(* 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.
+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/
+ lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_yle_plus_dx/
| #p #l #m #X #HX
lapply (lift_inv_gref2 … HX) -HX //
| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
[ #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/
+ lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, ylt_plus_dx1_trans, 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
+ elim (lift_inv_lref2 … Hi) -Hi * <yplus_inj #Hil2 #H destruct
+ [ -Hl12 lapply (ylt_inv_monotonic_plus_dx … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+ | -Hil1 >yplus_comm_23 in Hil2; #H lapply ( yle_inv_monotonic_plus_dx … H) -H #H
+ elim (yle_inv_plus_inj2 … H) -H >yminus_inj #Hl2im2 #H
+ lapply (yle_inv_inj … H) -H #Hm2i
+ lapply (yle_trans … Hl12 … Hl2im2) -Hl12 #Hl1im2
>le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
- /4 width=3 by lift_lref_ge, ex2_intro/
+ /3 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/
+ <yplus_succ1 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by yle_succ, lift_bind, 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
(* 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,m2,T2. ⬆[l1 + yinj 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/
+ >(lift_inv_lref2_lt … H) -H /3 width=3 by ylt_plus_dx1_trans, lift_lref_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/
+ elim (ylt_split (i+m1) (l1+m+m2)) #H0
+ [ elim (lift_inv_lref2_be … H) -H /3 width=2 by monotonic_yle_plus, yle_inj/
| >(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/ ]
+ lapply (yle_plus2_to_minus_inj2 … H0) #Hl1m21i
+ elim (yle_inv_plus_inj2 … H0) -H0 #Hl1m12 #Hm2im1
+ @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ]
+ @lift_lref_ge_minus_eq
+ [ <yminus_inj <yplus_inj >yplus_minus_assoc_inj /2 width=1 by yle_inj/
+ | /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 (lift_inv_bind2 … H) -H #V2 #T2 #HV2 <yplus_succ1 #HT2 #H destruct
+ elim (IHV1 … HV2) -V //
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
[ #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 (ylt_yle_trans … Hl12 Hil1) -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/
+ [ @(yle_trans … Hl21) -Hl21 /2 width=1 by monotonic_yle_plus_dx/
| -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
+ 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 *)
+ lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, yle_succ/ (**) (* 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
[ #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/
+ lapply (ylt_yle_trans … (l1+m2) ? Hil1) // #Him2
+ elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct
+ /4 width=3 by monotonic_ylt_plus_dx, monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_lt, 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/
+ lapply (yle_trans … Hl21 … Hil1) -Hl21 #Hil2
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by yle_plus_dx1_trans/ #HX destruct
+ >plus_plus_comm_23 /4 width=3 by monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_ge, 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/
+ elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, yle_succ, 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 //
[ #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 (ylt_yle_trans … (l1+m1) ? Hil1) // #Hil1m
+ lapply (ylt_yle_trans … (l2-m1) ? Hil1) /2 width=1 by yle_plus1_to_minus_inj2/ #Hil2m
+ lapply (ylt_yle_trans … Hlml Hil1m) -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/
+ elim (lift_inv_lref1 … HX) -HX * <yplus_inj #Himl #HX destruct
+ [ /4 width=3 by lift_lref_lt, lift_lref_ge, ylt_plus1_to_minus_inj2, ex2_intro/
+ | /4 width=3 by lift_lref_ge, yle_plus_dx1_trans, monotonic_yle_minus_dx, 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 (yle_inv_plus_inj2 … Hlml) #Hlm #Hml
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/
+ elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by yle_succ/ -Hlml
+ #T >yminus_succ1_inj /3 width=5 by lift_bind, 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 //
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.
+ m1 ≤ m2 → ⬆[l + yinj 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 //
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