+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⋕ ⋕ break [ term 46 T ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyEqAlt $T $L1 $L2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⋕ ⋕ break [ term 46 d , break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyEqAlt $d $T $L1 $L2 }.
(* *)
(**************************************************************************)
+include "ground_2/ynat/ynat_max.ma".
include "basic_2/notation/relations/extpsubst_6.ma".
include "basic_2/grammar/genv.ma".
include "basic_2/grammar/cl_shift.ma".
(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
(* activate genv *)
-inductive cpy: nat → nat → relation4 genv lenv term term ≝
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
| cpy_atom : ∀I,G,L,d,e. cpy d e G L (⓪{I}) (⓪{I})
-| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i + 1] V ≡ W → cpy d e G L (#i) W
+| cpy_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d+e →
+ ⇩[0, i] L ≡ K.ⓑ{I}V → ⇧[0, i+1] V ≡ W → cpy d e G L (#i) W
| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
- cpy d e G L V1 V2 → cpy (d + 1) e G (L.ⓑ{I}V2) T1 T2 →
+ cpy d e G L V1 V2 → cpy (⫯d) e G (L.ⓑ{I}V2) T1 T2 →
cpy d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
| cpy_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
cpy d e G L V1 V2 → cpy d e G L T1 T2 →
destruct
elim (lift_total V 0 (i+1)) #W #HVW
elim (lift_split … HVW i i)
- /3 width=5 by cpy_subst, le_n, ex2_2_intro/
+ /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
[ elim (IHU1 (L.ⓑ{J}W2) (d+1)) -IHU1
lemma cpy_weak: ∀G,L,T1,T2,d1,e1. ⦃G, L⦄ ⊢ T1 ▶×[d1, e1] T2 →
∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
⦃G, L⦄ ⊢ T1 ▶×[d2, e2] T2.
-#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1
-[ //
-| /3 width=5 by cpy_subst, transitive_le/
-| /4 width=3 by cpy_bind, le_to_lt_to_lt, le_S_S/
+#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
| /3 width=1 by cpy_flat/
]
qed-.
lemma cpy_weak_top: ∀G,L,T1,T2,d,e.
⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶×[d, |L| - d] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
+[ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
lapply (ldrop_fwd_length_lt2 … HLK)
- /3 width=5 by cpy_subst, lt_to_le_to_lt/
-| normalize /2 width=1 by cpy_bind/
+ /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
+| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
+ /2 width=1 by cpy_bind/
| /2 width=1 by cpy_flat/
]
qed-.
/2 width=2 by cpy_weak_top/
qed-.
-lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i - d] T & ⦃G, L⦄ ⊢ T ▶×[i, d + e - i] T2.
+lemma cpy_split_up: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[d, i-d] T & ⦃G, L⦄ ⊢ T ▶×[i, d+e-i] T2.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j) [ -Hide -Hjde | -Hdi -Hdj ]
- [ >(plus_minus_m_m j d) in ⊢ (%→?);
- /3 width=5 by cpy_subst, ex2_intro/
- | #Hij lapply (plus_minus_m_m … Hjde) -Hjde
- /3 width=9 by cpy_atom, cpy_subst, ex2_intro/
- ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
- -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V) ?) -HT1
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
]
qed-.
-lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 →
- ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d + e - i] T &
- ⦃G, L⦄ ⊢ T ▶×[d, i - d] T2.
+lemma cpy_split_down: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀i. i ≤ d + e →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶×[i, d+e-i] T & ⦃G, L⦄ ⊢ T ▶×[d, i-d] T2.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hdj #Hjde
- elim (lt_or_ge i j)
- [ -Hide -Hjde >(plus_minus_m_m j d) in ⊢ (% → ?); // -Hdj /3 width=9 by ex2_intro, cpy_atom, cpy_subst/
- | -Hdi -Hdj
- >(plus_minus_m_m (d+e) j) in Hide; // -Hjde /3 width=5 by ex2_intro, cpy_subst/
- ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i + 1)) -IHT12 /2 width=1 by le_S_S/
- -Hdi -Hide >arith_c1x #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 //
- -Hdi -Hide /3 width=5 by ex2_intro, cpy_flat/
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
]
qed-.
lemma cpy_append: ∀G,d,e. l_appendable_sn … (cpy d e G).
