--- /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/degree_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/drop.ma".
+include "basic_2/static/sd.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* activate genv *)
+inductive da (h:sh) (o:sd h): relation4 genv lenv term nat ≝
+| da_sort: ∀G,L,s,d. deg h o s d → da h o G L (⋆s) d
+| da_ldef: ∀G,L,K,V,i,d. ⬇[i] L ≡ K.ⓓV → da h o G K V d → da h o G L (#i) d
+| da_ldec: ∀G,L,K,W,i,d. ⬇[i] L ≡ K.ⓛW → da h o G K W d → da h o G L (#i) (d+1)
+| da_bind: ∀a,I,G,L,V,T,d. da h o G (L.ⓑ{I}V) T d → da h o G L (ⓑ{a,I}V.T) d
+| da_flat: ∀I,G,L,V,T,d. da h o G L T d → da h o G L (ⓕ{I}V.T) d
+.
+
+interpretation "degree assignment (term)"
+ 'Degree h o G L T d = (da h o G L T d).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact da_inv_sort_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d →
+ ∀s0. T = ⋆s0 → deg h o s0 d.
+#h #o #G #L #T #d * -G -L -T -d
+[ #G #L #s #d #Hkd #s0 #H destruct //
+| #G #L #K #V #i #d #_ #_ #s0 #H destruct
+| #G #L #K #W #i #d #_ #_ #s0 #H destruct
+| #a #I #G #L #V #T #d #_ #s0 #H destruct
+| #I #G #L #V #T #d #_ #s0 #H destruct
+]
+qed-.
+
+lemma da_inv_sort: ∀h,o,G,L,s,d. ⦃G, L⦄ ⊢ ⋆s ▪[h, o] d → deg h o s d.
+/2 width=5 by da_inv_sort_aux/ qed-.
+
+fact da_inv_lref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀j. T = #j →
+ (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨
+ (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 &
+ d = d0 + 1
+ ).
+#h #o #G #L #T #d * -G -L -T -d
+[ #G #L #s #d #_ #j #H destruct
+| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
+| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
+| #a #I #G #L #V #T #d #_ #j #H destruct
+| #I #G #L #V #T #d #_ #j #H destruct
+]
+qed-.
+
+lemma da_inv_lref: ∀h,o,G,L,j,d. ⦃G, L⦄ ⊢ #j ▪[h, o] d →
+ (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, o] d) ∨
+ (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, o] d0 & d = d0+1).
+/2 width=3 by da_inv_lref_aux/ qed-.
+
+fact da_inv_gref_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d → ∀p0. T = §p0 → ⊥.
+#h #o #G #L #T #d * -G -L -T -d
+[ #G #L #s #d #_ #p0 #H destruct
+| #G #L #K #V #i #d #_ #_ #p0 #H destruct
+| #G #L #K #W #i #d #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #d #_ #p0 #H destruct
+| #I #G #L #V #T #d #_ #p0 #H destruct
+]
+qed-.
+
+lemma da_inv_gref: ∀h,o,G,L,p,d. ⦃G, L⦄ ⊢ §p ▪[h, o] d → ⊥.
+/2 width=9 by da_inv_gref_aux/ qed-.
+
+fact da_inv_bind_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d →
+ ∀b,J,X,Y. T = ⓑ{b,J}Y.X → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d.
+#h #o #G #L #T #d * -G -L -T -d
+[ #G #L #s #d #_ #b #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct //
+| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct
+]
+qed-.
+
+lemma da_inv_bind: ∀h,o,b,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X ▪[h, o] d → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, o] d.
+/2 width=4 by da_inv_bind_aux/ qed-.
+
+fact da_inv_flat_aux: ∀h,o,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, o] d →
+ ∀J,X,Y. T = ⓕ{J}Y.X → ⦃G, L⦄ ⊢ X ▪[h, o] d.
+#h #o #G #L #T #d * -G -L -T -d
+[ #G #L #s #d #_ #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct
+| #I #G #L #V #T #d #HT #J #X #Y #H destruct //
+]
+qed-.
+
+lemma da_inv_flat: ∀h,o,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓕ{J}Y.X ▪[h, o] d → ⦃G, L⦄ ⊢ X ▪[h, o] d.
+/2 width=5 by da_inv_flat_aux/ 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/static/aaa_lift.ma".
+include "basic_2/static/da.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on atomic arity assignment for terms **************************)
+
+lemma aaa_da: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃d. ⦃G, L⦄ ⊢ T ▪[h, o] d.
+#h #o #G #L #T #A #H elim H -G -L -T -A
+[ #G #L #s elim (deg_total h o s) /3 width=2 by da_sort, ex_intro/
+| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
+| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
+]
+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/static/da_lift.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Main properties **********************************************************)
+
+theorem da_mono: ∀h,o,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, o] d1 →
+ ∀d2. ⦃G, L⦄ ⊢ T ▪[h, o] d2 → d1 = d2.
+#h #o #G #L #T #d1 #H elim H -G -L -T -d1
+[ #G #L #s #d1 #Hkd1 #d2 #H
+ lapply (da_inv_sort … H) -G -L #Hkd2
+ >(deg_mono … Hkd2 … Hkd1) -h -s -d2 //
+| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H
+ elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ]
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /2 width=1 by/
+| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H
+ elim (da_inv_lref … H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ]
+ lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /3 width=1 by eq_f/
+| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H
+ lapply (da_inv_bind … H) -H /2 width=1 by/
+| #I #G #L #V #T #d1 #_ #IHT #d2 #H
+ lapply (da_inv_flat … H) -H /2 width=1 by/
+]
+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/drop_drop.ma".
+include "basic_2/static/da.ma".
+
+(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
+
+(* Properties on relocation *************************************************)
+
+lemma da_lift: ∀h,o,G,L1,T1,d. ⦃G, L1⦄ ⊢ T1 ▪[h, o] d →
+ ∀L2,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T2. ⬆[l, k] T1 ≡ T2 →
+ ⦃G, L2⦄ ⊢ T2 ▪[h, o] d.
+#h #o #G #L1 #T1 #d #H elim H -G -L1 -T1 -d
+[ #G #L1 #s #d #Hkd #L2 #c #l #k #_ #X #H
+ >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
+| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_lref1 … H) * #Hil #H destruct
+ [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by da_ldef/
+ | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldef, drop_inv_gen/
+ ]
+| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_lref1 … H) * #Hil #H destruct
+ [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
+ /3 width=8 by da_ldec/
+ | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=8 by da_ldec, drop_inv_gen/
+ ]
+| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /4 width=5 by da_bind, drop_skip/
+| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
+ /3 width=5 by da_flat/
+]
+qed.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma da_inv_lift: ∀h,o,G,L2,T2,d. ⦃G, L2⦄ ⊢ T2 ▪[h, o] d →
+ ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 → ∀T1. ⬆[l, k] T1 ≡ T2 →
+ ⦃G, L1⦄ ⊢ T1 ▪[h, o] d.
+#h #o #G #L2 #T2 #d #H elim H -G -L2 -T2 -d
+[ #G #L2 #s #d #Hkd #L1 #c #l #k #_ #X #H
+ >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
+| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HV2 | -IHV2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
+ ]
+| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HW2 | -IHW2 ]
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
+ ]
+| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=5 by da_bind, drop_skip/
+| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=5 by da_flat/
+]
+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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ▪ break [ term 46 o , break term 46 h ] break term 46 d )"
+ non associative with precedence 45
+ for @{ 'Degree $h $o $G $L $T $d }.
]
qed-.
-(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f →
- ∃∃L. ⬇*[c, f1] L1 ≡ L & ⬇*[c, f2] L ≡ L2.
-#L1 #L2 #f #c #H elim H -L1 -L2 -f
-[ #f #Hc #f1 #f2 #Hf @(ex2_intro … (⋆)) @drops_atom
- #H lapply (Hc H) -c
- #H elim (after_inv_isid3 … Hf H) -f //
-| #I #L1 #L2 #V #f #HL12 #IHL12 #f1 #f2 #Hf elim (after_inv_xxS … Hf) -Hf *
- [ #g1 #g2 #Hf #H1 #H2 destruct elim (IHL12 … Hf) -f
- #L #HL1 #HL2 @(ex2_intro … (L.ⓑ{I}V)) /2 width=1 by drops_drop/
- @drops_skip //
- | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
- /3 width=5 by ex2_intro, drops_drop/
- ]
-| #I #L1 #L2 #V1 #V2 #f #_ #HV21 #IHL12 #f1 #f2 #Hf elim (after_inv_xxO … Hf) -Hf
- #g1 #g2 #Hf #H1 #H2 destruct elim (lifts_split_trans … HV21 … Hf) -HV21
- elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip/
-]
-qed-.
-
lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0.
/2 width=5 by drop_inv_length_eq/ 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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ * [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDropStar $m $L1 $L2 }.
--- /dev/null
+(* Removed theorems *********************************************************)
+
+include "basic_2/substitution/drop.ma".
+
+| fqu_drop : ∀G,L,K,T,U,k.
+ ⬇[⫯k] L ≡ K → ⬆[0, ⫯k] T ≡ U → fqu G L U G K T
+
+lemma fqu_drop_lt: ∀G,L,K,T,U,k. 0 < k →
+ ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
+#G #L #K #T #U #k #Hm lapply (ylt_inv_O1 … Hm) -Hm
+#Hm <Hm -Hm /2 width=3 by fqu_drop/
+qed.
+
+lemma fqu_lref_S_lt: ∀I,G,L,V,i. yinj 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(⫰i)⦄.
+/4 width=3 by drop_drop, lift_lref_pred, fqu_drop/
+qed.
--- /dev/null
+(* Removed theorems *********************************************************)
+
+lemma fquqa_drop: ∀G,L,K,T,U,k.
+ ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
+#G #L #K #T #U #k @(ynat_ind … k) -k /3 width=3 by fqu_drop, or_introl/
+#HLK #HTU >(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
+qed.
+
+inductive fquq: tri_relation genv lenv term ≝
+| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
+| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
+| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
+| fquq_drop : ∀G,L,K,T,U,k.
+ ⬇[k] L ≡ K → ⬆[0, k] T ≡ U → fquq G L U G K T
+.
--- /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( G ⊢ break term 46 L1 ⫃ ▪ break [ term 46 o, break term 46 h ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqD $h $o $G $L1 $L2 }.
--- /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/lrsubeqd_5.ma".
+include "basic_2/static/lsubr.ma".
+include "basic_2/static/da.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+inductive lsubd (h) (o) (G): relation lenv ≝
+| lsubd_atom: lsubd h o G (⋆) (⋆)
+| lsubd_pair: ∀I,L1,L2,V. lsubd h o G L1 L2 →
+ lsubd h o G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubd_beta: ∀L1,L2,W,V,d. ⦃G, L1⦄ ⊢ V ▪[h, o] d+1 → ⦃G, L2⦄ ⊢ W ▪[h, o] d →
+ lsubd h o G L1 L2 → lsubd h o G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (degree assignment)"
+ 'LRSubEqD h o G L1 L2 = (lsubd h o G L1 L2).
