx_lifts/ qed-.
-lemma drop_lpx_trans: ∀h,o,G. dedropable_sn (lpx h o G).
-/3 width=10 by lpx_sn_liftable_dedropable, cpx_lift/ qed-.
+(* Inversion lemmas with generic slicing for local environments *************)
-lemma lpx_drop_trans_O1: ∀h,o,G. dropable_dx (lpx h o G).
-/2 width=3 by lpx_sn_dropable/ qed-.
+lemma lfpx_drops_conf: ∀h,G. dropable_sn (cpx h G).
+/2 width=5 by lfxs_dropable_sn/ qed-.
+
+lemma lfpx_drops_trans: ∀h,G. dropable_dx (cpx h G).
+/2 width=5 by lfxs_dropable_dx/ qed-.
cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma
cpx.ma cpx_simple.ma cpx_drops.ma cpx_lsubr.ma
-lfpx.ma lfpx_length.ma lfpx_fqup.ma
+lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma
(* *)
(**************************************************************************)
+include "ground_2/relocation/nstream_coafter.ma".
include "basic_2/relocation/drops_drops.ma".
-include "basic_2/static/frees.ma".
+include "basic_2/static/frees_frees.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
]
qed-.
+(* Forward lemmas with generic slicing for local environments ***************)
+
+lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
+ ∀f,K. ⬇*[b, f] L ≡ K → ∀T. ⬆*[f] T ≡ U →
+ ∀f1. K ⊢ 𝐅*⦃T⦄ ≡ f1 → f ~⊚ f1 ≡ f2.
+/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
+
(* Inversion lemmas with generic slicing for local environments *************)
lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≡ f2 →
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/drops_ceq.ma".
+include "basic_2/relocation/drops_lexs.ma".
+include "basic_2/static/frees_fqup.ma".
+include "basic_2/static/frees_drops.ma".
+include "basic_2/static/lfxs.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 →
+ ∀K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
+ ∃∃L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
+
+definition dropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
+ ∃∃K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
+
+definition dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⦻*[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U →
+ ∃∃K1. ⬇*[b, f] L1 ≡ K1 & K1 ⦻*[R, T] K2.
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: includes: llpx_sn_lift_le llpx_sn_lift_ge *)
+lemma lfxs_liftable_dedropable: ∀R. (∀L. reflexive ? (R L)) →
+ d_liftable2 R → dedropable_sn R.
+#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
+elim (frees_total L1 U) #f2 #Hf2
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
+elim (lexs_liftable_co_dedropable … H1R … H2R … HLK1 … HK12 … Hf) -f1 -K1
+/3 width=6 by cfull_lift, ex3_intro, ex2_intro/
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: restricts: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
+(* Basic_2A1: was: llpx_sn_drop_conf_O *)
+lemma lfxs_dropable_sn: ∀R. dropable_sn R.
+#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
+elim (frees_total K1 T) #f1 #Hf1
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
+elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1
+/3 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: was: llpx_sn_drop_trans_O *)
+(* Note: the proof might be simplified *)
+lemma lfxs_dropable_dx: ∀R. dropable_dx R.
+#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
+elim (drops_isuni_ex … H1f L1) #K1 #HLK1
+elim (frees_total K1 T) #f1 #Hf1
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f
+elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2
+/4 width=9 by frees_inv_lifts, ex2_intro/
+qed-.
+
+(* Basic_2A1: was: llpx_sn_inv_lift_O *)
+lemma lfxs_inv_lift_bi: ∀R,L1,L2,U. L1 ⦻*[R, U] L2 →
+ ∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
+ ∀T. ⬆*[i] T ≡ U → K1 ⦻*[R, T] K2.
+#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU
+elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
+lapply (drops_mono … HY … HLK2) -L2 -i #H destruct //
+qed-.
]
*)
[ { "uncounted context-sensitive rt-transition" * } {
- [ "lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_fqup" * ]
+ [ "lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fqup" * ]
[ "cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_lsubr" * ]
}
]
}
]
[ { "generic extension on referred entries" * } {
- [ "lfxs ( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_fqup" + "lfxs_lfxs" * ]
+ [ "lfxs ( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_drops" + "lfxs_fqup" + "lfxs_lfxs" * ]
}
]
[ { "context-sensitive free variables" * } {
qed-.
fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_isfin2_fwd f1) →
- ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
+ ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
#i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
projects / helm.git / commitdiff