(* Main properties **********************************************************)
(* Basic_1: was: drop_mono *)
-theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
- ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
-#d #e #L #L1 #H elim H -d -e -L -L1
-[ #d #L2 #H elim (ldrop_inv_atom1 … H) -H //
-| #K #I #V #L2 #HL12 <(ldrop_inv_O2 … HL12) -L2 //
-| #L #K #I #V #e #_ #IHLK #L2 #H
- lapply (ldrop_inv_ldrop1 … H) -H /2 width=1 by/
-| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
+theorem ldrop_mono: ∀L,L1,s1,d,e. ⇩[s1, d, e] L ≡ L1 →
+ ∀L2,s2. ⇩[s2, d, e] L ≡ L2 → L1 = L2.
+#L #L1 #s1 #d #e #H elim H -L -L1 -d -e
+[ #d #e #He #L2 #s2 #H elim (ldrop_inv_atom1 … H) -H //
+| #I #K #V #L2 #s2 #HL12 <(ldrop_inv_O2 … HL12) -L2 //
+| #I #L #K #V #e #_ #IHLK #L2 #s2 #H
+ lapply (ldrop_inv_drop1 … H) -H /2 width=2 by/
+| #I #L #K1 #T #V1 #d #e #_ #HVT1 #IHLK1 #X #s2 #H
elim (ldrop_inv_skip1 … H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
>(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
>(IHLK1 … HLK2) -IHLK1 -HLK2 //
qed-.
(* Basic_1: was: drop_conf_ge *)
-theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
- ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
- ⇩[0, e2 - e1] L1 ≡ L2.
-#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1 //
-[ #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
- lapply (ldrop_inv_ldrop1_lt … H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
+theorem ldrop_conf_ge: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
+ ⇩[s2, 0, e2 - e1] L1 ≡ L2.
+#L #L1 #s1 #d1 #e1 #H elim H -L -L1 -d1 -e1 //
+[ #d #e #_ #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He destruct
+ @ldrop_atom #H >He // (**) (* explicit constructor *)
+| #I #L #K #V #e #_ #IHLK #L2 #s2 #e2 #H #He2
+ lapply (ldrop_inv_drop1_lt … H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
<minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
-| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
+| #I #L #K #V1 #V2 #d #e #_ #_ #IHLK #L2 #s2 #e2 #H #Hdee2
lapply (transitive_le 1 … Hdee2) // #He2
- lapply (ldrop_inv_ldrop1_lt … H ?) -H // -He2 #HL2
+ lapply (ldrop_inv_drop1_lt … H ?) -H // -He2 #HL2
lapply (transitive_le (1 + e) … Hdee2) // #Hee2
- @ldrop_ldrop_lt >minus_minus_comm /3 width=1 by O, lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
+ @ldrop_drop_lt >minus_minus_comm /3 width=1 by lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
]
qed.
(* Note: apparently this was missing in basic_1 *)
-theorem ldrop_conf_be: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
- ∃∃L. ⇩[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #L2 #e2 #H #Hd1 #_ elim (ldrop_inv_atom1 … H) -H #H1 #H2 destruct
- <(le_n_O_to_eq … Hd1) -d1 /2 width=3 by ldrop_atom, ex2_intro/
-| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+theorem ldrop_conf_be: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
+ ∀L2,e2. ⇩[e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+ ∃∃L. ⇩[s1, 0, d1 + e1 - e2] L2 ≡ L & ⇩[d1] L1 ≡ L.