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubd_fwd_lsubr: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L1 ⫃ L2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubd_inv_atom1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L1 = ⋆ → L2 = ⋆.
+#h #o #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubd_inv_atom1: ∀h,o,G,L2. G ⊢ ⋆ ⫃▪[h, o] L2 → L2 = ⋆.
+/2 width=6 by lsubd_inv_atom1_aux/ qed-.
+
+fact lsubd_inv_pair1_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
+ ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
+ G ⊢ K1 ⫃▪[h, o] K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+#h #o #G #L1 #L2 * -L1 -L2
+[ #J #K1 #X #H destruct
+| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
+]
+qed-.
+
+lemma lsubd_inv_pair1: ∀h,o,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃▪[h, o] L2 →
+ (∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
+ G ⊢ K1 ⫃▪[h, o] K2 &
+ I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
+/2 width=3 by lsubd_inv_pair1_aux/ qed-.
+
+fact lsubd_inv_atom2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 → L2 = ⋆ → L1 = ⋆.
+#h #o #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubd_inv_atom2: ∀h,o,G,L1. G ⊢ L1 ⫃▪[h, o] ⋆ → L1 = ⋆.
+/2 width=6 by lsubd_inv_atom2_aux/ qed-.
+
+fact lsubd_inv_pair2_aux: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
+ ∀I,K2,W. L2 = K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
+ G ⊢ K1 ⫃▪[h, o] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+#h #o #G #L1 #L2 * -L1 -L2
+[ #J #K2 #U #H destruct
+| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
+]
+qed-.
+
+lemma lsubd_inv_pair2: ∀h,o,I,G,L1,K2,W. G ⊢ L1 ⫃▪[h, o] K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, o] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, o] d &
+ G ⊢ K1 ⫃▪[h, o] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+/2 width=3 by lsubd_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsubd_refl: ∀h,o,G,L. G ⊢ L ⫃▪[h, o] L.
+#h #o #G #L elim L -L /2 width=1 by lsubd_pair/
+qed.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubd_drop_O1_conf: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
+ ∀K1,c,k. ⬇[c, 0, k] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃▪[h, o] K2 & ⬇[c, 0, k] L2 ≡ K2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K1 #c #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+ [ destruct
+ elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #c #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
+ [ destruct
+ elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: the constant 0 cannot be generalized *)
+lemma lsubd_drop_O1_trans: ∀h,o,G,L1,L2. G ⊢ L1 ⫃▪[h, o] L2 →
+ ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃▪[h, o] K2 & ⬇[c, 0, k] L1 ≡ K1.
+#h #o #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #K2 #c #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+ [ destruct
+ elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #c #k #H
+ elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
+ [ destruct
+ elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
+ <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
+ | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+]
+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/static/da_da.ma".
+include "basic_2/static/lsubd.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+(* Properties on degree assignment ******************************************)
+
+lemma lsubd_da_trans: ∀h,o,G,L2,T,d. ⦃G, L2⦄ ⊢ T ▪[h, o] d →
+ ∀L1. G ⊢ L1 ⫃▪[h, o] L2 → ⦃G, L1⦄ ⊢ T ▪[h, o] d.
+#h #o #G #L2 #T #d #H elim H -G -L2 -T -d
+[ /2 width=1 by da_sort/
+| #G #L2 #K2 #V #i #d #HLK2 #_ #IHV #L1 #HL12
+ elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubd_inv_pair2 … H) -H * #K1 [ | -IHV -HLK1 ]
+ [ #HK12 #H destruct /3 width=4 by da_ldef/
+ | #W #d0 #_ #_ #_ #H destruct
+ ]
+| #G #L2 #K2 #W #i #d #HLK2 #HW #IHW #L1 #HL12
+ elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
+ elim (lsubd_inv_pair2 … H) -H * #K1 [ -HW | -IHW ]
+ [ #HK12 #H destruct /3 width=4 by da_ldec/
+ | #V #d0 #HV #H0W #_ #_ #H destruct
+ lapply (da_mono … H0W … HW) -H0W -HW #H destruct /3 width=7 by da_ldef, da_flat/
+ ]
+| /4 width=1 by lsubd_pair, da_bind/
+| /3 width=1 by da_flat/
+]
+qed-.
+
+lemma lsubd_da_conf: ∀h,o,G,L1,T,d. ⦃G, L1⦄ ⊢ T ▪[h, o] d →
+ ∀L2. G ⊢ L1 ⫃▪[h, o] L2 → ⦃G, L2⦄ ⊢ T ▪[h, o] d.
+#h #o #G #L1 #T #d #H elim H -G -L1 -T -d
+[ /2 width=1 by da_sort/
+| #G #L1 #K1 #V #i #d #HLK1 #HV #IHV #L2 #HL12
+ elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+ elim (lsubd_inv_pair1 … H) -H * #K2 [ -HV | -IHV ]
+ [ #HK12 #H destruct /3 width=4 by da_ldef/
+ | #W0 #V0 #d0 #HV0 #HW0 #_ #_ #H1 #H2 destruct
+ lapply (da_inv_flat … HV) -HV #H0V0
+ lapply (da_mono … H0V0 … HV0) -H0V0 -HV0 #H destruct /2 width=4 by da_ldec/
+ ]
+| #G #L1 #K1 #W #i #d #HLK1 #HW #IHW #L2 #HL12
+ elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
+ elim (lsubd_inv_pair1 … H) -H * #K2 [ -HW | -IHW ]
+ [ #HK12 #H destruct /3 width=4 by da_ldec/
+ | #W0 #V0 #d0 #HV0 #HW0 #_ #H destruct
+ ]
+| /4 width=1 by lsubd_pair, da_bind/
+| /3 width=1 by da_flat/
+]
+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/static/lsubd_da.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
+
+(* Main properties **********************************************************)
+
+theorem lsubd_trans: ∀h,o,G. Transitive … (lsubd h o G).
+#h #o #G #L1 #L #H elim H -L1 -L
+[ #X #H >(lsubd_inv_atom1 … H) -H //
+| #I #L1 #L #Y #HL1 #IHL1 #X #H
+ elim (lsubd_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=1 by lsubd_pair/
+ | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct
+ /3 width=3 by lsubd_beta, lsubd_da_trans/
+ ]
+| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H
+ elim (lsubd_inv_pair1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/
+ | #W0 #V0 #d0 #_ #_ #_ #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 "ground_2/notation/functions/append_2.ma".
+include "basic_2/notation/functions/snbind2_3.ma".
+include "basic_2/notation/functions/snabbr_2.ma".
+include "basic_2/notation/functions/snabst_2.ma".
+include "basic_2/grammar/lenv.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+rec definition append L K on K ≝ match K with
+[ LAtom ⇒ L
+| LPair K I V ⇒ (append L K).ⓑ{I}V
+].
+
+interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
+
+interpretation "local environment tail binding construction (binary)"
+ 'SnBind2 I T L = (append (LPair LAtom I T) L).
+
+interpretation "tail abbreviation (local environment)"
+ 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
+
+interpretation "tail abstraction (local environment)"
+ 'SnAbst L T = (append (LPair LAtom Abst T) L).
+
+definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
+ ∀K,T1,T2. R K T1 T2 → ∀L. R (L@@K) T1 T2.
+
+(* Basic properties *********************************************************)
+
+lemma append_atom: ∀L. L @@ ⋆ = L.
+// qed.
+
+lemma append_pair: ∀I,L,K,V. L@@(K.ⓑ{I}V) = (L@@K).ⓑ{I}V.
+// qed.
+
+lemma append_atom_sn: ∀L. ⋆@@L = L.
+#L elim L -L //
+#L #I #V >append_pair //
+qed.
+
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 //
+qed.
+
+lemma append_shift: ∀L,K,I,V. L@@(ⓑ{I}V.K) = (L.ⓑ{I}V)@@K.
+#L #K #I #V <append_assoc //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma append_inj_sn: ∀K,L1,L2. L1@@K = L2@@K → L1 = L2.
+#K elim K -K //
+#K #I #V #IH #L1 #L2 >append_pair #H
+elim (destruct_lpair_lpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+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/grammar/lenv_length.ma".
+include "basic_2/grammar/append.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+(* Properties with length for local environments ****************************)
+
+lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
+#L1 #L2 elim L2 -L2 //
+#L2 #I #V2 >append_pair >length_pair >length_pair //
+qed.
+
+lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
+#I #L #V >append_length //
+qed.
+
+(* Basic_1: was just: chead_ctail *)
+lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
+#L elim L -L /2 width=5 by ex2_3_intro/
+#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
+#J #K #W #H #_ >H -H >ltail_length
+@(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
+qed-.
+
+(* Advanced inversion lemmas on length for local environments ***************)
+
+(* Basic_2A1: was: length_inv_pos_dx_ltail *)
+lemma length_inv_succ_dx_ltail: ∀L,l. |L| = ⫯l →
+ ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+(* Basic_2A1: was: length_inv_pos_sn_ltail *)
+lemma length_inv_succ_sn_ltail: ∀L,l. ⫯l = |L| →
+ ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
+#Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
+elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
+qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+(* Basic_2A1: was: append_inj_sn *)
+lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
+ #H destruct
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 #_ >length_atom >length_pair
+ #H destruct
+ | #K2 #I2 #V2 #L1 #L2 #H1 #H2
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
+ ]
+]
+qed-.
+
+(* Note: lemma 750 *)
+(* Basic_2A1: was: append_inj_dx *)
+lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
+ >length_pair >append_length >plus_n_Sm
+ #H elim (plus_xSy_x_false … H)
+| #K1 #I1 #V1 #IH *
+ [ #L1 #L2 >append_pair >append_atom #H destruct
+ >length_pair >append_length >plus_n_Sm #H
+ lapply (discr_plus_x_xy … H) -H #H destruct
+ | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
+ elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
+ ]
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma append_inj_dx: ∀L,K1,K2. L@@K1 = L@@K2 → K1 = K2.
+#L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L@@K = L → K = ⋆.
+#L #K #H elim (append_inj_dx … (⋆) … H) //
+qed-.
+
+lemma append_inv_pair_dx: ∀I,L,K,V. L@@K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+lemma lenv_ind_alt: ∀R:predicate lenv.
+ R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
+ ∀L. R L.
+#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
+#L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/
+qed-.
elim (destruct_lpair_lpair_aux … H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
+
+lemma discr_lpair_xy_x: ∀I,V,L. L.ⓑ{I}V = L→ ⊥.
+/2 width=4 by discr_lpair_x_xy/ 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 "ground_2/notation/functions/append_2.ma".
-include "basic_2/notation/functions/snbind2_3.ma".
-include "basic_2/notation/functions/snabbr_2.ma".
-include "basic_2/notation/functions/snabst_2.ma".
-include "basic_2/grammar/lenv_length.ma".
-
-(* LOCAL ENVIRONMENTS *******************************************************)
-
-let rec append L K on K ≝ match K with
-[ LAtom ⇒ L
-| LPair K I V ⇒ (append L K). ⓑ{I} V
-].
-
-interpretation "append (local environment)" 'Append L1 L2 = (append L1 L2).