-#G #d #e #K #T1 #T2 #H elim H -K -T1 -T2 -d -e
+#G #d #e #K #T1 #T2 #H elim H -G -K -T1 -T2 -d -e
/2 width=1 by cpy_atom, cpy_bind, cpy_flat/
#I #G #K #K0 #V #W #i #d #e #Hdi #Hide #HK0 #HVW #L
lapply (ldrop_fwd_length_lt2 … HK0) #H
fact cpy_inv_atom1_aux: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶×[d, e] T2 → ∀J. T1 = ⓪{J} →
T2 = ⓪{J} ∨
- ∃∃I,K,V,i. d ≤ i & i < d + e &
+ ∃∃I,K,V,i. d ≤ yinj i & i < d + e &
⇩[O, i] L ≡ K.ⓑ{I}V &
⇧[O, i + 1] V ≡ T2 &
J = LRef i.
lemma cpy_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶×[d, e] T2 →
T2 = ⓪{I} ∨
- ∃∃J,K,V,i. d ≤ i & i < d + e &
+ ∃∃J,K,V,i. d ≤ yinj i & i < d + e &
⇩[O, i] L ≡ K.ⓑ{J}V &
⇧[O, i + 1] V ≡ T2 &
I = LRef i.
fact cpy_inv_bind1_aux: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 →
∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
- ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ ⦃G, L. ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
U2 = ⓑ{a,I}V2.T2.
#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
[ #I #G #L #d #e #b #J #W1 #U1 #H destruct
lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶×[d, e] U2 →
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶×[d, e] V2 &
- ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[d + 1, e] T2 &
+ ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ▶×[⫯d, e] T2 &
U2 = ⓑ{a,I}V2.T2.
/2 width=3 by cpy_inv_bind1_aux/ qed-.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
[ //
| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- lapply (le_to_lt_to_lt … Hdi … Hide) -Hdi -Hide <plus_n_O #Hdd
- elim (lt_refl_false … Hdd)
+ elim (ylt_yle_false … Hdi) -Hdi //
| /3 width=1 by eq_f2/
| /3 width=1 by eq_f2/
]
elim (IH … HT10) -IH -HT10 #L2 #T2 #HL12 #H destruct
>append_length >HL12 -HL12
@(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] (**) (* explicit constructor *)
- /2 width=3 by trans_eq/
+ /2 width=3 by trans_eq/
]
qed-.
| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
elim (cpy_inv_lref1 … H1) -H1
[ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded
- [ -Hd1 -Hde2
- lapply (transitive_le … Hded Hd2) -Hded -Hd2 #H
- lapply (lt_to_le_to_lt … Hde1 H) -Hde1 -H #H
- elim (lt_refl_false … H)
- | -Hd2 -Hde1
- lapply (transitive_le … Hded Hd1) -Hded -Hd1 #H
- lapply (lt_to_le_to_lt … Hde2 H) -Hde2 -H #H
- elim (lt_refl_false … H)
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
+ [ elim (ylt_yle_false … Hde1) -Hde1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false … Hde2) -Hde2 /2 width=3 by yle_trans/
]
]
| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
[ -H #T #HT1 #HT2
lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V) ?) -HT1 /2 width=1 by lsuby_succ/
lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 >plus_plus_comm_23 >plus_plus_comm_23 in ⊢ (? ? %); elim H -H
- /3 width=1 by monotonic_le_plus_l, or_intror, or_introl/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
]
| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (cpy_inv_atom1 … H) -H
[ #H destruct //
| * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (lt_to_le_to_lt … (d+e) Hide2 ?) /2 width=5 by cpy_subst, monotonic_lt_plus_r/
+ lapply (ylt_yle_trans … (d+e) … Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
]
| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (cpy_weak … HVT2 0 (i +1) ? ?) -HVT2 /2 width=1 by le_S_S/ #HVT2
- <(cpy_inv_lift1_eq … HVT2 … HVW) -HVT2 /2 width=5 by cpy_subst/
+ lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
[ /2 width=3 by ex2_intro/
| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
- lapply (cpy_weak … HWT2 0 (i1 + 1) ? ?) -HWT2 normalize /2 width=1 by le_S/ -Hde2i1 #HWT2
- <(cpy_inv_lift1_eq … HWT2 … HVW) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+ lapply (yle_trans … Hde2d1 … Hdi1) -Hde2d1 #Hde2i1
+ lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
lapply (lsuby_cpy_trans … HT02 (L. ⓑ{I} V0) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 … HV02 ?) -IHV10 -HV02 // #V
- elim (IHT10 … HT02 ?) -T0 /2 width=1 by le_S_S/ #T #HT1 #HT2
+ elim (IHV10 … HV02) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
lapply (lsuby_cpy_trans … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1 by lsuby_succ/
