--- /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_2A/substitution/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Main properties ***********************************************************)
+
+(* Basic_1: was: lift_inj *)
+theorem lift_inj: ∀l,m,T1,U. ⬆[l,m] T1 ≡ U → ∀T2. ⬆[l,m] T2 ≡ U → T1 = T2.
+#l #m #T1 #U #H elim H -l -m -T1 -U
+[ #k #l #m #X #HX
+ lapply (lift_inv_sort2 … HX) -HX //
+| #i #l #m #Hil #X #HX
+ lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #l #m #Hli #X #HX
+ lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_le_plus_l/
+| #p #l #m #X #HX
+ lapply (lift_inv_gref2 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_gen_lift *)
+theorem lift_div_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+ ∀l2,m2,T2. ⬆[l2 + m1, m2] T2 ≡ T →
+ l1 ≤ l2 →
+ ∃∃T0. ⬆[l1, m1] T0 ≡ T2 & ⬆[l2, m2] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+ lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+ elim (lift_inv_lref2 … Hi) -Hi * #Hil2 #H destruct
+ [ -Hl12 lapply (lt_plus_to_lt_l … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+ | -Hil1 >plus_plus_comm_23 in Hil2; #H lapply (le_plus_to_le_r … H) -H #H
+ elim (le_inv_plus_l … H) -H #Hilm2 #Hm2i
+ lapply (transitive_le … Hl12 Hilm2) -Hl12 #Hl12
+ >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
+ /4 width=3 by lift_lref_ge, ex2_intro/
+ ]
+| #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
+| #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
+ lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+ elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+ >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
+ lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
+ elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Note: apparently this was missing in basic_1 *)
+theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+ ∀m,m2,T2. ⬆[l1 + m, m2] T2 ≡ T →
+ m ≤ m1 → m1 ≤ m + m2 →
+ ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+ >(lift_inv_lref2_lt … H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+ elim (lt_or_ge (i+m1) (l1+m+m2)) #Him1l1m2
+ [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
+ | >(lift_inv_lref2_ge … H ?) -H //
+ lapply (le_plus_to_minus … Him1l1m2) #Hl1m21i
+ elim (le_inv_plus_l … Him1l1m2) -Him1l1m2 #Hl1m12 #Hm2im1
+ @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ] -Hl1m12
+ @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
+ ]
+| #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
+ elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2) -V // >plus_plus_comm_23 in HT2; #HT2
+ elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
+| #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
+ elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2) -V //
+ elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
+#l #m #T #U1 #H elim H -l -m -T -U1
+[ #k #l #m #X #HX
+ lapply (lift_inv_sort1 … HX) -HX //
+| #i #l #m #Hil #X #HX
+ lapply (lift_inv_lref1_lt … HX ?) -HX //
+| #i #l #m #Hli #X #HX
+ lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #p #l #m #X #HX
+ lapply (lift_inv_gref1 … HX) -HX //
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
+]
+qed-.
+
+(* Basic_1: was: lift_free (left to right) *)
+theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+ ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
+ l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
+ >(lift_inv_sort1 … HT2) -HT2 //
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
+ lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2
+ lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2
+ [ @(transitive_le … Hl21 ?) -Hl21 /2 width=1 by monotonic_le_plus_l/
+ | -Hl21 /2 width=1 by lift_lref_ge/
+ ]
+| #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
+ >(lift_inv_gref1 … HT2) -HT2 //
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
+]
+qed.
+
+(* Basic_1: was: lift_d (right to left) *)
+theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+ ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
+ ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+ lapply (lt_to_le_to_lt … (l1+m2) Hil1 ?) // #Him2
+ elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
+ lapply (transitive_le … Hl21 Hil1) -Hl21 #Hil2
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
+ >plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Basic_1: was: lift_d (left to right) *)
+theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
+ ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
+ ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
+ lapply (lt_to_le_to_lt … (l1+m1) Hil1 ?) // #Hil1m
+ lapply (lt_to_le_to_lt … (l2-m1) Hil1 ?) /2 width=1 by le_plus_to_minus_r/ #Hil2m
+ lapply (lt_to_le_to_lt … Hil1m Hlml) -Hil1m -Hlml #Hil2
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+ elim (lift_inv_lref1 … HX) -HX * #Himl #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
+ <plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
+]
+qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
+ ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
+#T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
+elim (lift_total T1 0 m2) #T0 #HT10
+elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
+lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
+qed.
+
+lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
+ m1 ≤ m2 → ⬆[l + m1, m2 - m1] T1 ≡ T2.
+#T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
+elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
+>(lift_mono … H … HT1) -T //
+qed.
projects / helm.git / blobdiff