non associative with precedence 45
for @{ 'RLift $d $e $T1 $T2 }.
-notation "hvbox( T1 break ⊑ [ term 46 d , break term 46 e ] break term 46 T2 )"
+notation "hvbox( L1 break ⊑ [ term 46 d , break term 46 e ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'SubEq $T1 $d $e $T2 }.
+ for @{ 'SubEq $L1 $d $e $L2 }.
-notation "hvbox( â\89½ [ term 46 d , break term 46 e ] break term 46 T2 )"
+notation "hvbox( â\8a\92 [ term 46 d , break term 46 e ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'SubEqBottom $d $e $T2 }.
+ for @{ 'SubEqBottom $d $e $L2 }.
notation "hvbox( ⇩ [ term 46 e ] break term 46 L1 ≡ break term 46 L2 )"
non associative with precedence 45
--- /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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lsubr_lbotr.ma".
+include "basic_2/substitution/ldrop_ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Inversion lemmas about local env. full refinement for substitution *******)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ⊒[d, e] L →
+ d ≤ i → i < d + e → I = Abbr.
+#I #L elim L -L
+[ #K #V #i #H
+ lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L #J #W #IHL #K #V #i #H
+ elim (ldrop_inv_O1 … H) -H *
+ [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
+ lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
+ lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
+ elim (lsubr_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
+ | #Hi #HLK #d @(nat_ind_plus … d) -d
+ [ #e #H #_ #Hide
+ elim (lbotr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
+ @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
+ | #d #_ #e #H #Hdi #Hide
+ lapply (lbotr_inv_skip … H ?) -H // #HL
+ @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
+ ]
+ ]
+]
+qed-.
+
+(* Properties about local env. full refinement for substitution *************)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_ldrop: ∀L,d,e.
+ (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
+ ⊒[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d @(nat_ind_plus … d) -d
+[ #e @(nat_ind_plus … e) -e //
+ #e #_ #H0
+ >(H0 I L V 0 ? ? ?) //
+ /5 width=6 by lbotr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+| #d #_ #e #H0
+ /5 width=6 by lbotr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+]
+qed.
+
+lemma lbotr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+ dd + ee ≤ d → ⊒[dd, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
+elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
+elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
+@(lbotr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
+qed.
+
+lemma lbotr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+ ∀dd,ee. ⊒[dd, ee] L1 →
+ dd ≤ d + e → d + e ≤ dd + ee →
+ ⊒[d, dd + ee - d - e] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
+lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
+>commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
+qed.
+
+lemma lbotr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+ d + e ≤ dd → ⊒[dd - e, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
+>plus_minus in Hiddee; // #Hiddee
+lapply (transitive_le … Hdde Hddi) -Hdde #Hid
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
+qed.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lsubr_sfr.ma".
-include "basic_2/substitution/ldrop_ldrop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Inversion lemmas about local env. full refinement for substitution *******)
-
-(* Note: ldrop_ldrop not needed *)
-lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≽ [d, e] L →
- d ≤ i → i < d + e → I = Abbr.
-#I #L elim L -L
-[ #K #V #i #H
- lapply (ldrop_inv_atom1 … H) -H #H destruct
-| #L #J #W #IHL #K #V #i #H
- elim (ldrop_inv_O1 … H) -H *
- [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
- lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
- lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
- elim (lsubr_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
- | #Hi #HLK #d @(nat_ind_plus … d) -d
- [ #e #H #_ #Hide
- elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
- @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
- | #d #_ #e #H #Hdi #Hide
- lapply (sfr_inv_skip … H ?) -H // #HL
- @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
- ]
- ]
-]
-qed-.
-
-(* Properties about local env. full refinement for substitution *************)
-
-(* Note: ldrop_ldrop not needed *)
-lemma sfr_ldrop: ∀L,d,e.
- (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
- ≽ [d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d @(nat_ind_plus … d) -d
-[ #e @(nat_ind_plus … e) -e //
- #e #_ #H0
- >(H0 I L V 0 ? ? ?) //
- /5 width=6 by sfr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
-| #d #_ #e #H0
- /5 width=6 by sfr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
-]
-qed.