lapply (lsuby_cpy_trans … HT2 (L. ⓑ{I} V2) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
[ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
>(lift_mono … H1 … H2) -H1 -H2 //
| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj … Hid) -Hid #Hid
lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, ldrop_skip, le_S_S/
+ /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
elim (lift_inv_lref1 … H) -H * #Hid #H destruct
[ -Hdtd
- lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
+ lapply (ylt_yle_trans … (dt+et+e) … Hidet) // -Hidet #Hidete
elim (lift_trans_ge … HVW … HWU2) -W // <minus_plus #W #HVW #HWU2
elim (ldrop_trans_le … HLK … HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
>(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
| -Hdti
+ elim (yle_inv_inj2 … Hdtd) -Hdtd #dtt #Hdtd #H destruct
lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, transitive_le/
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid
+ /4 width=5 by cpy_subst, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
]
| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, ldrop_skip, le_S_S/ (**) (* auto a bit slow *)
+ /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
lemma cpy_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶×[dt, et] T2 →
∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
- d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶×[dt + e, et] U2.
+ d ≤ dt → ⦃G, L⦄ ⊢ U1 ▶×[dt+e, et] U2.
#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
[ #I #G #K #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
>(lift_mono … H1 … H2) -H1 -H2 //
| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
- lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
- lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+ lapply (yle_trans … Hddt … Hdti) -Hddt #Hid
+ elim (yle_inv_inj2 … Hid) -Hid #dd #Hddi #H0 destruct
+ lapply (lift_inv_lref1_ge … H … Hddi) -H #H destruct
lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=5 by cpy_subst, lt_minus_to_plus_r, monotonic_le_plus_l/
+ lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hddi
+ /3 width=5 by cpy_subst, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=5 by cpy_bind, ldrop_skip, le_minus_to_plus/
+ /4 width=5 by cpy_bind, ldrop_skip, yle_succ/
| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
]
| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
+ lapply (ylt_yle_trans … Hdetd … Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj … Hid) -Hid #Hid
lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
- elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW
?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
+ elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
| #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
elim (IHU12 … HTU1) -IHU12 -HTU1
- /3 width=5 by cpy_bind, ldrop_skip, lift_bind, le_S_S, ex2_intro/
+ /3 width=5 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
| #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1 //
lemma cpy_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
dt ≤ d → d + e ≤ dt + et →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, et-e] T2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
[ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
[ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
]
| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct
- [ -Hdtd -Hidet
- lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1 by le_plus_to_minus_r/ ] -Hdedet #Hidete
+ lapply (yle_fwd_plus_ge_inj … Hdtd Hdedet) #Heet
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
+ [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
elim (ldrop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
- | -Hdti -Hdedet
- lapply (transitive_le … (i - e) Hdtd ?) /2 width=1 by le_plus_to_minus_r/ -Hdtd #Hdtie
- elim (le_inv_plus_l … Hid) #Hdie #Hei
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+ /3 width=5 by cpy_subst, ex2_intro/
+ | elim (le_inv_plus_l … Hid) #Hdie #Hei
+ lapply (yle_trans … Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … Hdtie … HKV HV1) (**) (* explicit constructor *)
- >commutative_plus >plus_minus /2 width=1 by monotonic_lt_minus_l/
