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 }.
+ for @{ 'PSubstEvalAlt $G $L $T1 $T2 $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-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact cny_inv_subst_aux: ∀G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i+1] L ≡ K → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+#G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HW #HVW
+lapply (cny_narrow … HW (i+1) (⫰(d+e-i)) ? ?) -HW
+[ >yplus_SO2 <yplus_succ_swap >ylt_inv_O1
+ [ >ymax_pre_sn_comm /2 width=2 by ylt_fwd_le/
+ | lapply (monotonic_ylt_minus_dx … Hide i ?) //
+ ]
+| /2 width=3 by yle_trans/
+| #HW lapply (cny_lift_inv_ge … HW … HLK … HVW ?) // -L -W
+ >yplus_inj >yminus_refl //
+]
+qed-.
+
+lemma cny_inv_subst: ∀I,G,L,K,V,W,i,d,e. d ≤ yinj i → i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ▶[d, e] 𝐍⦃W⦄ →
+ ⇧[O, i+1] V ≡ W → ⦃G, K⦄ ⊢ ▶[O, ⫰(d+e-i)] 𝐍⦃V⦄.
+/3 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2/ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma
\ No newline at end of file
+lemma cpye_inv_sort1: ∀G,L,X,d,e,k. ⦃G, L⦄ ⊢ ⋆k ▶*[d, e] 𝐍⦃X⦄ → X = ⋆k.
+#G #L #X #d #e #k * /2 width=5 by cpys_inv_sort1/
+qed-.
+
+lemma cpye_inv_gref1: ∀G,L,X,d,e,p. ⦃G, L⦄ ⊢ §p ▶*[d, e] 𝐍⦃X⦄ → X = §p.
+#G #L #X #d #e #p * /2 width=5 by cpys_inv_gref1/
+qed-.
+
+lemma cpye_inv_bind1: ∀a,I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ &
+ X = ⓑ{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_bind1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_bind … H2) -H2
+/5 width=8 by lsuby_cny_conf, lsuby_succ, ex3_2_intro, conj/
+qed-.
+
+lemma cpye_inv_flat1: ∀I,G,L,V1,T1,X,d,e. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ▶*[d, e] 𝐍⦃X⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ & ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ &
+ X = ⓕ{I}V2.T2.
+#I #G #L #V1 #T1 #X #d #e * #H1 #H2 elim (cpys_inv_flat1 … H1) -H1
+#V2 #T2 #HV12 #HT12 #H destruct elim (cny_inv_flat … H2) -H2
+/3 width=5 by ex3_2_intro, conj/
+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/psubstevalalt_6.ma".
+include "basic_2/substitution/cpye_lift.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+
+(* Note: alternative definition of cpye *)
+inductive cpyea: ynat → ynat → relation4 genv lenv term term ≝
+| cpyea_sort : ∀G,L,d,e,k. cpyea d e G L (⋆k) (⋆k)
+| cpyea_free : ∀G,L,d,e,i. |L| ≤ i → cpyea d e G L (#i) (#i)
+| cpyea_top : ∀G,L,d,e,i. d + e ≤ yinj i → cpyea d e G L (#i) (#i)
+| cpyea_skip : ∀G,L,d,e,i. yinj i < d → cpyea d e G L (#i) (#i)
+| cpyea_subst: ∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d+e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → cpyea (yinj 0) (⫰(d+e-yinj i)) G K V1 V2 →
+ ⇧[0, i+1] V2 ≡ W2 → cpyea d e G L (#i) W2
+| cpyea_gref : ∀G,L,d,e,p. cpyea d e G L (§p) (§p)
+| cpyea_bind : ∀a,I,G,L,V1,V2,T1,T2,d,e.
+ cpyea d e G L V1 V2 → cpyea (⫯d) e G (L.ⓑ{I}V1) T1 T2 →
+ cpyea d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpyea_flat : ∀I,G,L,V1,V2,T1,T2,d,e.
+ cpyea d e G L V1 V2 → cpyea d e G L T1 T2 →
+ cpyea d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation
+ "evaluation for context-sensitive extended substitution (term) alternative"
+ 'PSubstEvalAlt G L T1 T2 d e = (cpyea d e G L T1 T2).
+
+(* Main properties **********************************************************)
+
+theorem cpye_cpyea: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃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 #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_sort1 … H) -H //
+| #i #HG #HL #HT #T2 #d #e #H destruct
+ elim (cpye_inv_lref1 … H) -H *
+ /4 width=7 by cpyea_subst, cpyea_free, cpyea_top, cpyea_skip, fqup_lref/
+| #p #_ #_ #_ #T2 #d #e #H -X -Y -Z >(cpye_inv_gref1 … H) -H //
+| #a #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+ elim (cpye_inv_bind1 … H) -H /3 width=1 by cpyea_bind/
+| #I #V1 #T1 #HG #HL #HT #T #d #e #H destruct
+ elim (cpye_inv_flat1 … H) -H /3 width=1 by cpyea_flat/
+]
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem cpyea_inv_cpye: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] 𝐍⦃T2⦄ → ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+/2 width=7 by cpye_subst, cpye_flat, cpye_bind, cpye_skip, cpye_top, cpye_free/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpye_ind_alt: ∀R:ynat→ynat→relation4 genv lenv term term.