-
-lemma sfr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 →
- dd + ee ≤ d → ≽ [dd, ee] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
-@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
-lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
-elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
-elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
-@(sfr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
-qed.
-
-lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
- ∀dd,ee. ≽ [dd, ee] L1 →
- dd ≤ d + e → d + e ≤ dd + ee →
- ≽ [d, dd + ee - d - e] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
-@sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
-lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
->commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
-lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi #HL1K2
-@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
-qed.
-
-lemma sfr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 →
- d + e ≤ dd → ≽ [dd - e, ee] L2.
-#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
-@sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
-elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
->plus_minus in Hiddee; // #Hiddee
-lapply (transitive_le … Hdde Hddi) -Hdde #Hid
-lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
-@(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
-qed.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lsubr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
+
+(* bottom element of the refinement *)
+definition lbotr: nat → nat → predicate lenv ≝
+ λd,e. NF_sn … (lsubr d e) (lsubr d e …).
+
+interpretation
+ "local environment full refinement (substitution)"
+ 'SubEqBottom d e L = (lbotr d e L).
+
+(* Basic properties *********************************************************)
+
+lemma lbotr_atom: ∀d,e. ⊒[d, e] ⋆.
+#d #e #L #H
+elim (lsubr_inv_atom2 … H) -H
+[ #H destruct //
+| * #H1 #H2 destruct //
+]
+qed.
+
+lemma lbotr_OO: ∀L. ⊒[0, 0] L.
+// qed.
+
+lemma lbotr_abbr: ∀L,V,e. ⊒[0, e] L → ⊒[0, e + 1] L.ⓓV.
+#L #V #e #HL #K #H
+elim (lsubr_inv_abbr2 … H ?) -H // <minus_plus_m_m #X #HLX #H destruct
+lapply (HL … HLX) -HL -HLX /2 width=1/
+qed.
+
+lemma lbotr_abbr_O: ∀L,V. ⊒[0,1] L.ⓓV.
+#L #V
+@(lbotr_abbr … 0) //
+qed.
+
+lemma lbotr_skip: ∀I,L,V,d,e. ⊒[d, e] L → ⊒[d + 1, e] L.ⓑ{I}V.
+#I #L #V #d #e #HL #K #H
+elim (lsubr_inv_skip2 … H ?) -H // <minus_plus_m_m #J #X #W #HLX #H destruct
+lapply (HL … HLX) -HL -HLX /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lbotr_inv_bind: ∀I,L,V,e. ⊒[0, e] L.ⓑ{I}V → 0 < e →
+ ⊒[0, e - 1] L ∧ I = Abbr.
+#I #L #V #e #HL #He
+lapply (HL (L.ⓓV) ?) /2 width=1/ #H
+elim (lsubr_inv_abbr2 … H ?) -H // #K #_ #H destruct
+@conj // #L #HKL
+lapply (HL (L.ⓓV) ?) -HL /2 width=1/ -HKL #H
+elim (lsubr_inv_abbr2 … H ?) -H // -He #X #HLX #H destruct //
+qed-.
+
+lemma lbotr_inv_skip: ∀I,L,V,d,e. ⊒[d, e] L.ⓑ{I}V → 0 < d → ⊒[d - 1, e] L.
+#I #L #V #d #e #HL #Hd #K #HLK
+lapply (HL (K.ⓑ{I}V) ?) -HL /2 width=1/ -HLK #H
+elim (lsubr_inv_skip2 … H ?) -H // -Hd #J #X #W #HKX #H destruct //
+qed-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lsubr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR SUBSTITUTION ****************************)
-
-(* bottom element of the refinement *)
-definition sfr: nat → nat → predicate lenv ≝
- λd,e. NF_sn … (lsubr d e) (lsubr d e …).
-
-interpretation
- "local environment full refinement (substitution)"
- 'SubEqBottom d e L = (sfr d e L).
-
-(* Basic properties *********************************************************)
-
-lemma sfr_atom: ∀d,e. ≽ [d, e] ⋆.
-#d #e #L #H
-elim (lsubr_inv_atom2 … H) -H
-[ #H destruct //
-| * #H1 #H2 destruct //
-]
-qed.
-
-lemma sfr_OO: ∀L. ≽ [0, 0] L.
-// qed.