+ @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+ >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
]
| #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
elim (IHU12 … HTU1) -U1
- [5: /2 width=2 by ldrop_skip/ |2: skip
- |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1 by le_S_S/
- |4: /2 width=1 by le_S_S/
- ]
- /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+ /3 width=5 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
| #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1 //
lemma cpy_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
d + e ≤ dt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt-e, et] T2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
[ * #G #L #i #dt #et #K #d #e #_ #T1 #H #_
[ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_sort, ex2_intro/
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpy_atom, lift_gref, ex2_intro/
]
| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
- lapply (transitive_le … Hdedt … Hdti) #Hdei
- elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
- elim (le_inv_plus_l … Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
- lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV10) /2 width=1 by monotonic_le_minus_l2/ (**) (* explicit constructor *)
- >plus_minus /2 width=1 by monotonic_lt_minus_l/
+ lapply (yle_trans … Hdedt … Hdti) #Hdei
+ elim (yle_inv_plus_inj2 … Hdedt) -Hdedt #_ #Hedt
+ elim (yle_inv_plus_inj2 … Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge … H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+ lapply (ldrop_conf_ge … HLK … HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+ elim (lift_split … HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+ ]
| #a #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (le_inv_plus_l … Hdetd) #_ #Hedt
+ elim (yle_inv_plus_inj2 … Hdetd) #_ #Hedt
elim (IHV12 … HLK … HWV1) -V1 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by le_S_S/ ]
- <plus_minus /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+ elim (IHU12 … HTU1) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1 by yle_succ/ ]
+ >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
| #I #G #L #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1 //
]
qed-.
-lemma cpy_inv_lift1_eq: ∀G,L,U1,U2,d,e.
- ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
-[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
- elim (lift_inv_lref2 … H) -H * #H
- [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
- elim (lt_refl_false … H)
- ]
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
- >IHV12 // >IHT12 //
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶×[d, e] U2 → U1 = U2.
+#G #T1 #U1 #d #e #H elim H -T1 -U1 -d -e
+[ #k #d #e #L #X #H >(cpy_inv_sort1 … H) -H //
+| #i #d #e #Hid #L #X #H elim (cpy_inv_lref1 … H) -H //
+ * #I #K #V #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/
+| #i #d #e #Hdi #L #X #H elim (cpy_inv_lref1 … H) -H //
+ * #I #K #V #_ #H elim (ylt_yle_false i d)
+ /2 width=2 by ylt_inv_monotonic_plus_dx, yle_inj/
+| #p #d #e #L #X #H >(cpy_inv_gref1 … H) -H //
+| #a #I #V1 #W1 #T1 #U1 #d #e #_ #_ #IHVW1 #IHTU1 #L #X #H elim (cpy_inv_bind1 … H) -H
+ #W2 #U2 #HW12 #HU12 #H destruct /3 width=2 by eq_f2/
+| #I #V1 #W1 #T1 #U1 #d #e #_ #_ #IHVW1 #IHTU1 #L #X #H elim (cpy_inv_flat1 … H) -H
+ #W2 #U2 #HW12 #HU12 #H destruct /3 width=2 by eq_f2/
]
qed-.
∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[d, dt + et - (d + e)] T2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
elim (cpy_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hddt -Hdtde #HU1
-lapply (cpy_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L // <minus_plus_m_m /2 width=3 by ex2_intro/
+lapply (cpy_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
qed-.
lemma cpy_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶×[dt, et] U2 →
∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
dt ≤ d → dt + et ≤ d + e →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d-dt] T2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (cpy_weak … HU12 dt (d + e - dt) ? ?) -HU12 /2 width=3 by transitive_le, le_n/ -Hdetde #HU12
+lapply (cpy_weak … HU12 dt (d+e-dt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
qed-.