+#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #He1 #L2 #e2 #H #Hd1 #_ elim (ldrop_inv_atom1 … H) -H #H #He2 destruct
+ >(He2 ?) in Hd1; // -He2 #Hd1 <(le_n_O_to_eq … Hd1) -d1
+ /4 width=3 by ldrop_atom, ex2_intro/
+| normalize #I #L #V #L2 #e2 #HL2 #_ #He2
lapply (le_n_O_to_eq … He2) -He2 #H destruct
lapply (ldrop_inv_O2 … HL2) -HL2 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| normalize #L0 #K0 #I #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
+| normalize #I #L0 #K0 #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
lapply (ldrop_inv_O1_pair1 … H) -H * * #He2 #HL20
- [ -IHLK0 -He21 destruct <minus_n_O /3 width=3 by ldrop_ldrop, ex2_intro/
+ [ -IHLK0 -He21 destruct <minus_n_O /3 width=3 by ldrop_drop, ex2_intro/
| -HLK0 <minus_le_minus_minus_comm //
elim (IHLK0 … HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
]
-| #L0 #K0 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
+| #I #L0 #K0 #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
elim (le_inv_plus_l … Hd1e2) #_ #He2
<minus_le_minus_minus_comm //
- lapply (ldrop_inv_ldrop1_lt … H ?) -H // #HL02
- elim (IHLK0 … HL02) -L0 /3 width=3 by ldrop_ldrop, monotonic_pred, ex2_intro/
+ lapply (ldrop_inv_drop1_lt … H ?) -H // #HL02
+ elim (IHLK0 … HL02) -L0 /3 width=3 by ldrop_drop, monotonic_pred, ex2_intro/
]
-qed.
+qed-.
(* Note: apparently this was missing in basic_1 *)
-theorem ldrop_conf_le: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
- ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
- ∃∃L. ⇩[0, e2] L1 ≡ L & ⇩[d1 - e2, e1] L2 ≡ L.
-#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #L2 #e2 #H
- elim (ldrop_inv_atom1 … H) -H #H destruct /2 width=3 by ldrop_atom, ex2_intro/
-| #K0 #I #V0 #L2 #e2 #H1 #H2
+theorem ldrop_conf_le: ∀L0,L1,s1,d1,e1. ⇩[s1, d1, e1] L0 ≡ L1 →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L0 ≡ L2 → e2 ≤ d1 →
+ ∃∃L. ⇩[s2, 0, e2] L1 ≡ L & ⇩[s1, d1 - e2, e1] L2 ≡ L.
+#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #He1 #L2 #s2 #e2 #H elim (ldrop_inv_atom1 … H) -H
+ #H #He2 #_ destruct /4 width=3 by ldrop_atom, ex2_intro/
+| #I #K0 #V0 #L2 #s2 #e2 #H1 #H2
lapply (le_n_O_to_eq … H2) -H2 #H destruct
lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| #K0 #K1 #I #V0 #e1 #HK01 #_ #L2 #e2 #H1 #H2
+| #I #K0 #K1 #V0 #e1 #HK01 #_ #L2 #s2 #e2 #H1 #H2
lapply (le_n_O_to_eq … H2) -H2 #H destruct
- lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by ldrop_ldrop, ex2_intro/
-| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #e2 #H #He2d1
+ lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by ldrop_drop, ex2_intro/
+| #I #K0 #K1 #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #s2 #e2 #H #He2d1
elim (ldrop_inv_O1_pair1 … H) -H *
[ -IHK01 -He2d1 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
| -HK01 -HV10 #He2 #HK0L2
elim (IHK01 … HK0L2) -IHK01 -HK0L2 /2 width=1 by monotonic_pred/
- >minus_le_minus_minus_comm /3 width=3 by ldrop_ldrop_lt, ex2_intro/
+ >minus_le_minus_minus_comm /3 width=3 by ldrop_drop_lt, ex2_intro/
]
]
-qed.
+qed-.
+(* Note: with "s2", the conclusion parameter is "s1 ∨ s2" *)
(* Basic_1: was: drop_trans_ge *)
-theorem ldrop_trans_ge: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
- ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L //
-[ /3 width=1/
-| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
+theorem ldrop_trans_ge: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀L2,e2. ⇩[e2] L ≡ L2 → d1 ≤ e2 → ⇩[s1, 0, e1 + e2] L1 ≡ L2.
