(* BASIC TERM RELOCATION ****************************************************)
-(* Main properies ***********************************************************)
+(* Main properties ***********************************************************)
(* Basic_1: was: lift_inj *)
theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U → T1 = T2.
| #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 width=1/
+ lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_le_plus_l/
| #p #d #e #X #HX
lapply (lift_inv_gref2 … HX) -HX //
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
]
qed-.
∃∃T0. ⇧[d1, e1] T0 ≡ T2 & ⇧[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 /3 width=3/
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
| #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 /2 width=3/ /3 width=3/
+ 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 #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
- [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
+ [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
| -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/
+ >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %);
+ /4 width=3 by lift_lref_ge, ex2_intro/
]
| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/
+ lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
| #a #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
- elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /3 width=5/
+ 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 #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
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/
+ elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
]
qed.
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/
+[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
+ >(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 #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/
+ [ elim (lift_inv_lref2_be … H) -H /2 width=1 by le_plus/
| >(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/ ]
+ @ex2_intro [2: /2 width=1/ | skip ] -Hd1e12
+ @lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
]
-| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
+| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
| #a #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/
+ 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 #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/
+ elim (IHV1 … HV2) -V //
+ elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
]
qed.
#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
+| #i #d #e #Hid #X #HX
lapply (lift_inv_lref1_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #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 //
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+ elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
]
qed-.
>(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 width=1/
+ lapply (lift_inv_lref1_lt … HT2 Hid2) /2 width=1 by lift_lref_lt/
| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1/
- | -Hd21 /2 width=1/
+ [ @(transitive_le … Hd21 ?) -Hd21 /2 width=1 by monotonic_le_plus_l/
+ | -Hd21 /2 width=1 by lift_lref_ge/
]
| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
>(lift_inv_gref1 … HT2) -HT2 //
| #a #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
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1/
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
| #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
lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) // /2 width=1/
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
]
qed.
∃∃T0. ⇧[d2, e2] T1 ≡ T0 & ⇧[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 width=3/
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
| #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 /3 width=3/ /4 width=3/
+ elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3/ #HX destruct
- >plus_plus_comm_23 /4 width=3/
+ 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 #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
| #a #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
- elim (IHV12 … HV20 ?) -IHV12 -HV20 //
- elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/
+ 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 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
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/
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
]
qed.
∃∃T0. ⇧[d2 - e1, e2] T1 ≡ T0 & ⇧[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 width=3/
+ >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
| #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 width=1/ #Hid2e
+ lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2 width=1 by le_plus_to_minus_r/ #Hid2e
lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e -Hded #Hid2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3/
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
+ elim (lift_inv_lref1 … HX) -HX * #Hied #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 #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3/
+ >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
| #a #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
- elim (IHV12 … HV20 ?) -IHV12 -HV20 //
- elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T
- <plus_minus /2 width=2/ /3 width=5/
+ 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 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
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/
+ elim (IHV12 … HV20) -IHV12 -HV20 //
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
]
qed.
∃∃T0. ⇧[0, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T2 ≡ T0.
#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
elim (lift_total T1 0 e2) #T0 #HT10
-elim (lift_trans_le … HT1 … HT10 ?) -HT1 // #X #HTX #HT20
-lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3/
+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,d,e1. ⇧[d, e1] T ≡ T1 → ∀T2,e2. ⇧[d, e2] T ≡ T2 →
e1 ≤ e2 → ⇧[d + e1, e2 - e1] T1 ≡ T2.
#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
-elim (lift_split … HT2 (d+e1) e1 ? ? ?) -HT2 // #X #H
+elim (lift_split … HT2 (d+e1) e1) -HT2 // #X #H
>(lift_mono … H … HT1) -T //
qed.
projects / helm.git / blobdiff