∃∃T2. ⦃G, K⦄ ⊢ T1 ▶×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
elim (cpy_split_up … HU12 d) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1 [2: >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak … HU2 d e ? ?) -HU2 // [ >commutative_plus <plus_minus_m_m // ] -Hddet -Hdetde #HU2
-lapply (cpy_inv_lift1_eq … HU2 … HTU) -L #H destruct /2 width=3 by ex2_intro/
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
+lapply (cpy_weak … HU2 d e ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
qed-.
>yminus_succ /3 width=5 by ldrop_ldrop_lt, ex2_3_intro/
]
| #I #L1 #L2 #V #d #e #_ #IHL12 #J1 #K1 #W1 #i #H #Hdi lapply (ylt_yle_trans 0 … Hdi ?) //
- #Hi <(ylt_inv_O1 … Hi) >yplus_succ1 >yminus_succ lapply (yle_fwd_succ1 … Hdi) -Hdi
- #Hdi #Hide lapply (ylt_inv_succ … Hide)
+ #Hi <(ylt_inv_O1 … Hi) >yplus_succ1 >yminus_succ elim (yle_inv_succ1 … Hdi) -Hdi
+ #Hdi #_ #Hide lapply (ylt_inv_succ … Hide)
#Hide lapply (ylt_inv_inj … Hi) -Hi
#Hi lapply (ldrop_inv_ldrop1_lt … H ?) -H //
#H elim (IHL12 … H) -IHL12 -H
(* *)
(**************************************************************************)
+include "ground_2/ynat/ynat_plus.ma".
include "basic_2/notation/relations/extlrsubeq_4.ma".
include "basic_2/relocation/ldrop.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-inductive lsuby: relation4 nat nat lenv lenv ≝
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
| lsuby_atom: ∀L,d,e. lsuby d e L (⋆)
| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
| lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
- lsuby 0 (e + 1) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+ lsuby 0 (⫯e) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
- lsuby d e L1 L2 → lsuby (d + 1) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
+ lsuby d e L1 L2 → lsuby (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
.
interpretation
(* Basic properties *********************************************************)
-lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, e-1] L2 → 0 < e →
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,e. L1 ⊑×[0, ⫰e] L2 → 0 < e →
L1.ⓑ{I1}V ⊑×[0, e] L2.ⓑ{I2}V.
-#I1 #I2 #L1 #L2 #V #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by lsuby_pair/
+#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
qed.
-lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[d-1, e] L2 → 0 < d →
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⊑×[⫰d, e] L2 → 0 < d →
L1.ⓑ{I1}V1 ⊑×[d, e] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2 width=1 by lsuby_succ/
+#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by lsuby_succ/
qed.
lemma lsuby_refl: ∀L,d,e. L ⊑×[d, e] L.
#L elim L -L //
-#L #I #V #IHL #d @(nat_ind_plus … d) -d /2 width=1 by lsuby_succ/
-#e @(nat_ind_plus … e) -e /2 width=2 by lsuby_pair, lsuby_zero/
+#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
+#Hd destruct /2 width=1 by lsuby_succ/
+#e elim (ynat_cases … e) [| * #x ]
+#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
qed.
-lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[0, 0] L2.
+lemma lsuby_length: ∀L1,L2. |L2| ≤ |L1| → L1 ⊑×[yinj 0, yinj 0] L2.
#L1 elim L1 -L1
[ #X #H lapply (le_n_O_to_eq … H) -H
#H lapply (length_inv_zero_sn … H) #H destruct /2 width=1 by lsuby_atom/
#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
/3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_
- <plus_n_Sm #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_
- <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx … H)
]
qed-.
fact lsuby_inv_pair1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
L2 = ⋆ ∨
- ∃∃J2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{J2}W.
