--- /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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PSubstEval $G $L $T1 $T2 $d $e }.
--- /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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ▶ ▶ * break [ term 46 d , break term 46 e ] break 𝐍 ⦃ term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedEval $G $L $T1 $T2 $d $e }.
(* Basic properties *********************************************************)
+lemma lsuby_cny_conf: ∀G,d,e.
+ ∀L1,T. ⦃G, L1⦄ ⊢ ▶[d, e] 𝐍⦃T⦄ →
+ ∀L2. L1 ⊑×[d, e] L2 → ⦃G, L2⦄ ⊢ ▶[d, e] 𝐍⦃T⦄.
+#G #d #e #L1 #T1 #HT1 #L2 #HL12 #T2 #HT12
+@HT1 /3 width=3 by lsuby_cpy_trans/
+qed-.
+
lemma cny_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃⋆k⦄.
#G #L #d #e #k #X #H elim (cpy_inv_sort1 … H) -H //
qed.
elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
>(HV1 … HV12) -V2 >(HT1 … HT12) -T2 //
qed.
+
+lemma cny_narrow: ∀G,L,T,d1,e1. ⦃G, L⦄ ⊢ ▶[d1, e1] 𝐍⦃T⦄ →
+ ∀d2,e2. d1 ≤ d2 → d2 + e2 ≤ d1 + e1 → ⦃G, L⦄ ⊢ ▶[d2, e2] 𝐍⦃T⦄.
+#G #L #T1 #d1 #e1 #HT1 #d2 #e2 #Hd12 #Hde21 #T2 #HT12
+@HT1 /2 width=5 by cpy_weak/ qed-.
>ymax_pre_sn // -Hedt #HU12
lapply (HU1 … HU12) -L /2 width=5 by lift_inj/
qed-.
+
+(* Advanced properties ******************************************************)
+
+fact cny_subst_aux: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HV #HVW
+lapply (cny_lift_be … HV … HLK … HVW ? ?) // -HV -HLK -HVW
+#HW @(cny_narrow … HW) -HW //
+qed-.
+
+lemma cny_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄.
+/3 width=13 by cny_subst_aux, ldrop_fwd_drop2/ 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/notation/relations/psubsteval_6.ma".
+include "basic_2/relocation/cny.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+definition cpye: ynat → ynat → relation4 genv lenv term term ≝
+ λd,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 ∧ ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive extended substitution (term)"
+ 'PSubstEval G L T1 T2 d e = (cpye d e G L T1 T2).
+
+(* Basic_properties *********************************************************)
+
+lemma cpye_sort: ∀G,L,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃⋆k⦄.
+/3 width=5 by cny_sort, conj/ qed.
+
+lemma cpye_free: ∀G,L,d,e,i. |L| ≤ i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_free, conj/ qed.
+
+lemma cpye_top: ∀G,L,d,e,i. d + e ≤ yinj i → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_top, conj/ qed.
+
+lemma cpye_skip: ∀G,L,d,e,i. yinj i < d → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃#i⦄.
+/3 width=6 by cny_lref_skip, conj/ qed.
+
+lemma cpye_gref: ∀G,L,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃§p⦄.
+/3 width=5 by cny_gref, conj/ qed.
+
+lemma cpye_bind: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃ⓑ{a,I}V2.T2⦄.
+#G #L #V1 #V2 #d #e * #HV12 #HV2 #I #T1 #T2 *
+/5 width=8 by cpys_bind, cny_bind, lsuby_cny_conf, lsuby_succ, conj/
+qed.
+
+lemma cpye_flat: ∀G,L,V1,V2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ∀T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃ⓕ{I}V2.T2⦄.
+#G #L #V1 #V2 #d #e * #HV12 #HV2 #T1 #T2 *
+/3 width=7 by cpys_flat, cny_flat, conj/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma
\ No newline at end of file
--- /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/relocation/cny_lift.ma".
+include "basic_2/substitution/fqup.ma".
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/cpye.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+lemma cpye_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] 𝐍⦃V2⦄ →
+ ⇧[O, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃W2⦄.
+#I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK *
+/4 width=13 by cpys_subst, cny_subst_aux, ldrop_fwd_drop2, conj/
+qed.
+
+lemma cpys_total: ∀G,L,T1,d,e. ∃T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 @(fqup_wf_ind_eq … G L T1) -G -L -T1
+#Z #Y #X #IH #G #L * *
+[ #k #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
+| #i #HG #HL #HT #d #e destruct
+ elim (ylt_split i d) /3 width=2 by cpye_skip, ex_intro/
+ elim (ylt_split i (d+e)) /3 width=2 by cpye_top, ex_intro/
+ elim (lt_or_ge i (|L|)) /3 width=2 by cpye_free, ex_intro/
+ #Hi #Hide #Hdi elim (ldrop_O1_lt L i) // -Hi
+ #I #K #V1 #HLK elim (IH G K V1 … 0 (⫰(d+e-i))) -IH /2 width=2 by fqup_lref/
+ #V2 elim (lift_total V2 0 (i+1)) /3 width=8 by ex_intro, cpye_subst/
+| #p #HG #HL #HT #d #e destruct -IH /2 width=2 by ex_intro/
+| #a #I #V1 #T1 #HG #HL #HT #d #e destruct
+ elim (IH G L V1 … d e) // elim (IH G (L.ⓑ{I}V1) T1 … (⫯d) e) //
+ /3 width=2 by cpye_bind, ex_intro/
+| #I #V1 #T1 #HG #HL #HT #d #e destruct
+ elim (IH G L V1 … d e) // elim (IH G L T1 … d e) //
+ /3 width=2 by cpye_flat, ex_intro/
+]
+qed-.
[ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" + "lleq_ext" * ]
}
]
- [ { "normal forms for context-sensitive extended substitution" * } {
- [ "cny ( ⦃?,?⦄ ⊢ ▶[?,?] 𝐍⦃?⦄ )" "cny_lift" * ]
- }
- ]
[ { "contxt-sensitive extended multiple substitution" * } {
- [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*×[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*×[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
+ [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
}
]
[ { "iterated structural successor for closures" * } {
]
class "orange"
[ { "relocation" * } {
+ [ { "normal forms for context-sensitive extended substitution" * } {
+ [ "cny ( ⦃?,?⦄ ⊢ ▶[?,?] 𝐍⦃?⦄ )" "cny_lift" * ]
+ }
+ ]
[ { "contxt-sensitive extended ordinary substitution" * } {
- [ "cpy ( ⦃?,?⦄ ⊢ ? ▶×[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
+ [ "cpy ( ⦃?,?⦄ ⊢ ? ▶[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
}
]
[ { "local env. ref. for extended substitution" * } {
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