-
-interpretation "local environment tail binding construction (binary)"
- 'SnBind2 I T L = (append (LPair LAtom I T) L).
-
-interpretation "tail abbreviation (local environment)"
- 'SnAbbr T L = (append (LPair LAtom Abbr T) L).
-
-interpretation "tail abstraction (local environment)"
- 'SnAbst L T = (append (LPair LAtom Abst T) L).
-
-definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
- ∀K,T1,T2. R K T1 T2 → ∀L. R (L @@ K) T1 T2.
-
-(* Basic properties *********************************************************)
-
-lemma append_atom: ∀L. L @@ ⋆ = L.
-// qed.
-
-lemma append_pair: ∀I,L,K,V. L @@ (K.ⓑ{I}V) = (L @@ K).ⓑ{I}V.
-// qed.
-
-lemma append_atom_sn: ∀L. ⋆ @@ L = L.
-#L elim L -L //
-#L #I #V >append_pair //
-qed.
-
-lemma append_assoc: associative … append.
-#L1 #L2 #L3 elim L3 -L3 //
-qed.
-
-lemma append_length: ∀L1,L2. |L1 @@ L2| = |L1| + |L2|.
-#L1 #L2 elim L2 -L2 //
-#L2 #I #V2 >append_pair >length_pair >length_pair //
-qed.
-
-lemma ltail_length: ∀I,L,V. |ⓑ{I}V.L| = ⫯|L|.
-#I #L #V >append_length //
-qed.
-
-(* Basic_1: was just: chead_ctail *)
-lemma lpair_ltail: ∀L,I,V. ∃∃J,K,W. L.ⓑ{I}V = ⓑ{J}W.K & |L| = |K|.
-#L elim L -L /2 width=5 by ex2_3_intro/
-#L #Z #X #IHL #I #V elim (IHL Z X) -IHL
-#J #K #W #H #_ >H -H >ltail_length
-@(ex2_3_intro … J (K.ⓑ{I}V) W) /2 width=1 by length_pair/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma append_inj_sn: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |K1| = |K2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #V2 #L1 #L2 #_ >length_atom >length_pair
- #H destruct
-| #K1 #I1 #V1 #IH *
- [ #L1 #L2 #_ >length_atom >length_pair
- #H destruct
- | #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /2 width=1 by conj/
- ]
-]
-qed-.
-
-(* Note: lemma 750 *)
-lemma append_inj_dx: ∀K1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 → |L1| = |L2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #V2 #L1 #L2 >append_atom >append_pair #H destruct
- >length_pair >append_length >plus_n_Sm
- #H elim (plus_xSy_x_false … H)
-| #K1 #I1 #V1 #IH *
- [ #L1 #L2 >append_pair >append_atom #H destruct
- >length_pair >append_length >plus_n_Sm #H
- lapply (discr_plus_x_xy … H) -H #H destruct
- | #K2 #I2 #V2 #L1 #L2 >append_pair >append_pair #H1 #H2
- elim (destruct_lpair_lpair_aux … H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /2 width=1 by conj/
- ]
-]
-qed-.
-
-lemma append_inv_refl_dx: ∀L,K. L @@ K = L → K = ⋆.
-#L #K #H elim (append_inj_dx … (⋆) … H) //
-qed-.
-
-lemma append_inv_pair_dx: ∀I,L,K,V. L @@ K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
-#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
-qed-.
-
-lemma length_inv_pos_dx_ltail: ∀L,l. |L| = ⫯l →
- ∃∃I,K,V. |K| = l & L = ⓑ{I}V.K.
-#Y #l #H elim (length_inv_succ_dx … H) -H #I #L #V #Hl #HLK destruct
-elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
-qed-.
-
-lemma length_inv_pos_sn_ltail: ∀L,l. ⫯l = |L| →
- ∃∃I,K,V. l = |K| & L = ⓑ{I}V.K.
-#Y #l #H elim (length_inv_succ_sn … H) -H #I #L #V #Hl #HLK destruct
-elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
-qed-.
-
-(* Basic eliminators ********************************************************)
-
-(* Basic_1: was: c_tail_ind *)
-lemma lenv_ind_alt: ∀R:predicate lenv.
- R (⋆) → (∀I,L,T. R L → R (ⓑ{I}T.L)) →
- ∀L. R L.
-#R #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * // -IH1
-#L #I #V #H destruct elim (lpair_ltail L I V) /4 width=1 by/
-qed-.
d : term degree
e : reserved: future use (\lambda\delta 3)
f,g : local reference transforming map
-h : sort degree parameter
+h : sort hierarchy parameter
i,j : local reference depth (de Bruijn's)
k,l : global reference level
m :
n : type iterations
-o : sort hierarchy parameter
+o : sort degree parameter
p,q : binder polarity
r : reduction kind parameter (true = ordinary, false = extended)
s : sort index
notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ÷ break term 46 A )"
non associative with precedence 45
- for @{ 'BinaryArity $o $L $T $A }.
+ for @{ 'BinaryArity $h $L $T $A }.
notation "hvbox( h ⊢ break term 46 L1 ÷ ⫃ break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEqB $o $L1 $L2 }.
+ for @{ 'LRSubEqB $h $L1 $L2 }.
notation "hvbox( L1 ⊢ ⬌ break term 46 L2 )"
non associative with precedence 45
notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : break term 46 T2 )"
non associative with precedence 45
- for @{ 'NativeType $o $L $T1 $T2 }.
+ for @{ 'NativeType $h $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : : break term 46 T2 )"
non associative with precedence 45
- for @{ 'NativeTypeAlt $o $L $T1 $T2 }.
+ for @{ 'NativeTypeAlt $h $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T1 : * break term 46 T2 )"
non associative with precedence 45
- for @{ 'NativeTypeStar $o $L $T1 $T2 }.
+ for @{ 'NativeTypeStar $h $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'BTPRed $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'BTPRed $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ ≽ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'BTPRedAlt $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'BTPRedAlt $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'BTPRedProper $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'BTPRedProper $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'BTPRedStar $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'BTPRedStar $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ ≥ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'BTPRedStarAlt $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'BTPRedStarAlt $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
notation "hvbox( ⦥ [ term 46 o, break term 46 h ] break ⦃ term 46 G, break term 46 L, break term 46 T ⦄ )"
non associative with precedence 45
- for @{ 'BTSN $o $h $G $L $T }.
+ for @{ 'BTSN $h $o $G $L $T }.
notation "hvbox( ⦥ ⦥ [ term 46 o, break term 46 h ] break ⦃ term 46 G, break term 46 L, break term 46 T ⦄ )"
non associative with precedence 45
- for @{ 'BTSNAlt $o $h $G $L $T }.
+ for @{ 'BTSNAlt $h $o $G $L $T }.
notation "hvbox( G ⊢ ~ ⬊ * break [ term 46 o , break term 46 h , break term 46 f ] break term 46 L )"
non associative with precedence 45
- for @{ 'CoSN $o $h $f $G $L }.
+ for @{ 'CoSN $h $o $f $G $L }.
+++ /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 T ▪ break [ term 46 o , break term 46 h ] break term 46 d )"
- non associative with precedence 45
- for @{ 'Degree $o $h $G $L $T $d }.
notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • * ⬌ * break [ term 46 o , break term 46 h , break term 46 n1 , break term 46 n2 ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'DPConvStar $o $h $n1 $n2 $G $L $T1 $T2 }.
+ for @{ 'DPConvStar $h $o $n1 $n2 $G $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 • * ➡ * break [ term 46 o , break term 46 h , break term 46 n ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'DPRedStar $o $h $n $G $L $T1 $T2 }.
+ for @{ 'DPRedStar $h $o $n $G $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ >≡ break [ term 46 o, break term 46 h ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'LazyBTPRedStarProper $o $h $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'LazyBTPRedStarProper $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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( G ⊢ break term 46 L1 ⫃ ▪ break [ term 46 o, break term 46 h ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqD $o $h $G $L1 $L2 }.
notation "hvbox( G ⊢ break term 46 L1 ⫃ ¡ break [ term 46 o, break term 46 h ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEqV $o $h $G $L1 $L2 }.
+ for @{ 'LRSubEqV $h $o $G $L1 $L2 }.
notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 o, break term 46 h ] )"
non associative with precedence 45
- for @{ 'NativeValid $o $h $G $L $T }.
+ for @{ 'NativeValid $h $o $G $L $T }.
notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 o , break term 46 h , break term 46 d ] )"
non associative with precedence 45
- for @{ 'NativeValid $o $h $d $G $L $T }.
+ for @{ 'NativeValid $h $o $d $G $L $T }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ break [ term 46 o , break term 46 h ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'PRed $o $h $G $L $T1 $T2 }.
+ for @{ 'PRed $h $o $G $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break [ term 46 o , break term 46 h ] break 𝐍 ⦃ term 46 T2 ⦄ )"
non associative with precedence 45
- for @{ 'PRedEval $o $h $G $L $T1 $T2 }.
+ for @{ 'PRedEval $h $o $G $L $T1 $T2 }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐍 break ⦃ term 46 T ⦄ )"
non associative with precedence 45
- for @{ 'PRedNormal $o $h $G $L $T }.
+ for @{ 'PRedNormal $h $o $G $L $T }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐈 break ⦃ term 46 T ⦄ )"
non associative with precedence 45
- for @{ 'PRedNotReducible $o $h $G $L $T }.
+ for @{ 'PRedNotReducible $h $o $G $L $T }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] 𝐑 break ⦃ term 46 T ⦄ )"
non associative with precedence 45
- for @{ 'PRedReducible $o $h $G $L $T }.
+ for @{ 'PRedReducible $h $o $G $L $T }.
notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ break [ term 46 o , break term 46 h ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'PRedSn $o $h $G $L1 $L2 }.
+ for @{ 'PRedSn $h $o $G $L1 $L2 }.
notation "hvbox( ⦃ term 46 G, break term 46 L1 ⦄ ⊢ ➡ * break [ term 46 o , break term 46 h ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'PRedSnStar $o $h $G $L1 $L2 }.
+ for @{ 'PRedSnStar $h $o $G $L1 $L2 }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * break [ term 46 o , break term 46 h ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'PRedStar $o $h $G $L $T1 $T2 }.
+ for @{ 'PRedStar $h $o $G $L $T1 $T2 }.
--- /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 i ] break term 46 L1 ≡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RDropStar $i $L1 $L2 }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬊ * break [ term 46 o, break term 46 h ] break term 46 T )"
non associative with precedence 45
- for @{ 'SN $o $h $G $L $T }.
+ for @{ 'SN $h $o $G $L $T }.
notation "hvbox( G ⊢ ⬊ * break [ term 46 o , break term 46 h , break term 46 T , break term 46 f ] break term 46 L )"
non associative with precedence 45
- for @{ 'SN $o $h $T $f $G $L }.
+ for @{ 'SN $h $o $T $f $G $L }.
notation "hvbox( ⦃ term 46 G, break term 46 L ⦄ ⊢ ⬊ ⬊ * break [ term 46 o , break term 46 h ] break term 46 T )"
non associative with precedence 45
- for @{ 'SNAlt $o $h $G $L $T }.
