(* *)
(**************************************************************************)
+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
"local environment refinement (extended substitution)"
'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
-definition lsuby_trans: ∀S. predicate (lenv → relation S) ≝ λS,R.
- ∀L2,s1,s2. R L2 s1 s2 →
- ∀L1,d,e. L1 ⊑×[d, e] L2 → R L1 s1 s2.
-
(* 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/
]
qed.
-lemma TC_lsuby_trans: ∀S,R. lsuby_trans S R → lsuby_trans S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -s2 /3 width=7 by step, inj/
-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_length/
+| #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 ***************************************************)
#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-.
lemma lsuby_inv_pair2: ∀I2,K2,L1,V,e. L1 ⊑×[0, e] K2.ⓑ{I2}V → 0 < e →
- ∃∃I1,K1. K1 ⊑×[0, e-1] K2 & L1 = K1.ⓑ{I1}V.
+ ∃∃I1,K1. K1 ⊑×[0, ⫰e] K2 & L1 = K1.ⓑ{I1}V.
/2 width=6 by lsuby_inv_pair2_aux/ 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 *****************************************************)
qed-.
lemma lsuby_fwd_ldrop2_be: ∀L1,L2,d,e. L1 ⊑×[d, e] L2 →
- ∀I2,K2,W,i. ⇩[0, i] L2 ≡ K2.ⓑ{I2}W →
+ ∀I2,K2,W,s,i. ⇩[s, 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 & ⇩[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
- 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/
+| #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 #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_drop_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 #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_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_drop_lt, ex2_2_intro/
]
qed-.
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