+ ∃∃J2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
/3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_
- <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
]
qed-.
lemma lsuby_inv_pair1: ∀I1,K1,L2,V,e. K1.ⓑ{I1}V ⊑×[0, e] L2 → 0 < e →
L2 = ⋆ ∨
- ∃∃I2,K2. K1 ⊑×[0, e-1] K2 & L2 = K2.ⓑ{I2}V.
+ ∃∃I2,K2. K1 ⊑×[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-
fact lsuby_inv_succ1_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊑×[d-1, e] K2 & L2 = K2.ⓑ{J2}W2.
+ ∃∃J2,K2,W2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
/3 width=5 by ex2_3_intro, or_intror/
]
lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⊑×[d, e] L2 → 0 < d →
L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊑×[d - 1, e] K2 & L2 = K2.ⓑ{I2}V2.
+ ∃∃I2,K2,V2. K1 ⊑×[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
/2 width=5 by lsuby_inv_succ1_aux/ qed-.
fact lsuby_inv_zero2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
[ #L1 #d #e #J2 #K2 #W1 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
/2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_
- <plus_n_Sm #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_
- <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+ elim (ysucc_inv_O_dx … H)
]
qed-.
fact lsuby_inv_pair2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
- ∃∃J1,K1. K1 ⊑×[0, e-1] K2 & L1 = K1.ⓑ{J1}W.
+ ∃∃J1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
#L1 #L2 #d #e * -L1 -L2 -d -e
[ #L1 #d #e #J2 #K2 #W #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
/2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_
- <plus_n_Sm #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
]
qed-.
fact lsuby_inv_succ2_aux: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
- ∃∃J1,K1,W1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{J1}W1.
+ ∃∃J1,K1,W1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
#L1 #L2 #d #e * -L1 -L2 -d -e
[ #L1 #d #e #J2 #K2 #W2 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
- elim (lt_zero_false … H)
+ elim (ylt_yle_false … H) //
| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
/2 width=5 by ex2_3_intro/
]
qed-.
lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,d,e. L1 ⊑×[d, e] K2.ⓑ{I2}V2 → 0 < d →
- ∃∃I1,K1,V1. K1 ⊑×[d-1, e] K2 & L1 = K1.ⓑ{I1}V1.
+ ∃∃I1,K1,V1. K1 ⊑×[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
/2 width=5 by lsuby_inv_succ2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
lemma lsuby_fwd_ldrop2_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
d ≤ i → i < d + e →
- ∃∃I1,K1. K1 ⊑×[0, d+e-i-1] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
+ ∃∃I1,K1. K1 ⊑×[0, ⫰(d+e-i)] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
[ #L1 #d #e #J2 #K2 #W #i #H
elim (ldrop_inv_atom1 … H) -H #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
- elim (lt_zero_false … H)
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ #Hie
- elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1
- [ -IHL12 -Hie destruct normalize -I2
- <minus_n_O <minus_plus_m_m /2 width=4 by ldrop_pair, ex2_2_intro/
- | -HL12
- elim (IHL12 … HLK1) -IHL12 -HLK1 // [2: /2 width=1 by lt_plus_to_minus/ ] -Hie normalize
- >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ >yplus_O_sn
+ elim (ldrop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct -I2 >ypred_succ
+ /2 width=4 by ldrop_pair, ex2_2_intro/
+ | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+ #H <H -H #H lapply (ylt_inv_succ … H) -H
+ #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
+ >yminus_succ <yminus_inj /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #H #Hdi >plus_plus_comm_23 #Hide
- elim (le_inv_plus_l … Hdi) #_ #Hi
- lapply (ldrop_inv_ldrop1_lt … H ?) -H // #HLK1
- elim (IHL12 … HLK1) -IHL12 -HLK1
- [2,3: /2 width=1 by lt_plus_to_minus, monotonic_pred/ ] -Hdi -Hide
- >minus_minus_comm >arith_b1 /3 width=4 by ldrop_ldrop_lt, ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #HLK2 #Hdi
+ elim (yle_inv_succ1 … Hdi) -Hdi
+ #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
+ #Hide lapply (ldrop_inv_ldrop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+ #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+ /4 width=4 by ylt_O, ldrop_ldrop_lt, ex2_2_intro/
]
qed-.