+#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
+[ #d1 #e1 #He1 #L2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He2 destruct /4 width=1 by ldrop_atom, eq_f2/
+| /2 width=1 by ldrop_gen/
+| /3 width=1 by ldrop_drop/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 #L #e2 #H #Hde2
lapply (lt_to_le_to_lt 0 … Hde2) // #He2
lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
- lapply (ldrop_inv_ldrop1_lt … H ?) -H // #HL2
- @ldrop_ldrop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
+ lapply (ldrop_inv_drop1_lt … H ?) -H // #HL2
+ @ldrop_drop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
]
qed.
(* Basic_1: was: drop_trans_le *)
-theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
- ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 →
- ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
-[ #d #e2 #L2 #H
- elim (ldrop_inv_atom1 … H) -H /2 width=3 by ldrop_atom, ex2_intro/
-| #K #I #V #e2 #L2 #HL2 #H
- lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3 by ldrop_pair, ex2_intro/
-| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
- lapply (le_n_O_to_eq … H) -H #H destruct
- elim (IHL12 … HL2) -IHL12 -HL2 // #L0 #H #HL0
- lapply (ldrop_inv_O2 … H) -H #H destruct /3 width=5 by ldrop_pair, ldrop_ldrop, ex2_intro/
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
+theorem ldrop_trans_le: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀L2,s2,e2. ⇩[s2, 0, e2] L ≡ L2 → e2 ≤ d1 →
+ ∃∃L0. ⇩[s2, 0, e2] L1 ≡ L0 & ⇩[s1, d1 - e2, e1] L0 ≡ L2.
+#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
+[ #d1 #e1 #He1 #L2 #s2 #e2 #H #_ elim (ldrop_inv_atom1 … H) -H
+ #H #He2 destruct /4 width=3 by ldrop_atom, ex2_intro/
+| #I #K #V #L2 #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+ #H destruct /2 width=3 by ldrop_pair, ex2_intro/
+| #I #L1 #L2 #V #e #_ #IHL12 #L #s2 #e2 #HL2 #H lapply (le_n_O_to_eq … H) -H
+ #H destruct elim (IHL12 … HL2) -IHL12 -HL2 //
+ #L0 #H #HL0 lapply (ldrop_inv_O2 … H) -H #H destruct
+ /3 width=5 by ldrop_pair, ldrop_drop, ex2_intro/
+| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV12 #IHL12 #L #s2 #e2 #H #He2d
elim (ldrop_inv_O1_pair1 … H) -H *
[ -He2d -IHL12 #H1 #H2 destruct /3 width=5 by ldrop_pair, ldrop_skip, ex2_intro/
| -HL12 -HV12 #He2 #HL2
- elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by ldrop_ldrop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
+ elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by ldrop_drop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
]
]
-qed.
-
-(* Basic_1: was: drop_conf_rev *)
-axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
- ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
+qed-.
(* Advanced properties ******************************************************)
lemma l_liftable_llstar: ∀R. l_liftable R → ∀l. l_liftable (llstar … R l).
#R #HR #l #K #T1 #T2 #H @(lstar_ind_r … l T2 H) -l -T2
-[ #L #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
+[ #L #s #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
>(lift_mono … HTU2 … HTU1) -T1 -U2 -d -e //
-| #l #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T d e) /3 width=11 by lstar_dx/
+| #l #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /3 width=12 by lstar_dx/
]
-qed.
+qed-.
(* Basic_1: was: drop_conf_lt *)
-lemma ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
- ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
+lemma ldrop_conf_lt: ∀L,L1,s1,d1,e1. ⇩[s1, d1, e1] L ≡ L1 →
+ ∀I,K2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ K2.ⓑ{I}V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 &
- ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
-#d1 #e1 #L #L1 #H1 #e2 #K2 #I #V2 #H2 #He2d1
+ ∃∃K1,V1. ⇩[s2, 0, e2] L1 ≡ K1.ⓑ{I}V1 &
+ ⇩[s1, d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
+#L #L1 #s1 #d1 #e1 #H1 #I #K2 #V2 #s2 #e2 #H2 #He2d1
elim (ldrop_conf_le … H1 … H2) -L /2 width=2 by lt_to_le/ #K #HL1K #HK2
-elim (ldrop_inv_skip1 … HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/ #K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed.