+ for @{ 'SNAlt $h $o $G $L $T }.
notation "hvbox( G ⊢ ⬊ ⬊ * break [ term 46 o , break term 46 h , break term 46 T , break term 46 f ] break term 46 L )"
non associative with precedence 45
- for @{ 'SNAlt $o $h $T $f $G $L }.
+ for @{ 'SNAlt $h $o $T $f $G $L }.
notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T1 •* break [ term 46 o , break term 46 n ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'StaticTypeStar $o $G $L $n $T1 $T2 }.
+ for @{ 'StaticTypeStar $h $G $L $n $T1 $T2 }.
+++ /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 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⊐⸮ break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermOptAlt $G1 $L1 $T1 $G2 $L2 $T2 }.
(* Basic properties *********************************************************)
-lemma cir_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆k⦄.
+lemma cir_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐈⦃⋆s⦄.
/2 width=4 by crr_inv_sort/ qed.
lemma cir_gref: ∀G,L,p. ⦃G, L⦄ ⊢ ➡ 𝐈⦃§p⦄.
(* Properties on relocation *************************************************)
-lemma cir_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄.
+lemma cir_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄.
/3 width=8 by crr_inv_lift/ qed.
-lemma cir_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄.
+lemma cir_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐈⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐈⦃T⦄.
/3 width=8 by crr_lift/ qed-.
(* IRREDUCIBLE TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ***************)
definition cix: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⊥.
+ λh,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⊥.
interpretation "irreducibility for context-sensitive extended reduction (term)"
- 'PRedNotReducible h g G L T = (cix h g G L T).
+ 'PRedNotReducible h o G L T = (cix h o G L T).
(* Basic inversion lemmas ***************************************************)
-lemma cix_inv_sort: ∀h,g,G,L,k,d. deg h g k (d+1) → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄ → ⊥.
+lemma cix_inv_sort: ∀h,o,G,L,s,d. deg h o s (d+1) → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄ → ⊥.
/3 width=2 by crx_sort/ qed-.
-lemma cix_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄ → ⊥.
+lemma cix_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄ → ⊥.
/3 width=4 by crx_delta/ qed-.
-lemma cix_inv_ri2: ∀h,g,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃②{I}V.T⦄ → ⊥.
+lemma cix_inv_ri2: ∀h,o,I,G,L,V,T. ri2 I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃②{I}V.T⦄ → ⊥.
/3 width=1 by crx_ri2/ qed-.
-lemma cix_inv_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+lemma cix_inv_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄.
/4 width=1 by crx_ib2_sn, crx_ib2_dx, conj/ qed-.
-lemma cix_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & ib2 a I.
-#h #g #a * [ elim a -a ]
+lemma cix_inv_bind: ∀h,o,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & ib2 a I.
+#h #o #a * [ elim a -a ]
#G #L #V #T #H [ elim H -H /3 width=1 by crx_ri2, or_introl/ ]
elim (cix_inv_ib2 … H) -H /3 width=1 by and3_intro, or_introl/
qed-.
-lemma cix_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄.
-#h #g #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/
+lemma cix_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄.
+#h #o #G #L #V #T #HVT @and3_intro /3 width=1 by crx_appl_sn, crx_appl_dx/
generalize in match HVT; -HVT elim T -T //
* // #a * #U #T #_ #_ #H elim H -H /2 width=1 by crx_beta, crx_theta/
qed-.
-lemma cix_inv_flat: ∀h,g,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓕ{I}V.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
-#h #g * #G #L #V #T #H
+lemma cix_inv_flat: ∀h,o,I,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓕ{I}V.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
+#h #o * #G #L #V #T #H
[ elim (cix_inv_appl … H) -H /2 width=1 by and4_intro/
| elim (cix_inv_ri2 … H) -H //
]
(* Basic forward lemmas *****************************************************)
-lemma cix_inv_cir: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
+lemma cix_inv_cir: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T⦄.
/3 width=1 by crr_crx/ qed-.
(* Basic properties *********************************************************)
-lemma cix_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃⋆k⦄.
-#h #g #G #L #k #Hk #H elim (crx_inv_sort … H) -L #d #Hkd
-lapply (deg_mono … Hk Hkd) -h -
k <plus_n_Sm #H destruct
+lemma cix_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃⋆s⦄.
+#h #o #G #L #s #Hk #H elim (crx_inv_sort … H) -L #d #Hkd
+lapply (deg_mono … Hk Hkd) -h -
s <plus_n_Sm #H destruct
qed.
-lemma tix_lref: ∀h,g,G,i. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄.
-#h #g #G #i #H elim (trx_inv_atom … H) -H #k #d #_ #H destruct
+lemma tix_lref: ∀h,o,G,i. ⦃G, ⋆⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄.
+#h #o #G #i #H elim (trx_inv_atom … H) -H #s #d #_ #H destruct
qed.
-lemma cix_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃§p⦄.
-#h #g #G #L #p #H elim (crx_inv_gref … H)
+lemma cix_gref: ∀h,o,G,L,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃§p⦄.
+#h #o #G #L #p #H elim (crx_inv_gref … H)
qed.
-lemma cix_ib2: ∀h,g,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓑ{a,I}V.T⦄.
-#h #g #a #I #G #L #V #T #HI #HV #HT #H
+lemma cix_ib2: ∀h,o,a,I,G,L,V,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ → ⦃G, L.ⓑ{I}V⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓑ{a,I}V.T⦄.
+#h #o #a #I #G #L #V #T #HI #HV #HT #H
elim (crx_inv_ib2 … HI H) -HI -H /2 width=1 by/
qed.
-lemma cix_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃ⓐV.T⦄.
-#h #g #G #L #V #T #HV #HT #H1 #H2
+lemma cix_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃ⓐV.T⦄.
+#h #o #G #L #V #T #HV #HT #H1 #H2
elim (crx_inv_appl … H2) -H2 /2 width=1 by/
qed.
(* Advanced properties ******************************************************)
-lemma cix_lref: ∀h,g,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃#i⦄.
-#h #g #G #L #i #HL #H elim (crx_inv_lref … H) -h #I #K #V #HLK
+lemma cix_lref: ∀h,o,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃#i⦄.
+#h #o #G #L #i #HL #H elim (crx_inv_lref … H) -h #I #K #V #HLK
lapply (drop_mono … HLK … HL) -L -i #H destruct
qed.
(* Properties on relocation *************************************************)
-lemma cix_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃U⦄.
+lemma cix_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃U⦄.
/3 width=8 by crx_inv_lift/ qed.
-lemma cix_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+lemma cix_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, o] 𝐈⦃T⦄.
/3 width=8 by crx_lift/ qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: nf2_sort *)
-lemma cnr_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆k⦄.
-#G #L #k #X #H
+lemma cnr_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐍⦃⋆s⦄.
+#G #L #s #X #H
>(cpr_inv_sort1 … H) //
qed.
(* Relocation properties ****************************************************)
(* Basic_1: was: nf2_lift *)
-lemma cnr_lift: ∀G,L0,L,T,T0,s,l,m. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ →
- ⬇[s, l, m] L0 ≡ L → ⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄.
-#G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
+lemma cnr_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄ →
+ ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄.
+#G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H
elim (cpr_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
<(HLT … HT1) in HT0; -L #HT0
>(lift_mono … HT10 … HT0) -T1 -X //
qed.
(* Note: this was missing in basic_1 *)
-lemma cnr_inv_lift: ∀G,L0,L,T,T0,s,l,m. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ →
- ⬇[s, l, m] L0 ≡ L → ⬆[l, m] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
-#G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X l m) #X0 #HX0
+lemma cnr_inv_lift: ∀G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡ 𝐍⦃T0⦄ →
+ ⬇[c, l, k] L0 ≡ L → ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+#G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l k) #X0 #HX0
lapply (cpr_lift … H … HL0 … HT0 … HX0) -L #HTX0
>(HLT0 … HTX0) in HX0; -L0 -X0 #H
->(lift_inj … H … HT0) -T0 -X -s -l -m //
+>(lift_inj … H … HT0) -T0 -X -c -l -k //
qed-.
(* NORMAL TERMS FOR CONTEXT-SENSITIVE EXTENDED REDUCTION ********************)
definition cnx: ∀h. sd h → relation3 genv lenv term ≝
- λh,g,G,L. NF … (cpx h g G L) (eq …).
+ λh,o,G,L. NF … (cpx h o G L) (eq …).
interpretation
"normality for context-sensitive extended reduction (term)"
- 'PRedNormal h g L T = (cnx h g L T).
+ 'PRedNormal h o L T = (cnx h o L T).
(* Basic inversion lemmas ***************************************************)
-lemma cnx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄ → deg h g k 0.
-#h #g #G #L #k #H elim (deg_total h g k)
+lemma cnx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄ → deg h o s 0.
+#h #o #G #L #s #H elim (deg_total h o s)
#d @(nat_ind_plus … d) -d // #d #_ #Hkd
-lapply (H (⋆(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H
+lapply (H (⋆(next h s)) ?) -H /2 width=2 by cpx_st/ -L -d #H
lapply (destruct_tatom_tatom_aux … H) -H #H (**) (* destruct lemma needed *)
lapply (destruct_sort_sort_aux … H) -H #H (**) (* destruct lemma needed *)
-lapply (next_lt h k) >H -H #H elim (lt_refl_false … H)
+lapply (next_lt h s) >H -H #H elim (lt_refl_false … H)
qed-.
-lemma cnx_inv_delta: ∀h,g,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄ → ⊥.
-#h #g #I #G #L #K #V #i #HLK #H
+lemma cnx_inv_delta: ∀h,o,I,G,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄ → ⊥.
+#h #o #I #G #L #K #V #i #HLK #H
elim (lift_total V 0 (i+1)) #W #HVW
lapply (H W ?) -H [ /3 width=7 by cpx_delta/ ] -HLK #H destruct
elim (lift_inv_lref2_be … HVW) -HVW /2 width=1 by ylt_inj/
qed-.
-lemma cnx_inv_abst: ∀h,g,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}V.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-#h #g #a #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abst: ∀h,o,a,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}V.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓛV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
+#h #o #a #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (ⓛ{a}V2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓛ{a}V1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
]
qed-.
-lemma cnx_inv_abbr: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃-ⓓV.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-#h #g #G #L #V1 #T1 #HVT1 @conj
+lemma cnx_inv_abbr: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃-ⓓV.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ ∧ ⦃G, L.ⓓV⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
+#h #o #G #L #V1 #T1 #HVT1 @conj
[ #V2 #HV2 lapply (HVT1 (-ⓓV2.T1) ?) -HVT1 /2 width=2 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (-ⓓV1.T2) ?) -HVT1 /2 width=2 by cpx_bind/ -HT2 #H destruct //
]
qed-.
-lemma cnx_inv_zeta: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃+ⓓV.T⦄ → ⊥.
-#h #g #G #L #V #T #H elim (is_lift_dec T 0 1)
+lemma cnx_inv_zeta: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃+ⓓV.T⦄ → ⊥.
+#h #o #G #L #V #T #H elim (is_lift_dec T 0 1)
[ * #U #HTU
lapply (H U ?) -H /2 width=3 by cpx_zeta/ #H destruct
elim (lift_inv_pair_xy_y … HTU)
]
qed-.