+ (∀G,L,d,e,k. R d e G L (⋆k) (⋆k)) →
+ (∀G,L,d,e,i. |L| ≤ i → R d e G L (#i) (#i)) →
+ (∀G,L,d,e,i. d + e ≤ yinj i → R d e G L (#i) (#i)) →
+ (∀G,L,d,e,i. yinj i < d → R d e G L (#i) (#i)) →
+ (∀I,G,L,K,V1,V2,W2,i,d,e. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ▶*[yinj O, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ →
+ ⇧[O, i+1] V2 ≡ W2 → R (yinj O) (⫰(d+e-yinj i)) G K V1 V2 → R d e G L (#i) W2
+ ) →
+ (∀G,L,d,e,p. R d e G L (§p) (§p)) →
+ (∀a,I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶*[⫯d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+ R (⫯d) e G (L.ⓑ{I}V1) T1 T2 → R d e G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ ) →
+ (∀I,G,L,V1,V2,T1,T2,d,e. ⦃G, L⦄ ⊢ V1 ▶*[d, e] 𝐍⦃V2⦄ →
+ ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L V1 V2 →
+ R d e G L T1 T2 → R d e G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) →
+ ∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] 𝐍⦃T2⦄ → R d e G L T1 T2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #d #e #G #L #T1 #T2 #H elim (cpye_cpyea … H) -G -L -T1 -T2 -d -e
+/3 width=8 by cpyea_inv_cpye/
+qed-.
(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED SUBSTITUTION ON TERMS **********)
+(* Advanced properties ******************************************************)
+
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⦄.
/3 width=2 by cpye_flat, ex_intro/
]
qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpye_inv_lref1: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∨∨ |L| ≤ i ∧ T2 = #i
+ | d + e ≤ yinj i ∧ T2 = #i
+ | yinj i < d ∧ T2 = #i
+ | ∃∃I,K,V1,V2. d ≤ yinj i & yinj i < d + e &
+ ⇩[i] L ≡ K.ⓑ{I}V1 &
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄ &
+ ⇧[O, i+1] V2 ≡ T2.
+#G #L #T2 #i #d #e * #H1 #H2 elim (cpys_inv_lref1 … H1) -H1
+[ #H destruct elim (cny_inv_lref … H2) -H2
+ /3 width=1 by or4_intro0, or4_intro1, or4_intro2, conj/
+| * #I #K #V1 #V2 #Hdi #Hide #HLK #HV12 #HVT2
+ @or4_intro3 @(ex5_4_intro … HLK … HVT2) (**) (* explicit constructor *)
+ /4 width=13 by cny_inv_subst_aux, ldrop_fwd_drop2, conj/
+]
+qed-.
+
+lemma cpye_inv_lref1_free: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ (∨∨ |L| ≤ i | d + e ≤ yinj i | yinj i < d) → T2 = #i.
+#G #L #T2 #d #e #i #H * elim (cpye_inv_lref1 … H) -H * //
+#I #K #V1 #V2 #Hdi #Hide #HLK #_ #_ #H
+[ elim (lt_refl_false i) -d
+ @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/ (**) (* full auto slow: 19s *)
+]
+elim (ylt_yle_false … H) //
+qed-.
+
+lemma cpye_inv_lref1_subst: ∀G,L,T2,d,e,i. ⦃G, L⦄ ⊢ #i ▶*[d, e] 𝐍⦃T2⦄ →
+ ∀I,K,V1,V2. d ≤ yinj i → yinj i < d + e →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[O, i+1] V2 ≡ T2 →
+ ⦃G, K⦄ ⊢ V1 ▶*[yinj 0, ⫰(d+e-yinj i)] 𝐍⦃V2⦄.
+#G #L #T2 #d #e #i #H #I #K #V1 #V2 #Hdi #Hide #HLK #HVT2 elim (cpye_inv_lref1 … H) -H *
+[ #H elim (lt_refl_false i) -V2 -T2 -d
+ @(lt_to_le_to_lt … H) -H /2 width=5 by ldrop_fwd_length_lt2/
+|2,3: #H elim (ylt_yle_false … H) //
+| #Z #Y #X1 #X2 #_ #_ #HLY #HX12 #HXT2
+ lapply (ldrop_mono … HLY … HLK) -HLY -HLK #H destruct
+ lapply (lift_inj … HXT2 … HVT2) -HXT2 -HVT2 #H destruct //
+]
+qed-.
(* Basic inversion lemmas ***************************************************)
-lemma cpysa_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
+lemma cpysa_inv_cpys: ∀G,L,T1,T2,d,e. ⦃G, L⦄ ⊢ T1 ▶▶*[d, e] T2 → ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2.
#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
qed-.
+(* Advanced eliminators *****************************************************)
+
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 ≤ yinj i → i < d + e →
) →
∀d,e,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*[d, e] T2 → R d e G L T1 T2.
#R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa … H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpysa_cpys/
+/3 width=8 by cpysa_inv_cpys/
qed-.
]
class "cyan"
[ { "computation" * } {
- [ { "context-sensitive extended evaluation" * } {
+ [ { "evaluation for context-sensitive extended reduction" * } {
[ "cpxe ( ⦃?,?⦄ ⊢ ➡*[?,?] 𝐍⦃?⦄ )" * ]
}
]
- [ { "context-sensitive evaluation" * } {
+ [ { "evaluation for context-sensitive reduction" * } {
[ "cpre ( ⦃?,?⦄ ⊢ ➡* 𝐍⦃?⦄ )" "cpre_cpre" * ]
}
]
[ "lleq ( ? ⋕[?,?] ? )" "lleq_alt ( ? ⋕⋕[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" + "lleq_ext" * ]
}
]
+ [ { "evaluation for contxt-sensitive extended substitution" * } {
+ [ "cpye ( ⦃?,?⦄ ⊢ ? ▶*[?,?] 𝐍⦃?⦄ )" "cpye_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] 𝐍⦃?⦄ )" "cpye_lift" * ]
+ }
+ ]
[ { "contxt-sensitive extended multiple substitution" * } {
[ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
}
projects / helm.git / commitdiff