+elim (ldrop_inv_skip1 … HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/
+#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
(* Note: apparently this was missing in basic_1 *)
-lemma ldrop_trans_lt: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
- ∀e2,L2,I,V2. ⇩[0, e2] L ≡ L2.ⓑ{I}V2 →
+lemma ldrop_trans_lt: ∀L1,L,s1,d1,e1. ⇩[s1, d1, e1] L1 ≡ L →
+ ∀I,L2,V2,s2,e2. ⇩[s2, 0, e2] L ≡ L2.ⓑ{I}V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃L0,V0. ⇩[0, e2] L1 ≡ L0.ⓑ{I}V0 &
- ⇩[d, e1] L0 ≡ L2 & ⇧[d, e1] V2 ≡ V0.
-#d1 #e1 #L1 #L #HL1 #e2 #L2 #I #V2 #HL2 #Hd21
+ ∃∃L0,V0. ⇩[s2, 0, e2] L1 ≡ L0.ⓑ{I}V0 &
+ ⇩[s1, d, e1] L0 ≡ L2 & ⇧[d, e1] V2 ≡ V0.
+#L1 #L #s1 #d1 #e1 #HL1 #I #L2 #V2 #s2 #e2 #HL2 #Hd21
elim (ldrop_trans_le … HL1 … HL2) -L /2 width=1 by lt_to_le/ #L0 #HL10 #HL02
elim (ldrop_inv_skip2 … HL02) -HL02 /2 width=1 by lt_plus_to_minus_r/ #L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
- ⇩[d1, e1] L1 ≡ L → ⇩[0, e2] L ≡ L2 → d1 ≤ e2 →
- ⇩[0, e2 + e1] L1 ≡ L2.
-#e1 #e1 #e2 >commutative_plus /2 width=5 by ldrop_trans_ge/
+lemma ldrop_trans_ge_comm: ∀L1,L,L2,s1,d1,e1,e2.
+ ⇩[s1, d1, e1] L1 ≡ L → ⇩[e2] L ≡ L2 → d1 ≤ e2 →
+ ⇩[s1, 0, e2 + e1] L1 ≡ L2.
+#L1 #L #L2 #s1 #d1 #e1 #e2
+>commutative_plus /2 width=5 by ldrop_trans_ge/
qed.
-lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[0, e1] L ≡ K. ⓑ{I1} V1 →
- ∀I2,V2,e2. ⇩[0, e2] L ≡ K. ⓑ{I2} V2 →
+lemma ldrop_conf_div: ∀I1,L,K,V1,e1. ⇩[e1] L ≡ K.ⓑ{I1}V1 →
+ ∀I2,V2,e2. ⇩[e2] L ≡ K.ⓑ{I2}V2 →
∧∧ e1 = e2 & I1 = I2 & V1 = V2.
#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2
elim (le_or_ge e1 e2) #He
(* Advanced forward lemmas **************************************************)
-lemma ldrop_fwd_be: ∀L,K,d,e,i. ⇩[d, e] L ≡ K → |K| ≤ i → i < d → |L| ≤ i.
-#L #K #d #e #i #HLK #HK #Hd elim (lt_or_ge i (|L|)) //
+lemma ldrop_fwd_be: ∀L,K,s,d,e,i. ⇩[s, d, e] L ≡ K → |K| ≤ i → i < d → |L| ≤ i.
+#L #K #s #d #e #i #HLK #HK #Hd elim (lt_or_ge i (|L|)) //
#HL elim (ldrop_O1_lt … HL) #I #K0 #V #HLK0 -HL
elim (ldrop_conf_lt … HLK … HLK0) // -HLK -HLK0 -Hd
#K1 #V1 #HK1 #_ #_ lapply (ldrop_fwd_length_lt2 … HK1) -I -K1 -V1
projects / helm.git / blobdiff