-lemma cnx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄ →
- ∧∧ ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ & 𝐒⦃T⦄.
-#h #g #G #L #V1 #T1 #HVT1 @and3_intro
+lemma cnx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄ →
+ ∧∧ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ & ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+#h #o #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2 #H destruct //
| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
]
qed-.
-lemma cnx_inv_eps: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓝV.T⦄ → ⊥.
-#h #g #G #L #V #T #H lapply (H T ?) -H
+lemma cnx_inv_eps: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓝV.T⦄ → ⊥.
+#h #o #G #L #V #T #H lapply (H T ?) -H
/2 width=4 by cpx_eps, discr_tpair_xy_y/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cnx_fwd_cnr: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
-#h #g #G #L #T #H #U #HTU
+lemma cnx_fwd_cnr: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡ 𝐍⦃T⦄.
+#h #o #G #L #T #H #U #HTU
@H /2 width=1 by cpr_cpx/ (**) (* auto fails because a δ-expansion gets in the way *)
qed-.
(* Basic properties *********************************************************)
-lemma cnx_sort: ∀h,g,G,L,k. deg h g k 0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆k⦄.
-#h #g #G #L #k #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_
+lemma cnx_sort: ∀h,o,G,L,s. deg h o s 0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆s⦄.
+#h #o #G #L #s #Hk #X #H elim (cpx_inv_sort1 … H) -H // * #d #Hkd #_
lapply (deg_mono … Hkd Hk) -h -L <plus_n_Sm #H destruct
qed.
-lemma cnx_sort_iter: ∀h,g,G,L,k,d. deg h g k d → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃⋆((next h)^d k)⦄.
-#h #g #G #L #k #d #Hkd
+lemma cnx_sort_iter: ∀h,o,G,L,s,d. deg h o s d → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃⋆((next h)^d s)⦄.
+#h #o #G #L #s #d #Hkd
lapply (deg_iter … d Hkd) -Hkd <minus_n_n /2 width=6 by cnx_sort/
qed.
-lemma cnx_lref_free: ∀h,g,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
-#h #g #G #L #i #Hi #X #H elim (cpx_inv_lref1 … H) -H // *
+lemma cnx_lref_free: ∀h,o,G,L,i. |L| ≤ i → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄.
+#h #o #G #L #i #Hi #X #H elim (cpx_inv_lref1 … H) -H // *
#I #K #V1 #V2 #HLK lapply (drop_fwd_length_lt2 … HLK) -HLK
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed.
-lemma cnx_lref_atom: ∀h,g,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃#i⦄.
-#h #g #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
+lemma cnx_lref_atom: ∀h,o,G,L,i. ⬇[i] L ≡ ⋆ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃#i⦄.
+#h #o #G #L #i #HL @cnx_lref_free >(drop_fwd_length … HL) -HL //
qed.
-lemma cnx_abst: ∀h,g,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓛ{a}W.T⦄.
-#h #g #a #G #L #W #T #HW #HT #X #H
+lemma cnx_abst: ∀h,o,a,G,L,W,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃W⦄ → ⦃G, L.ⓛW⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓛ{a}W.T⦄.
+#h #o #a #G #L #W #T #HW #HT #X #H
elim (cpx_inv_abst1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
>(HW … HW0) -W0 >(HT … HT0) -T0 //
qed.
-lemma cnx_appl_simple: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃ⓐV.T⦄.
-#h #g #G #L #V #T #HV #HT #HS #X #H
+lemma cnx_appl_simple: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃V⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃ⓐV.T⦄.
+#h #o #G #L #V #T #HV #HT #HS #X #H
elim (cpx_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
>(HV … HV0) -V0 >(HT … HT0) -T0 //
qed.
-axiom cnx_dec: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T1⦄ ∨
- ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & (T1 = T2 → ⊥).
+axiom cnx_dec: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T1⦄ ∨
+ ∃∃T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & (T1 = T2 → ⊥).
(* Main properties on irreducibility ****************************************)
-theorem cix_cnx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
+theorem cix_cnx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
/2 width=6 by cpx_fwd_cix/ qed.
(* Main inversion lemmas on irreducibility **********************************)
-theorem cnx_inv_cix: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T⦄.
+theorem cnx_inv_cix: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T⦄.
/2 width=7 by cnx_inv_crx/ qed-.
(* Advanced inversion lemmas on reducibility ********************************)
(* Note: this property is unusual *)
-lemma cnx_inv_crx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⊥.
-#h #g #G #L #T #H elim H -L -T
-[ #L #k #d #Hkd #H
+lemma cnx_inv_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⊥.
+#h #o #G #L #T #H elim H -L -T
+[ #L #s #d #Hkd #H
lapply (cnx_inv_sort … H) -H #H
- lapply (deg_mono … H Hkd) -h -L -
k <plus_n_Sm #H destruct
+ lapply (deg_mono … H Hkd) -h -L -
s <plus_n_Sm #H destruct
| #I #L #K #V #i #HLK #H
elim (cnx_inv_delta … HLK H)
| #L #V #T #_ #IHV #H
(* Relocation properties ****************************************************)
-lemma cnx_lift: ∀h,g,G,L0,L,T,T0,s,l,m. ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → ⬇[s, l, m] L0 ≡ L →
- ⬆[l, m] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄.
-#h #g #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
+lemma cnx_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⬇[c, l, k] L0 ≡ L →
+ ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄.
+#h #o #G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H
elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
<(HLT … HT1) in HT0; -L #HT0
>(lift_mono … HT10 … HT0) -T1 -X //
qed.
-lemma cnx_inv_lift: ∀h,g,G,L0,L,T,T0,s,l,m. ⦃G, L0⦄ ⊢ ➡[h, g] 𝐍⦃T0⦄ → ⬇[s, l, m] L0 ≡ L →
- ⬆[l, m] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄.
-#h #g #G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X l m) #X0 #HX0
+lemma cnx_inv_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄ → ⬇[c, l, k] L0 ≡ L →
+ ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
+#h #o #G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l k) #X0 #HX0
lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0
>(HLT0 … HTX0) in HX0; -L0 -X0 #H
->(lift_inj … H … HT0) -T0 -X -l -m //
+>(lift_inj … H … HT0) -T0 -X -l -k //
qed-.
/2 width=3 by cpr_inv_atom1_aux/ qed-.
(* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡ T2 → T2 = ⋆k.
-#G #L #T2 #k #H
+lemma cpr_inv_sort1: ∀G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡ T2 → T2 = ⋆s.
+#G #L #T2 #s #H
elim (cpr_inv_atom1 … H) -H //
* #K #V #V2 #i #_ #_ #_ #H destruct
qed-.
(* Basic_1: includes: pr0_lift pr2_lift *)
lemma cpr_lift: ∀G. d_liftable (cpr G).
#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
+[ #I #G #K #L #c #l #k #_ #U1 #H1 #U2 #H2
>(lift_mono … H1 … H2) -H1 -H2 //
-| #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+| #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #c #l #k #HLK #U1 #H #U2 #HWU2
elim (lift_inv_lref1 … H) * #Hil #H destruct
[ elim (lift_trans_ge … HVW2 … HWU2) -W2 /2 width=1 by ylt_fwd_le_succ1/ #W2 #HVW2 #HWU2
elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H
| lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil /3 width=6 by cpr_delta, drop_inv_gen/
]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #c #l #k #HLK #U1 #H1 #U2 #H2
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpr_bind, drop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #c #l #k #HLK #U1 #H1 #U2 #H2
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpr_flat/
-| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #c #l #k #HLK #U1 #H #U2 #HTU2
elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpr_zeta, drop_skip/
-| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T2 #_ #IHT12 #L #c #l #k #HLK #U1 #H #U2 #HTU2
elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpr_eps/
-| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #c #l #k #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpr_beta, drop_skip/
-| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #c #l #k #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
lemma cpr_inv_lift1: ∀G. d_deliftable_sn (cpr G).
#G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #i #G #L #K #s #l #m #_ #T1 #H
+[ * #i #G #L #K #c #l #k #_ #T1 #H
[ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
| elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+| #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #c #l #k #HLK #T1 #H
elim (lift_inv_lref2 … H) -H * #Hil #H destruct
[ elim (drop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
| elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
lapply (yle_inv_inj … Hmi) -Hmi #Hmi
lapply (drop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 l (i - m + 1)) -HVW2 [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim
+ elim (lift_split … HVW2 l (i - k + 1)) -HVW2 [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O
/3 width=8 by cpr_delta, ex2_intro/
]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpr_bind, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1
elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=6 by cpr_flat, lift_flat, ex2_intro/
-| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU1 (K.ⓓW1) s … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
+ elim (IHU1 (K.ⓓW1) c … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
elim (lift_div_le … HU2 … HTU) -U /3 width=6 by cpr_zeta, ex2_intro/
-| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
+| #G #L #V #U1 #U2 #_ #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpr_eps, ex2_intro/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #c #l #k #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 … HLK … HV01) -V1
- elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by drop_skip/
+ elim (IHT12 (K.ⓛW0) c … HT01) -T1 /2 width=1 by drop_skip/
elim (IHW12 … HLK … HW01) -W1 /4 width=7 by cpr_beta, lift_flat, lift_bind, ex2_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #c #l #k #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
- elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+ elim (IHT12 (K.ⓓW0) c … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
elim (IHW12 … HLK … HW01) -W1
elim (lift_trans_le … HV3 … HV2) -V
/4 width=9 by cpr_theta, lift_flat, lift_bind, ex2_intro/
(* Properties on lazy sn pointwise extensions *******************************)
lemma cpr_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R →
- ∀G. s_r_confluent1 … (cpr G) (llpx_sn R 0).
+ ∀G. c_r_confluent1 … (cpr G) (llpx_sn R 0).
#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
-| #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1l #HLKd #HV1s #HV1sd
+| #G #Ls #Ks #V1c #V2c #W2c #i #HLKs #_ #HVW2c #IHV12c #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1l #HLKd #HV1c #HV1sd
lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2c) -HLKs -HLKd -HVW2c /2 width=1 by/ (**) (* full auto too slow *)
| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
/4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
(* avtivate genv *)
-inductive cpx (h) (g): relation4 genv lenv term term ≝
-| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
-| cpx_st : ∀G,L,k,d. deg h g k (d+1) → cpx h g G L (⋆k) (⋆(next h k))
+inductive cpx (h) (o): relation4 genv lenv term term ≝
+| cpx_atom : ∀I,G,L. cpx h o G L (⓪{I}) (⓪{I})
+| cpx_st : ∀G,L,s,d. deg h o s (d+1) → cpx h o G L (⋆s) (⋆(next h s))
| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
- ⬇[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
- ⬆[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
+ ⬇[i] L ≡ K.ⓑ{I}V → cpx h o G K V V2 →
+ ⬆[0, i+1] V2 ≡ W2 → cpx h o G L (#i) W2
| cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
- cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
- cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+ cpx h o G L V1 V2 → cpx h o G (L.ⓑ{I}V1) T1 T2 →
+ cpx h o G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
| cpx_flat : ∀I,G,L,V1,V2,T1,T2.
