(* *)
(**************************************************************************)
-include "Basic-2/substitution/lift.ma".
+include "Basic_2/substitution/lift.ma".
-(* RELOCATION ***************************************************************)
+(* BASIC TERM RELOCATION ****************************************************)
(* Main properies ***********************************************************)
-(* Basic-1: was: lift_inj *)
-theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
-#d #e #T1 #U #H elim H -H d e T1 U
+(* Basic_1: was: lift_inj *)
+theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
+#d #e #T1 #U #H elim H -d -e -T1 -U
[ #k #d #e #X #HX
lapply (lift_inv_sort2 … HX) -HX //
-| #i #d #e #Hid #X #HX
+| #i #d #e #Hid #X #HX
lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref2_ge … HX ?) -HX /2/
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/
+| #p #d #e #X #HX
+ lapply (lift_inv_gref2 … HX) -HX //
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
]
-qed.
+qed-.
-(* Basic-1: was: lift_gen_lift *)
-theorem lift_div_le: â\88\80d1,e1,T1,T. â\86\91[d1, e1] T1 ≡ T →
- â\88\80d2,e2,T2. â\86\91[d2 + e1, e2] T2 ≡ T →
+(* Basic_1: was: lift_gen_lift *)
+theorem lift_div_le: â\88\80d1,e1,T1,T. â\87§[d1, e1] T1 ≡ T →
+ â\88\80d2,e2,T2. â\87§[d2 + e1, e2] T2 ≡ T →
d1 ≤ d2 →
- â\88\83â\88\83T0. â\86\91[d1, e1] T0 â\89¡ T2 & â\86\91[d2, e2] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+ â\88\83â\88\83T0. â\87§[d1, e1] T0 â\89¡ T2 & â\87§[d2, e2] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /2 width=3/ /3 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
- [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
- | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
- @(ex2_1_intro … #(i - e2))
- [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
- | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
- ]
+ elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
+ [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
+ | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
+ elim (le_inv_plus_l … H) -H #Hide2 #He2i
+ lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
+ >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
+ /4 width=3/
]
+| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/
| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
- elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
+ 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 ?) /2 width=1/ /3 width=5/
| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
- elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- elim (IHU … HU2 ?) /3 width = 5/
+ 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/
+]
+qed.
+
+(* Note: apparently this was missing in Basic_1 *)
+theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
+ ∀e,e2,T2. ⇧[d1 + e, e2] T2 ≡ T →
+ e ≤ e1 → e1 ≤ e + e2 →
+ ∃∃T0. ⇧[d1, e] T0 ≡ T2 & ⇧[d1, e + e2 - e1] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+ elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+ [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/
+ | >(lift_inv_lref2_ge … H ?) -H //
+ lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
+ elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
+ @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12
+ @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ]
+ ]
+| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ 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 // -He1 -He1e2 /3 width=5/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+ elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ elim (IHV1 … HV2 ? ?) -V //
+ elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
]
qed.
-theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
-#d #e #T #U1 #H elim H -H d e T U1
+theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 → U1 = U2.
+#d #e #T #U1 #H elim H -d -e -T -U1
[ #k #d #e #X #HX
lapply (lift_inv_sort1 … HX) -HX //
| #i #d #e #Hid #X #HX
lapply (lift_inv_lref1_lt … HX ?) -HX //
| #i #d #e #Hdi #X #HX
lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #p #d #e #X #HX
+ lapply (lift_inv_gref1 … HX) -HX //
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+ elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
]
-qed.
+qed-.
-(* Basic-1: was: lift_free (left to right) *)
-theorem lift_trans_be: â\88\80d1,e1,T1,T. â\86\91[d1, e1] T1 ≡ T →
- â\88\80d2,e2,T2. â\86\91[d2, e2] T ≡ T2 →
- d1 â\89¤ d2 â\86\92 d2 â\89¤ d1 + e1 â\86\92 â\86\91[d1, e1 + e2] T1 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+(* Basic_1: was: lift_free (left to right) *)
+theorem lift_trans_be: â\88\80d1,e1,T1,T. â\87§[d1, e1] T1 ≡ T →
+ â\88\80d2,e2,T2. â\87§[d2, e2] T ≡ T2 →
+ d1 â\89¤ d2 â\86\92 d2 â\89¤ d1 + e1 â\86\92 â\87§[d1, e1 + e2] T1 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
>(lift_inv_sort1 … HT2) -HT2 //
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
+ lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(transitive_le … Hd21 ?) -Hd21 /2/
- | -Hd21 /2/
+ [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
+ | -Hd21 /2 width=1/
]
+| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_gref1 … HT2) -HT2 //
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2/
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2 width=1/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2/
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
+ lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) // /2 width=1/
]
qed.
-(* Basic-1: was: lift_d (right to left) *)
-theorem lift_trans_le: â\88\80d1,e1,T1,T. â\86\91[d1, e1] T1 ≡ T →
- â\88\80d2,e2,T2. â\86\91[d2, e2] T ≡ T2 → d2 ≤ d1 →
- â\88\83â\88\83T0. â\86\91[d2, e2] T1 â\89¡ T0 & â\86\91[d1 + e2, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+(* Basic_1: was: lift_d (right to left) *)
+theorem lift_trans_le: â\88\80d1,e1,T1,T. â\87§[d1, e1] T1 ≡ T →
+ â\88\80d2,e2,T2. â\87§[d2, e2] T ≡ T2 → d2 ≤ d1 →
+ â\88\83â\88\83T0. â\87§[d2, e2] T1 â\89¡ T0 & â\87§[d1 + e2, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2/
+ >(lift_inv_sort1 … HX) -HX /2 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
- elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
+ elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /3 width=3/ /4 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
- >plus_plus_comm_23 /4/
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
+ >plus_plus_comm_23 /4 width=3/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ 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/ /3 width=5/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ 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/
]
qed.
-(* Basic-1: was: lift_d (left to right) *)
-theorem lift_trans_ge: â\88\80d1,e1,T1,T. â\86\91[d1, e1] T1 ≡ T →
- â\88\80d2,e2,T2. â\86\91[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
- â\88\83â\88\83T0. â\86\91[d2 - e1, e2] T1 â\89¡ T0 & â\86\91[d1, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+(* Basic_1: was: lift_d (left to right) *)
+theorem lift_trans_ge: â\88\80d1,e1,T1,T. â\87§[d1, e1] T1 ≡ T →
+ â\88\80d2,e2,T2. â\87§[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
+ â\88\83â\88\83T0. â\87§[d2 - e1, e2] T1 â\89¡ T0 & â\87§[d1, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2/
+ >(lift_inv_sort1 … HX) -HX /2 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
- lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
- lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
+ lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1/ #Hid2e
+ lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
- [2: >plus_plus_comm_23] /4/
+ elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_gref1 … HX) -HX /2 width=3/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
- <plus_minus /3 width=5/
+ 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/ #T
+ <plus_minus /2 width=2/ /3 width=5/
| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+ 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/
]
qed.
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