(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-definition cpys: nat → nat → relation4 genv lenv term term ≝
+definition cpys: ynat → ynat → relation4 genv lenv term term ≝
λd,e,G. LTC … (cpy d e G).
interpretation "context-sensitive extended multiple substritution (term)"
(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
(* alternative definition of cpys *)
-inductive cpysa: nat → nat → relation4 genv lenv term term ≝
+inductive cpysa: ynat → ynat → relation4 genv lenv term term ≝
| cpysa_atom : ∀I,G,L,d,e. cpysa d e G L (⓪{I}) (⓪{I})
-| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+| cpysa_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d+e →
⇩[0, i] L ≡ K.ⓑ{I}V1 → cpysa 0 (d+e-i-1) G K V1 V2 →
⇧[0, i+1] V2 ≡ W2 → cpysa d e G L (#i) W2
| cpysa_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
| #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
lapply (ldrop_fwd_ldrop2 … HLK) #H0LK
lapply (cpy_weak … H 0 (d+e) ? ?) -H // #H
- elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2 /3 width=7 by cpysa_subst/
+ elim (cpy_inv_lift1_be … H … H0LK … HVW2) -H -H0LK -HVW2
+ /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
| #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /2 width=1 by lsuby_succ/ #HT2
/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
qed-.
-lemma cpys_ind_alt: ∀R:nat→nat→relation4 genv lenv term term.
+lemma cpys_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
(∀I,G,L,d,e. R d e G L (⓪{I}) (⓪{I})) →
- (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ i → i < d + e →
+ (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
⇩[O, i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*×[O, d+e-i-1] V2 →
⇧[O, i + 1] V2 ≡ W2 → R O (d+e-i-1) G K V1 V2 → R d e G L (#i) W2
) →
#G #L #T1 #T2 #d #e #H #i #Hdi #Hide @(cpys_ind … H) -T2
[ /2 width=3 by ex2_intro/
| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (cpy_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
- elim (cpys_strap1_down … HT3 … HT0 ?) -T /3 width=5 by cpys_strap1, ex2_intro/
- >commutative_plus /2 width=1 by le_minus_to_plus_r/
+ elim (cpy_split_up … HT12 … Hide) -HT12 -Hide #T0 #HT0 #HT02
+ elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //
]
qed-.
⇧[d, e] T2 ≡ U2.
#G #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
elim (cpys_split_up … HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1 by le_minus_to_plus_r/ ] -Hddt -Hdtde #HU1
+lapply (cpys_weak … HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
-elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3 by ex2_intro/
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
qed-.
(* Main properties **********************************************************)
(* Advanced properties ******************************************************)
lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
- d ≤ i → i < d + e →
- ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, d+e-i-1] U1 →
- ∀U2. ⇧[0, i + 1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
+ d ≤ yinj i → i < d + e →
+ ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 →
+ ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
#I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
[ /3 width=5 by cpy_cpys, cpy_subst/
| #U #U1 #_ #HU1 #IHU #U2 #HU12
elim (lift_total U 0 (i+1)) #U0 #HU0
lapply (IHU … HU0) -IHU #H
lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
- lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // normalize #HU02
- lapply (cpy_weak … HU02 d e ? ?) -HU02 [2,3: /2 width=3 by cpys_strap1, le_S/ ]
- >minus_plus >commutative_plus /2 width=1 by le_minus_to_plus_r/
+ lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+ lapply (cpy_weak … HU02 d e ? ?) -HU02
+ [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+ >yplus_O_sn <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
]
qed.