- cpx h g G L V1 V2 → cpx h g G L T1 T2 →
- cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
- ⬆[0, 1] T2 ≡ T → cpx h g G L (+ⓓV.T1) T2
-| cpx_eps : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
-| cpx_ct : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
+ cpx h o G L V1 V2 → cpx h o G L T1 T2 →
+ cpx h o G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h o G (L.ⓓV) T1 T →
+ ⬆[0, 1] T2 ≡ T → cpx h o G L (+ⓓV.T1) T2
+| cpx_eps : ∀G,L,V,T1,T2. cpx h o G L T1 T2 → cpx h o G L (ⓝV.T1) T2
+| cpx_ct : ∀G,L,V1,V2,T. cpx h o G L V1 V2 → cpx h o G L (ⓝV1.T) V2
| cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
- cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
- cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
+ cpx h o G L V1 V2 → cpx h o G L W1 W2 → cpx h o G (L.ⓛW1) T1 T2 →
+ cpx h o G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
| cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- cpx h g G L V1 V → ⬆[0, 1] V ≡ V2 → cpx h g G L W1 W2 →
- cpx h g G (L.ⓓW1) T1 T2 →
- cpx h g G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
+ cpx h o G L V1 V → ⬆[0, 1] V ≡ V2 → cpx h o G L W1 W2 →
+ cpx h o G (L.ⓓW1) T1 T2 →
+ cpx h o G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
.
interpretation
"context-sensitive extended parallel reduction (term)"
- 'PRed h g G L T1 T2 = (cpx h g G L T1 T2).
+ 'PRed h o G L T1 T2 = (cpx h o G L T1 T2).
(* Basic properties *********************************************************)
-lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
-#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+lemma lsubr_cpx_trans: ∀h,o,G. lsub_trans … (cpx h o G) lsubr.
+#h #o #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
[ //
| /2 width=2 by cpx_st/
| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
qed-.
(* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
-lemma cpx_refl: ∀h,g,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
-#h #g #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
+lemma cpx_refl: ∀h,o,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, o] T.
+#h #o #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
qed.
-lemma cpr_cpx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2
+lemma cpr_cpx: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2.
+#h #o #G #L #T1 #T2 #H elim H -L -T1 -T2
/2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
qed.
-lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
- ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
-#h #g * /2 width=1 by cpx_bind, cpx_flat/
+lemma cpx_pair_sn: ∀h,o,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+ ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, o] ②{I}V2.T.
+#h #o * /2 width=1 by cpx_bind, cpx_flat/
qed.
-lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⬆[l, 1] T ≡ T2.
-#h #g #I #G #K #V #T1 elim T1 -T1
+lemma cpx_delift: ∀h,o,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & ⬆[l, 1] T ≡ T2.
+#h #o #I #G #K #V #T1 elim T1 -T1
[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
destruct
(* Basic inversion lemmas ***************************************************)
-fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
+fact cpx_inv_atom1_aux: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J}
- | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
- | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ | ∃∃s,d. deg h o s (d+1) & T2 = ⋆(next h s) & J = Sort s
+ | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
⬆[O, i+1] V2 ≡ T2 & J = LRef i.
-#G #h #g #L #T1 #T2 * -L -T1 -T2
+#G #h #o #L #T1 #T2 * -L -T1 -T2
[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
-| #G #L #k #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #G #L #s #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
]
qed-.
-lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
+lemma cpx_inv_atom1: ∀h,o,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, o] T2 →
∨∨ T2 = ⓪{J}
- | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
- | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ | ∃∃s,d. deg h o s (d+1) & T2 = ⋆(next h s) & J = Sort s
+ | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
⬆[O, i+1] V2 ≡ T2 & J = LRef i.
/2 width=3 by cpx_inv_atom1_aux/ qed-.
-lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
- ∃∃d. deg h g k (d+1) & T2 = ⋆(next h k).
-#h #g #G #L #T2 #k #H
+lemma cpx_inv_sort1: ∀h,o,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h, o] T2 → T2 = ⋆s ∨
+ ∃∃d. deg h o s (d+1) & T2 = ⋆(next h s).
+#h #o #G #L #T2 #s #H
elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
+[ #s0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
-lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
+lemma cpx_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, o] T2 →
T2 = #i ∨
- ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
⬆[O, i+1] V2 ≡ T2.
-#h #g #G #L #T2 #i #H
+#h #o #G #L #T2 #i #H
elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k #d #_ #_ #H destruct
+[ #s #d #_ #_ #H destruct
| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
-lemma cpx_inv_lref1_ge: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 → |L| ≤ i → T2 = #i.
-#h #g #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
+lemma cpx_inv_lref1_ge: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, o] T2 → |L| ≤ i → T2 = #i.
+#h #o #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
#I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed-.
-lemma cpx_inv_gref1: ∀h,g,G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
-#h #g #G #L #T2 #p #H
+lemma cpx_inv_gref1: ∀h,o,G,L,T2,p. ⦃G, L⦄ ⊢ §p ➡[h, o] T2 → T2 = §p.
+#h #o #G #L #T2 #p #H
elim (cpx_inv_atom1 … H) -H // *
-[ #k #d #_ #_ #H destruct
+[ #s #d #_ #_ #H destruct
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
-fact cpx_inv_bind1_aux: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
+fact cpx_inv_bind1_aux: ∀h,o,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, o] U2 →
∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, o] T2 &
U2 = ⓑ{a,J}V2.T2
) ∨
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+ ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T &
a = true & J = Abbr.
-#h #g #G #L #U1 #U2 * -L -U1 -U2
+#h #o #G #L #U1 #U2 * -L -U1 -U2
[ #I #G #L #b #J #W #U1 #H destruct
-| #G #L #k #d #_ #b #J #W #U1 #H destruct
+| #G #L #s #d #_ #b #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
]
qed-.
-lemma cpx_inv_bind1: ∀h,g,a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, g] U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_bind1: ∀h,o,a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, o] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, o] T2 &
U2 = ⓑ{a,I} V2. T2
) ∨
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+ ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T &
a = true & I = Abbr.
/2 width=3 by cpx_inv_bind1_aux/ qed-.
-lemma cpx_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g] U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_abbr1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, o] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T2 &
U2 = ⓓ{a} V2. T2
) ∨
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T & a = true.
-#h #g #a #G #L #V1 #T1 #U2 #H
+ ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T & a = true.
+#h #o #a #G #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_abst1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_abst1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, o] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, o] T2 &
U2 = ⓛ{a} V2. T2.
-#h #g #a #G #L #V1 #T1 #U2 #H
+#h #o #a #G #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H *
[ /3 width=5 by ex3_2_intro/
| #T #_ #_ #_ #H destruct
]
qed-.
-fact cpx_inv_flat1_aux: ∀h,g,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
+fact cpx_inv_flat1_aux: ∀h,o,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, o] U2 →
∀J,V1,U1. U = ⓕ{J}V1.U1 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
U2 = ⓕ{J}V2.T2
- | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ J = Cast)
- | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ J = Cast)
- | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | (⦃G, L⦄ ⊢ U1 ➡[h, o] U2 ∧ J = Cast)
+ | (⦃G, L⦄ ⊢ V1 ➡[h, o] U2 ∧ J = Cast)
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
-#h #g #G #L #U #U2 * -L -U -U2
+#h #o #G #L #U #U2 * -L -U -U2
[ #I #G #L #J #W #U1 #H destruct
-| #G #L #k #d #_ #J #W #U1 #H destruct
+| #G #L #s #d #_ #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
]
qed-.
-lemma cpx_inv_flat1: ∀h,g,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, g] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_flat1: ∀h,o,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, o] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
U2 = ⓕ{I} V2. T2
- | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ I = Cast)
- | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ I = Cast)
- | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | (⦃G, L⦄ ⊢ U1 ➡[h, o] U2 ∧ I = Cast)
+ | (⦃G, L⦄ ⊢ V1 ➡[h, o] U2 ∧ I = Cast)
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
/2 width=3 by cpx_inv_flat1_aux/ qed-.
-lemma cpx_inv_appl1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_appl1: ∀h,o,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, o] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
U2 = ⓐ V2. T2
- | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
- ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
-#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+#h #o #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
[ /3 width=5 by or3_intro0, ex3_2_intro/
|2,3: #_ #H destruct
| /3 width=11 by or3_intro1, ex5_6_intro/
qed-.
(* Note: the main property of simple terms *)
-lemma cpx_inv_appl1_simple: ∀h,g,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_appl1_simple: ∀h,o,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, o] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 &
U = ⓐV2.T2.
-#h #g #G #L #V1 #T1 #U #H #HT1
+#h #o #G #L #V1 #T1 #U #H #HT1
elim (cpx_inv_appl1 … H) -H *
[ /2 width=5 by ex3_2_intro/
| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
]
qed-.
-lemma cpx_inv_cast1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_cast1: ∀h,o,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, o] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
U2 = ⓝ V2. T2
- | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
- | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
-#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+ | ⦃G, L⦄ ⊢ U1 ➡[h, o] U2
+ | ⦃G, L⦄ ⊢ V1 ➡[h, o] U2.
+#h #o #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
[ /3 width=5 by or3_intro0, ex3_2_intro/
|2,3: /2 width=1 by or3_intro1, or3_intro2/
| #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,g,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, g] T → ∀b.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, g] ⓑ{b,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,o,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, o] T → ∀b.
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, o] ⓑ{b,I}V2.T2 &
T = -ⓑ{I}V2.T2.
-#h #g #I #G #L #V1 #T1 #T #H #b
+#h #o #I #G #L #V1 #T1 #T #H #b
elim (cpx_inv_bind1 … H) -H *
[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpx_bind, ex2_2_intro/
| #T2 #_ #_ #H destruct
(* Advanced forward lemmas on irreducibility ********************************)
-lemma cpx_fwd_cix: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ ➡[h, g] 𝐈⦃T1⦄ → T2 = T1.
-#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
+lemma cpx_fwd_cix: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐈⦃T1⦄ → T2 = T1.
+#h #o #G #L #T1 #T2 #H elim H -G -L -T1 -T2
[ //
-| #G #L #k #d #Hkd #H elim (cix_inv_sort … Hkd H)
+| #G #L #s #d #Hkd #H elim (cix_inv_sort … Hkd H)
| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
elim (cix_inv_delta … HLK) //
| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
(* Advanced properties ******************************************************)
-fact sta_cpx_aux: ∀h,g,G,L,T1,T2,d2,d1. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → d2 = 1 →
- ⦃G, L⦄ ⊢ T1 ▪[h, g] d1+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2
-[ #G #L #d2 #k #H0 destruct normalize
+fact sta_cpx_aux: ∀h,o,G,L,T1,T2,d2,d1. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 → d2 = 1 →
+ ⦃G, L⦄ ⊢ T1 ▪[h, o] d1+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2.
+#h #o #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2
+[ #G #L #d2 #s #H0 destruct normalize
/3 width=4 by cpx_st, da_inv_sort/
| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct
elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
]
qed-.
-lemma sta_cpx: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
- ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
+lemma sta_cpx: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
+ ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2.
/2 width=3 by sta_cpx_aux/ qed.
