(**************************************************************************)
include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/extlrsubeq_4.ma".
+include "basic_2/notation/relations/lrsubeq_4.ma".
include "basic_2/relocation/ldrop.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
| lsuby_pair: ∀I1,I2,L1,L2,V,e. lsuby 0 e L1 L2 →
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) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
+ lsuby d e L1 L2 → lsuby (⫯d) e (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
.
interpretation
"local environment refinement (extended substitution)"
- 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
+ 'LRSubEq L1 d e L2 = (lsuby d e L1 L2).
(* Basic properties *********************************************************)
-lemma lsuby_pair_lt: â\88\80I1,I2,L1,L2,V,e. L1 â\8a\91Ã\97[0, ⫰e] L2 → 0 < e →
- L1.â\93\91{I1}V â\8a\91Ã\97[0, e] L2.ⓑ{I2}V.
+lemma lsuby_pair_lt: â\88\80I1,I2,L1,L2,V,e. L1 â\8a\86[0, ⫰e] L2 → 0 < e →
+ L1.â\93\91{I1}V â\8a\86[0, e] L2.ⓑ{I2}V.
#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by lsuby_pair/
qed.
-lemma lsuby_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1 â\8a\91Ã\97[⫰d, e] L2 → 0 < d →
- L1.â\93\91{I1}V1 â\8a\91Ã\97[d, e] L2. ⓑ{I2}V2.
+lemma lsuby_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1 â\8a\86[⫰d, e] L2 → 0 < d →
+ L1.â\93\91{I1}V1 â\8a\86[d, e] L2. ⓑ{I2}V2.
#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.
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+ ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: ∀L,d,e. L ⊆[d, e] L.
#L elim L -L //
#L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
#Hd destruct /2 width=1 by lsuby_succ/
#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
qed.
-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 #I1 #V1 #IHL1 * normalize
- /4 width=2 by lsuby_zero, le_S_S_to_le/
+lemma lsuby_O2: ∀L2,L1,d. |L2| ≤ |L1| → L1 ⊆[d, yinj 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
+[ #d #H elim (le_plus_xSy_O_false … H)
+| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r … H) -H #HL12
+ elim (ynat_cases d) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
]
qed.
+lemma lsuby_sym: ∀d,e,L1,L2. L1 ⊆[d, e] L2 → |L1| = |L2| → L2 ⊆[d, e] L1.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2
+[ #L1 #d #e #H >(length_inv_zero_dx … H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l … H)
+ /3 width=1 by lsuby_succ/
+]
+qed-.
+
(* Basic inversion lemmas ***************************************************)
-fact lsuby_inv_atom1_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 → L1 = ⋆ → L2 = ⋆.
+fact lsuby_inv_atom1_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 → L1 = ⋆ → L2 = ⋆.
#L1 #L2 #d #e * -L1 -L2 -d -e //
[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
]
qed-.
-lemma lsuby_inv_atom1: â\88\80L2,d,e. â\8b\86 â\8a\91Ã\97[d, e] L2 → L2 = ⋆.
+lemma lsuby_inv_atom1: â\88\80L2,d,e. â\8b\86 â\8a\86[d, e] L2 → L2 = ⋆.
/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-fact lsuby_inv_zero1_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_zero1_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
L2 = ⋆ ∨
- â\88\83â\88\83J2,K2,W2. K1 â\8a\91Ã\97[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+ â\88\83â\88\83J2,K2,W2. K1 â\8a\86[0, 0] 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 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
/3 width=5 by ex2_3_intro, or_intror/
]
qed-.
-lemma lsuby_inv_zero1: â\88\80I1,K1,L2,V1. K1.â\93\91{I1}V1 â\8a\91Ã\97[0, 0] L2 →
+lemma lsuby_inv_zero1: â\88\80I1,K1,L2,V1. K1.â\93\91{I1}V1 â\8a\86[0, 0] L2 →
L2 = ⋆ ∨
- â\88\83â\88\83I2,K2,V2. K1 â\8a\91Ã\97[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+ â\88\83â\88\83I2,K2,V2. K1 â\8a\86[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
/2 width=9 by lsuby_inv_zero1_aux/ qed-.
-fact lsuby_inv_pair1_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_pair1_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J1,K1,W. L1 = K1.ⓑ{J1}W → d = 0 → 0 < e →
L2 = ⋆ ∨
- â\88\83â\88\83J2,K2. K1 â\8a\91Ã\97[0, ⫰e] K2 & L2 = K2.ⓑ{J2}W.
+ â\88\83â\88\83J2,K2. K1 â\8a\86[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 (ylt_yle_false … H) //
]
qed-.
-lemma lsuby_inv_pair1: â\88\80I1,K1,L2,V,e. K1.â\93\91{I1}V â\8a\91Ã\97[0, e] L2 → 0 < e →
+lemma lsuby_inv_pair1: â\88\80I1,K1,L2,V,e. K1.â\93\91{I1}V â\8a\86[0, e] L2 → 0 < e →
L2 = ⋆ ∨
- â\88\83â\88\83I2,K2. K1 â\8a\91Ã\97[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
+ â\88\83â\88\83I2,K2. K1 â\8a\86[0, ⫰e] K2 & L2 = K2.ⓑ{I2}V.