(* Advanced inverion lemmas *************************************************)
-lemma cpys_inv_atom1: ∀G,L,T2,I,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
+lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
T2 = ⓪{I} ∨
- ∃∃J,K,V1,V2,i. d ≤ i & i < d + e &
+ ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
⇩[O, i] L ≡ K.ⓑ{J}V1 &
⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
- ⇧[O, i + 1] V2 ≡ T2 &
+ ⇧[O, i+1] V2 ≡ T2 &
I = LRef i.
-#G #L #T2 #I #d #e #H @(cpys_ind … H) -T2
+#I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
[ /2 width=1 by or_introl/
| #T #T2 #_ #HT2 *
[ #H destruct
elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
| * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
lapply (ldrop_fwd_ldrop2 … HLK) #H
- elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) normalize -HT2 -H -HVT [2,3,4: /2 width=1 by le_S/ ]
- <minus_plus /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+ elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
+ [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+ /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
]
]
qed-.
T2 = #i ∨
∃∃I,K,V1,V2. d ≤ i & i < d + e &
⇩[O, i] L ≡ K.ⓑ{I}V1 &
- ⦃G, K⦄ ⊢ V1 ▶*×[0, d + e - i - 1] V2 &
- ⇧[O, i + 1] V2 ≡ T2.
+ ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
+ ⇧[O, i+1] V2 ≡ T2.
#G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
* #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
qed-.
(* Relocation properties ****************************************************)
lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
- ∀L,U1,d,e. dt + et ≤ d → ⇩[d, e] L ≡ K →
+ ∀L,U1,d,e. dt + et ≤ yinj d → ⇩[d, e] L ≡ K →
⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2.
#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
qed-.
lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
- ∀L,U1,d,e. dt ≤ d → d ≤ dt + et →
+ ∀L,U1,d,e. dt ≤ yinj d → d ≤ dt + et →
⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2.
#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
qed-.
lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
- ∀L,U1,d,e. d ≤ dt → ⇩[d, e] L ≡ K →
+ ∀L,U1,d,e. yinj d ≤ dt → ⇩[d, e] L ≡ K →
⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
- ⦃G, L⦄ ⊢ U1 ▶*×[dt + e, et] U2.
+ ⦃G, L⦄ ⊢ U1 ▶*×[dt+e, et] U2.
#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
[ #U2 #H >(lift_mono … HTU1 … H) -H //
| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
[ /2 width=3 by ex2_intro/
| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge … HU2 … HLK … HTU ?) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+ elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
]
qed-.
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2,d,e.
+lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2 //
#U #U2 #_ #HU2 #IHU destruct
-<(cpy_inv_lift1_eq … HU2 … HTU1) -HU2 -HTU1 //
+<(cpy_inv_lift1_eq … HTU1 … HU2) -HU2 -HTU1 //
qed-.
lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
(* Basic forward lemmas *****************************************************)
-lemma ylt_inv_gen: ∀x,y. x < y → ∃m. x = yinj m.
+lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
#x #y * -x -y /2 width=2 by ex_intro/
qed-.
+lemma ylt_fwd_le_succ: ∀x,y. x < y → ⫯x ≤ y.
+#x #y * -x -y /2 width=1 by yle_inj/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
qed-.
lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
-#n #H elim (ylt_inv_gen … H) -H
+#n #H elim (ylt_fwd_gen … H) -H
#y #H destruct
qed-.
#m >yminus_inj //
qed.
-lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
+lemma yminus_succ1_inj: ∀n:nat. ∀m:ynat. n ≤ m → ⫯m - n = ⫯(m - n).
#n *
[ #m #Hmn >yminus_inj >yminus_inj
/4 width=1 by yle_inv_inj, plus_minus, eq_f/
]
qed-.
+lemma yminus_succ2: ∀y,x. x - ⫯y = ⫰(x-y).
+* //
+qed.
+
(* Properties on order ******************************************************)
lemma yle_minus_sn: ∀n,m. m - n ≤ m.
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