(* Relocation properties ****************************************************)
-lemma cpx_lift: ∀h,g,G. d_liftable (cpx h g G).
-#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
+lemma cpx_lift: ∀h,o,G. d_liftable (cpx h o G).
+#h #o #G #K #T1 #T2 #H elim H -G -K -T1 -T2
+[ #I #G #K #L #c #l #k #_ #U1 #H1 #U2 #H2
>(lift_mono … H1 … H2) -H1 -H2 //
-| #G #K #k #d #Hkd #L #s #l #m #_ #U1 #H1 #U2 #H2
+| #G #K #s #d #Hkd #L #c #l #k #_ #U1 #H1 #U2 #H2
>(lift_inv_sort1 … H1) -U1
>(lift_inv_sort1 … H2) -U2 /2 width=2 by cpx_st/
-| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #c #l #k #HLK #U1 #H #U2 #HWU2
elim (lift_inv_lref1 … H) * #Hil #H destruct
[ elim (lift_trans_ge … HVW2 … HWU2) -W2 /2 width=1 by ylt_fwd_le_succ1/ #W2 #HVW2 #HWU2
elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H
| lapply (lift_trans_be … HVW2 … HWU2 ? ?) -W2 /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
lapply (drop_trans_ge_comm … HLK … HKV ?) -K /3 width=7 by cpx_delta, drop_inv_gen/
]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #c #l #k #HLK #U1 #H1 #U2 #H2
elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpx_bind, drop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #c #l #k #HLK #U1 #H1 #U2 #H2
elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpx_flat/
-| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #c #l #k #HLK #U1 #H #U2 #HTU2
elim (lift_inv_bind1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 … HTU2 … HT2) -T2 /4 width=6 by cpx_zeta, drop_skip/
-| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T2 #_ #IHT12 #L #c #l #k #HLK #U1 #H #U2 #HTU2
elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_eps/
-| #G #K #V1 #V2 #T #_ #IHV12 #L #s #l #m #HLK #U1 #H #U2 #HVU2
+| #G #K #V1 #V2 #T #_ #IHV12 #L #c #l #k #HLK #U1 #H #U2 #HVU2
elim (lift_inv_flat1 … H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ct/
-| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #c #l #k #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
elim (lift_inv_flat1 … HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpx_beta, drop_skip/
-| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #c #l #k #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
]
qed.
-lemma cpx_inv_lift1: ∀h,g,G. d_deliftable_sn (cpx h g G).
-#h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #i #G #L #K #s #l #m #_ #T1 #H
+lemma cpx_inv_lift1: ∀h,o,G. d_deliftable_sn (cpx h o G).
+#h #o #G #L #U1 #U2 #H elim H -G -L -U1 -U2
+[ * #i #G #L #K #c #l #k #_ #T1 #H
[ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_sort, ex2_intro/
| elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
]
-| #G #L #k #d #Hkd #K #s #l #m #_ #T1 #H
+| #G #L #s #d #Hkd #K #c #l #k #_ #T1 #H
lapply (lift_inv_sort2 … H) -H #H destruct /3 width=3 by cpx_st, lift_sort, ex2_intro/
-| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #c #l #k #HLK #T1 #H
elim (lift_inv_lref2 … H) -H * #Hil #H destruct
[ elim (drop_conf_lt … HLK … HLV) -L // #L #U #HKL #HLV #HUV
elim (IHV2 … HLV … HUV) -V #U2 #HUV2 #HU2
| elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
lapply (yle_inv_inj … Hmi) -Hmi #Hmi
lapply (drop_conf_ge … HLK … HLV ?) -L // #HKLV
- elim (lift_split … HVW2 l (i - m + 1)) -HVW2 /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ -Hil -Hlim
+ elim (lift_split … HVW2 l (i - k + 1)) -HVW2 /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ -Hil -Hlim
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -IHV12 #W2 #HW12 #HWV2
elim (IHU12 … HTU1) -IHU12 -HTU1 /3 width=6 by cpx_bind, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -V1
elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
-| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
- elim (IHU1 (K.ⓓW1) s … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
+ elim (IHU1 (K.ⓓW1) c … HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
elim (lift_div_le … HU2 … HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
+| #G #L #V #U1 #U2 #_ #IHU12 #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU12 … HLK … HTU1) -L -U1 /3 width=3 by cpx_eps, ex2_intro/
-| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #l #m #HLK #X #H
+| #G #L #V1 #V2 #U1 #_ #IHV12 #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 … HLK … HWV1) -L -V1 /3 width=3 by cpx_ct, ex2_intro/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #c #l #k #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 … HLK … HV01) -V1 #V3 #HV32 #HV03
- elim (IHT12 (K.ⓛW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+ elim (IHT12 (K.ⓛW0) c … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
elim (IHW12 … HLK … HW01) -W1
/4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #c #l #k #HLK #X #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV1 … HLK … HV01) -V1 #V3 #HV3 #HV03
- elim (IHT12 (K.ⓓW0) s … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
+ elim (IHT12 (K.ⓓW0) c … HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
elim (IHW12 … HLK … HW01) -W1 #W3 #HW32 #HW03
elim (lift_trans_le … HV3 … HV2) -V
/4 width=9 by cpx_theta, lift_bind, lift_flat, ex2_intro/
(* Properties on supclosure *************************************************)
-lemma fqu_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
[ #I #G #L #V2 #U2 #HVU2
elim (lift_total U2 0 1)
/4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
-| #G #L #K #T1 #U1 #m #HLK1 #HTU1 #T2 #HTU2
- elim (lift_total T2 0 (m+1))
+| #G #L #K #T1 #U1 #k #HLK1 #HTU1 #T2 #HTU2
+ elim (lift_total T2 0 (k+1))
/3 width=11 by cpx_lift, fqu_drop, ex2_intro/
]
qed-.
-lemma fqu_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+lemma fqu_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 →
- ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d+1 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+ ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-.
-lemma fquq_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
+lemma fquq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fquq_inv_gen … H) -H
[ #HT12 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fquq_sta_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+lemma fquq_sta_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 •*[h, 1] U2 →
- ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, g] d+1 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+ ∀d. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d+1 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-.
-lemma fqup_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma fqup_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
/3 width=3 by fqu_fqup, ex2_intro/
| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
]
qed-.
-lemma fqus_cpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
+lemma fqus_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 elim (fqus_inv_gen … H) -H
[ #HT12 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
]
qed-.
-lemma fqu_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
#U2 #HVU2 @(ex3_intro … U2)
[1,3: /3 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop/
[1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
| #H0 destruct /2 width=1 by/
]
-| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
+| #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
#U2 #HTU2 @(ex3_intro … U2)
[1,3: /2 width=10 by cpx_lift, fqu_drop/
| #H0 destruct /3 width=5 by lift_inj/
]
qed-.
-lemma fquq_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+lemma fquq_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
[ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fquq, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
]
qed-.
-lemma fqup_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+lemma fqup_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
]
qed-.
-lemma fqus_cpx_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, g] U2 → (T2 = U2 → ⊥) →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+lemma fqus_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h, o] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
[ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
/3 width=4 by fqup_fqus, ex3_intro/
| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
(* Properties on lazy equivalence for local environments ********************)
-lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
+lemma lleq_cpx_trans: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+ ∀L1. L1 ≡[T1, 0] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2.
+#h #o #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_st/
[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
[ #H elim (ylt_yle_false … H) //
| * /3 width=7 by cpx_delta/
]
qed-.
-lemma cpx_lleq_conf: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
+ ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2.
/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
-lemma cpx_lleq_conf_sn: ∀h,g,G. s_r_confluent1 … (cpx h g G) (lleq 0).
+lemma cpx_lleq_conf_sn: ∀h,o,G. c_r_confluent1 … (cpx h o G) (lleq 0).
/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
-lemma cpx_lleq_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, o] T2 →
∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.
(* Note: lemma 1000 *)
lemma cpx_llpx_sn_conf: ∀R. (∀L. reflexive … (R L)) → d_liftable R → d_deliftable_sn R →
- ∀h,g,G. s_r_confluent1 … (cpx h g G) (llpx_sn R 0).
-#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
+ ∀h,o,G. c_r_confluent1 … (cpx h o G) (llpx_sn R 0).
+#R #H1R #H2R #H3R #h #o #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
-| #I #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
- #Kd #V1l #HLKd #HV1s #HV1sd
+| #I #G #Ls #Ks #V1c #V2c #W2c #i #HLKs #_ #HVW2c #IHV12c #Ld #H elim (llpx_sn_inv_lref_ge_sn … H … HLKs) // -H
+ #Kd #V1l #HLKd #HV1c #HV1sd
lapply (drop_fwd_drop2 … HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 … HLKd) -HLKd #HLKd
- @(llpx_sn_lift_le … HLKs HLKd … HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
+ @(llpx_sn_lift_le … HLKs HLKd … HVW2c) -HLKs -HLKd -HVW2c /2 width=1 by/ (**) (* full auto too slow *)
| #a #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_bind_O … H) -H
/4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
| #I #G #Ls #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #Ld #H elim (llpx_sn_inv_flat … H) -H
(* Properties on equivalence for local environments *************************)
-lemma lreq_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) (lreq 0 (∞)).
-#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+lemma lreq_cpx_trans: ∀h,o,G. lsub_trans … (cpx h o G) (lreq 0 (∞)).
+#h #o #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
[ //
| /2 width=2 by cpx_st/
| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
(* Basic inversion lemmas ***************************************************)
-fact crr_inv_sort_aux: ∀G,L,T,k. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⋆k → ⊥.
-#G #L #T #k0 * -L -T
+fact crr_inv_sort_aux: ∀G,L,T,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = ⋆s → ⊥.
+#G #L #T #s0 * -L -T
[ #L #K #V #i #HLK #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
]
qed-.
-lemma crr_inv_sort: ∀G,L,k. ⦃G, L⦄ ⊢ ➡ 𝐑⦃⋆k⦄ → ⊥.
+lemma crr_inv_sort: ∀G,L,s. ⦃G, L⦄ ⊢ ➡ 𝐑⦃⋆s⦄ → ⊥.
/2 width=6 by crr_inv_sort_aux/ qed-.
fact crr_inv_lref_aux: ∀G,L,T,i. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → T = #i →
(* Properties on relocation *************************************************)
-lemma crr_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄.
+lemma crr_lift: ∀G,K,T. ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄.
#G #K #T #H elim H -K -T
-[ #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+[ #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H
elim (lift_inv_lref1 … H) -H * #Hil #H destruct
[ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crr_delta/
| lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=4 by crr_delta, drop_inv_gen/
]
-| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
+| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
-| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
+| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
-| #I #K #V #T #HI #L #s #l #m #_ #X #H
+| #I #K #V #T #HI #L #c #l #k #_ #X #H
elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crr_ri2/
-| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
-| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
-| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+| #a #K #V #V0 #T #L #c #l #k #_ #X #H
elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_beta/
-| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+| #a #K #V #V0 #T #L #c #l #k #_ #X #H
elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_theta/
]
qed.
-lemma crr_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄.
+lemma crr_inv_lift: ∀G,L,U. ⦃G, L⦄ ⊢ ➡ 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡ 𝐑⦃T⦄.