/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-fact lsuby_inv_succ1_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_succ1_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
L2 = ⋆ ∨
- â\88\83â\88\83J2,K2,W2. K1 â\8a\91Ã\97[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
+ â\88\83â\88\83J2,K2,W2. K1 â\8a\86[⫰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 (ylt_yle_false … H) //
]
qed-.
-lemma lsuby_inv_succ1: â\88\80I1,K1,L2,V1,d,e. K1.â\93\91{I1}V1 â\8a\91Ã\97[d, e] L2 → 0 < d →
+lemma lsuby_inv_succ1: â\88\80I1,K1,L2,V1,d,e. K1.â\93\91{I1}V1 â\8a\86[d, e] L2 → 0 < d →
L2 = ⋆ ∨
- â\88\83â\88\83I2,K2,V2. K1 â\8a\91Ã\97[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
+ â\88\83â\88\83I2,K2,V2. K1 â\8a\86[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
/2 width=5 by lsuby_inv_succ1_aux/ qed-.
-fact lsuby_inv_zero2_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_zero2_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → d = 0 → e = 0 →
- â\88\83â\88\83J1,K1,W1. K1 â\8a\91Ã\97[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+ â\88\83â\88\83J1,K1,W1. K1 â\8a\86[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
#L1 #L2 #d #e * -L1 -L2 -d -e
[ #L1 #d #e #J2 #K2 #W1 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
]
qed-.
-lemma lsuby_inv_zero2: â\88\80I2,K2,L1,V2. L1 â\8a\91Ã\97[0, 0] K2.ⓑ{I2}V2 →
- â\88\83â\88\83I1,K1,V1. K1 â\8a\91Ã\97[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+lemma lsuby_inv_zero2: â\88\80I2,K2,L1,V2. L1 â\8a\86[0, 0] K2.ⓑ{I2}V2 →
+ â\88\83â\88\83I1,K1,V1. K1 â\8a\86[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
/2 width=9 by lsuby_inv_zero2_aux/ qed-.
-fact lsuby_inv_pair2_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_pair2_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J2,K2,W. L2 = K2.ⓑ{J2}W → d = 0 → 0 < e →
- â\88\83â\88\83J1,K1. K1 â\8a\91Ã\97[0, ⫰e] K2 & L1 = K1.ⓑ{J1}W.
+ â\88\83â\88\83J1,K1. K1 â\8a\86[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
]
qed-.
-lemma lsuby_inv_pair2: â\88\80I2,K2,L1,V,e. L1 â\8a\91Ã\97[0, e] K2.ⓑ{I2}V → 0 < e →
- â\88\83â\88\83I1,K1. K1 â\8a\91Ã\97[0, e-1] K2 & L1 = K1.ⓑ{I1}V.
+lemma lsuby_inv_pair2: â\88\80I2,K2,L1,V,e. L1 â\8a\86[0, e] K2.ⓑ{I2}V → 0 < e →
+ â\88\83â\88\83I1,K1. K1 â\8a\86[0, â«°e] K2 & L1 = K1.ⓑ{I1}V.
/2 width=6 by lsuby_inv_pair2_aux/ qed-.
-fact lsuby_inv_succ2_aux: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 →
+fact lsuby_inv_succ2_aux: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 →
∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < d →
- â\88\83â\88\83J1,K1,W1. K1 â\8a\91Ã\97[⫰d, e] K2 & L1 = K1.ⓑ{J1}W1.
+ â\88\83â\88\83J1,K1,W1. K1 â\8a\86[⫰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
]
qed-.
-lemma lsuby_inv_succ2: â\88\80I2,K2,L1,V2,d,e. L1 â\8a\91Ã\97[d, e] K2.ⓑ{I2}V2 → 0 < d →
- â\88\83â\88\83I1,K1,V1. K1 â\8a\91Ã\97[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
+lemma lsuby_inv_succ2: â\88\80I2,K2,L1,V2,d,e. L1 â\8a\86[d, e] K2.ⓑ{I2}V2 → 0 < d →
+ â\88\83â\88\83I1,K1,V1. K1 â\8a\86[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
/2 width=5 by lsuby_inv_succ2_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lsuby_fwd_length: â\88\80L1,L2,d,e. L1 â\8a\91Ã\97[d, e] L2 → |L2| ≤ |L1|.
+lemma lsuby_fwd_length: â\88\80L1,L2,d,e. L1 â\8a\86[d, e] L2 → |L2| ≤ |L1|.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
qed-.
-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)] K2 & ⇩[0, i] L1 ≡ K1.ⓑ{I1}W.
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_ldrop_trans_be: ∀L1,L2,d,e. L1 ⊆[d, e] L2 →
+ ∀I2,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+ d ≤ i → i < d + e →
+ ∃∃I1,K1. K1 ⊆[0, ⫰(d+e-i)] K2 & ⇩[s, 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
+[ #L1 #d #e #J2 #K2 #W #s #i #H
elim (ldrop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #i #_ #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #i #H #_ >yplus_O_sn
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
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/
+ >yminus_succ <yminus_inj /3 width=4 by ldrop_drop_lt, ex2_2_intro/
]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #i #HLK2 #Hdi
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #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/
+ #Hide lapply (ldrop_inv_drop1_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/
+ /4 width=4 by ylt_O, ldrop_drop_lt, ex2_2_intro/
]
qed-.
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