-
-lemma sfr_abbr: ∀L,V,e. ≽ [0, e] L → ≽ [0, e + 1] L.ⓓV.
-#L #V #e #HL #K #H
-elim (lsubr_inv_abbr2 … H ?) -H // <minus_plus_m_m #X #HLX #H destruct
-lapply (HL … HLX) -HL -HLX /2 width=1/
-qed.
-
-lemma sfr_abbr_O: ∀L,V. ≽[0,1] L.ⓓV.
-#L #V
-@(sfr_abbr … 0) //
-qed.
-
-lemma sfr_skip: ∀I,L,V,d,e. ≽ [d, e] L → ≽ [d + 1, e] L.ⓑ{I}V.
-#I #L #V #d #e #HL #K #H
-elim (lsubr_inv_skip2 … H ?) -H // <minus_plus_m_m #J #X #W #HLX #H destruct
-lapply (HL … HLX) -HL -HLX /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma sfr_inv_bind: ∀I,L,V,e. ≽ [0, e] L.ⓑ{I}V → 0 < e →
- ≽ [0, e - 1] L ∧ I = Abbr.
-#I #L #V #e #HL #He
-lapply (HL (L.ⓓV) ?) /2 width=1/ #H
-elim (lsubr_inv_abbr2 … H ?) -H // #K #_ #H destruct
-@conj // #L #HKL
-lapply (HL (L.ⓓV) ?) -HL /2 width=1/ -HKL #H
-elim (lsubr_inv_abbr2 … H ?) -H // -He #X #HLX #H destruct //
-qed-.
-
-lemma sfr_inv_skip: ∀I,L,V,d,e. ≽ [d, e] L.ⓑ{I}V → 0 < d → ≽ [d - 1, e] L.
-#I #L #V #d #e #HL #Hd #K #HLK
-lapply (HL (K.ⓑ{I}V) ?) -HL /2 width=1/ -HLK #H
-elim (lsubr_inv_skip2 … H ?) -H // -Hd #J #X #W #HKX #H destruct //
-qed-.
(* *)
(**************************************************************************)
-include "basic_2/substitution/ldrop_sfr.ma".
+include "basic_2/substitution/ldrop_lbotr.ma".
include "basic_2/unfold/tpss_lift.ma".
include "basic_2/unfold/delift.ma".
>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
qed.
-lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L →
- ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
+lemma lbotr_delift: ∀L,T1,d,e. d + e ≤ |L| → ⊒[d, e] L →
+ ∃T2. L ⊢ ▼*[d, e] T1 ≡ T2.
#L #T1 @(f2_ind … fw … L T1) -L -T1
#n #IH #L * * /2 width=2/
[ #i #H #d #e #Hde #HL destruct
elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
lapply (lt_to_le_to_lt … Hide Hde) #Hi
elim (ldrop_O1_lt … Hi) -Hi #I #K #V1 #HLK
- lapply (sfr_inv_ldrop … HLK … HL ? ?) // #H destruct
+ lapply (lbotr_inv_ldrop … HLK … HL ? ?) // #H destruct
lapply (ldrop_pair2_fwd_fw … HLK (#i)) #HKL
lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
lapply (ldrop_fwd_O1_length … HLK0) #H
- lapply (sfr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
+ lapply (lbotr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
[1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
elim (IH … HKL … HK) -IH -HKL -HK
[2: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
}
]
[ { "basic local env. slicing" * } {
- [ "ldrop ( ⇩[?,?] ? ≡ ? )" "ldrop_append" + "ldrop_lpx" + "ldrop_sfr" + "ldrop_ldrop" * ]
+ [ "ldrop ( ⇩[?,?] ? ≡ ? )" "ldrop_append" + "ldrop_lpx" + "ldrop_lbotr" + "ldrop_ldrop" * ]
}
]
[ { "local env. ref. for substitution" * } {
- [ "lsubr ( ? ⊑[?,?] ? )" "(lsubr_lsubr)" "lsubr_sfr ( ≽[?,?] ? )" * ]
+ [ "lsubr ( ? ⊑[?,?] ? )" "(lsubr_lsubr)" "lsubr_lbotr ( ⊒[?,?] ? )" * ]
}
]
[ { "restricted structural predecessor for closures" * } {
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