#G #L #U #H elim H -L -U
-[ #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+[ #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H
elim (lift_inv_lref2 … H) -H * #Hil #H destruct
[ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crr_delta/
| lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crr_delta/
]
-| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
+| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
-| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
+| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
-| #I #L #W #U #HI #K #s #l #m #_ #X #H
+| #I #L #W #U #HI #K #c #l #k #_ #X #H
elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crr_ri2/
-| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
-| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
-| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+| #a #L #W #W0 #U #K #c #l #k #_ #X #H
elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_beta/
-| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+| #a #L #W #W0 #U #K #c #l #k #_ #X #H
elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_theta/
]
(* activate genv *)
(* extended reducible terms *)
-inductive crx (h) (g) (G:genv): relation2 lenv term ≝
-| crx_sort : ∀L,k,d. deg h g k (d+1) → crx h g G L (⋆k)
-| crx_delta : ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → crx h g G L (#i)
-| crx_appl_sn: ∀L,V,T. crx h g G L V → crx h g G L (ⓐV.T)
-| crx_appl_dx: ∀L,V,T. crx h g G L T → crx h g G L (ⓐV.T)
-| crx_ri2 : ∀I,L,V,T. ri2 I → crx h g G L (②{I}V.T)
-| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h g G L V → crx h g G L (ⓑ{a,I}V.T)
-| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h g G (L.ⓑ{I}V) T → crx h g G L (ⓑ{a,I}V.T)
-| crx_beta : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓛ{a}W.T)
-| crx_theta : ∀a,L,V,W,T. crx h g G L (ⓐV. ⓓ{a}W.T)
+inductive crx (h) (o) (G:genv): relation2 lenv term ≝
+| crx_sort : ∀L,s,d. deg h o s (d+1) → crx h o G L (⋆s)
+| crx_delta : ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → crx h o G L (#i)
+| crx_appl_sn: ∀L,V,T. crx h o G L V → crx h o G L (ⓐV.T)
+| crx_appl_dx: ∀L,V,T. crx h o G L T → crx h o G L (ⓐV.T)
+| crx_ri2 : ∀I,L,V,T. ri2 I → crx h o G L (②{I}V.T)
+| crx_ib2_sn : ∀a,I,L,V,T. ib2 a I → crx h o G L V → crx h o G L (ⓑ{a,I}V.T)
+| crx_ib2_dx : ∀a,I,L,V,T. ib2 a I → crx h o G (L.ⓑ{I}V) T → crx h o G L (ⓑ{a,I}V.T)
+| crx_beta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓛ{a}W.T)
+| crx_theta : ∀a,L,V,W,T. crx h o G L (ⓐV. ⓓ{a}W.T)
.
interpretation
"reducibility for context-sensitive extended reduction (term)"
- 'PRedReducible h g G L T = (crx h g G L T).
+ 'PRedReducible h o G L T = (crx h o G L T).
(* Basic properties *********************************************************)
-lemma crr_crx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
-#h #g #G #L #T #H elim H -L -T
+lemma crr_crx: ∀h,o,G,L,T. ⦃G, L⦄ ⊢ ➡ 𝐑⦃T⦄ → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄.
+#h #o #G #L #T #H elim H -L -T
/2 width=4 by crx_delta, crx_appl_sn, crx_appl_dx, crx_ri2, crx_ib2_sn, crx_ib2_dx, crx_beta, crx_theta/
qed.
(* Basic inversion lemmas ***************************************************)
-fact crx_inv_sort_aux: ∀h,g,G,L,T,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⋆k →
- ∃d. deg h g k (d+1).
-#h #g #G #L #T #k0 * -L -T
-[ #L #k #d #Hkd #H destruct /2 width=2 by ex_intro/
+fact crx_inv_sort_aux: ∀h,o,G,L,T,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⋆s →
+ ∃d. deg h o s (d+1).
+#h #o #G #L #T #s0 * -L -T
+[ #L #s #d #Hkd #H destruct /2 width=2 by ex_intro/
| #I #L #K #V #i #HLK #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
]
qed-.
-lemma crx_inv_sort: ∀h,g,G,L,k. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃⋆k⦄ → ∃d. deg h g k (d+1).
+lemma crx_inv_sort: ∀h,o,G,L,s. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃⋆s⦄ → ∃d. deg h o s (d+1).
/2 width=5 by crx_inv_sort_aux/ qed-.
-fact crx_inv_lref_aux: ∀h,g,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = #i →
+fact crx_inv_lref_aux: ∀h,o,G,L,T,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = #i →
∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
-#h #g #G #L #T #j * -L -T
-[ #L #k #d #_ #H destruct
+#h #o #G #L #T #j * -L -T
+[ #L #s #d #_ #H destruct
| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
]
qed-.
-lemma crx_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
+lemma crx_inv_lref: ∀h,o,G,L,i. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃#i⦄ → ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I}V.
/2 width=6 by crx_inv_lref_aux/ qed-.
-fact crx_inv_gref_aux: ∀h,g,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = §p → ⊥.
-#h #g #G #L #T #q * -L -T
-[ #L #k #d #_ #H destruct
+fact crx_inv_gref_aux: ∀h,o,G,L,T,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = §p → ⊥.
+#h #o #G #L #T #q * -L -T
+[ #L #s #d #_ #H destruct
| #I #L #K #V #i #HLK #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
]
qed-.
-lemma crx_inv_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃§p⦄ → ⊥.
+lemma crx_inv_gref: ∀h,o,G,L,p. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃§p⦄ → ⊥.
/2 width=8 by crx_inv_gref_aux/ qed-.
-lemma trx_inv_atom: ∀h,g,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, g] 𝐑⦃⓪{I}⦄ →
- ∃∃k,d. deg h g k (d+1) & I = Sort k.
-#h #g * #i #G #H
+lemma trx_inv_atom: ∀h,o,I,G. ⦃G, ⋆⦄ ⊢ ➡[h, o] 𝐑⦃⓪{I}⦄ →
+ ∃∃s,d. deg h o s (d+1) & I = Sort s.
+#h #o * #i #G #H
[ elim (crx_inv_sort … H) -H /2 width=4 by ex2_2_intro/
| elim (crx_inv_lref … H) -H #I #L #V #H
elim (drop_inv_atom1 … H) -H #H destruct
]
qed-.
-fact crx_inv_ib2_aux: ∀h,g,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ →
- T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃U⦄.
-#h #g #b #J #G #L #W0 #U #T #HI * -L -T
-[ #L #k #d #_ #H destruct
+fact crx_inv_ib2_aux: ∀h,o,a,I,G,L,W,U,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ →
+ T = ⓑ{a,I}W.U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃U⦄.
+#h #o #b #J #G #L #W0 #U #T #HI * -L -T
+[ #L #s #d #_ #H destruct
| #I #L #K #V #i #_ #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
]
qed-.
-lemma crx_inv_ib2: ∀h,g,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓑ{a,I}W.T⦄ →
- ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
+lemma crx_inv_ib2: ∀h,o,a,I,G,L,W,T. ib2 a I → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓑ{a,I}W.T⦄ →
+ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ ∨ ⦃G, L.ⓑ{I}W⦄ ⊢ ➡[h, o] 𝐑⦃T⦄.
/2 width=5 by crx_inv_ib2_aux/ qed-.
-fact crx_inv_appl_aux: ∀h,g,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → T = ⓐW.U →
- ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
-#h #g #G #L #W0 #U #T * -L -T
-[ #L #k #d #_ #H destruct
+fact crx_inv_appl_aux: ∀h,o,G,L,W,U,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → T = ⓐW.U →
+ ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃W⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ | (𝐒⦃U⦄ → ⊥).
+#h #o #G #L #W0 #U #T * -L -T
+[ #L #s #d #_ #H destruct
| #I #L #K #V #i #_ #H destruct
| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/
| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/
]
qed-.
-lemma crx_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃ⓐV.T⦄ →
- ∨∨ ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
+lemma crx_inv_appl: ∀h,o,G,L,V,T. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃ⓐV.T⦄ →
+ ∨∨ ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃V⦄ | ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ | (𝐒⦃T⦄ → ⊥).
/2 width=3 by crx_inv_appl_aux/ qed-.
(* Properties on relocation *************************************************)
-lemma crx_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ ➡[h, g] 𝐑⦃T⦄ → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄.
-#h #g #G #K #T #H elim H -K -T
-[ #K #k #d #Hkd #L #s #l #m #_ #X #H
+lemma crx_lift: ∀h,o,G,K,T. ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄ → ∀L,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀U. ⬆[l, k] T ≡ U → ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄.
+#h #o #G #K #T #H elim H -K -T
+[ #K #s #d #Hkd #L #c #l #k #_ #X #H
>(lift_inv_sort1 … H) -X /2 width=2 by crx_sort/
-| #I #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+| #I #K #K0 #V #i #HK0 #L #c #l #k #HLK #X #H
elim (lift_inv_lref1 … H) -H * #Hil #H destruct
[ elim (drop_trans_lt … HLK … HK0) -K /2 width=4 by crx_delta/
| lapply (drop_trans_ge … HLK … HK0 ?) -K /3 width=5 by crx_delta, drop_inv_gen/
]
-| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
+| #K #V #T #_ #IHV #L #c #l #k #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
-| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
+| #K #V #T #_ #IHT #L #c #l #k #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
-| #I #K #V #T #HI #L #s #l #m #_ #X #H
+| #I #K #V #T #HI #L #c #l #k #_ #X #H
elim (lift_fwd_pair1 … H) -H #W #U #_ #H destruct /2 width=1 by crx_ri2/
-| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHV #L #c #l #k #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
-| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHT #L #c #l #k #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
-| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+| #a #K #V #V0 #T #L #c #l #k #_ #X #H
elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_beta/
-| #a #K #V #V0 #T #L #s #l #m #_ #X #H
+| #a #K #V #V0 #T #L #c #l #k #_ #X #H
elim (lift_inv_flat1 … H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 … H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_theta/
]
qed.
-lemma crx_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ ➡[h, g] 𝐑⦃U⦄ → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀T. ⬆[l, m] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, g] 𝐑⦃T⦄.
-#h #g #G #L #U #H elim H -L -U
-[ #L #k #d #Hkd #K #s #l #m #_ #X #H
+lemma crx_inv_lift: ∀h,o,G,L,U. ⦃G, L⦄ ⊢ ➡[h, o] 𝐑⦃U⦄ → ∀K,c,l,k. ⬇[c, l, k] L ≡ K →
+ ∀T. ⬆[l, k] T ≡ U → ⦃G, K⦄ ⊢ ➡[h, o] 𝐑⦃T⦄.
+#h #o #G #L #U #H elim H -L -U
+[ #L #s #d #Hkd #K #c #l #k #_ #X #H
>(lift_inv_sort2 … H) -X /2 width=2 by crx_sort/
-| #I #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+| #I #L #L0 #W #i #HK0 #K #c #l #k #HLK #X #H
elim (lift_inv_lref2 … H) -H * #Hil #H destruct
[ elim (drop_conf_lt … HLK … HK0) -L /2 width=4 by crx_delta/
| lapply (drop_conf_ge … HLK … HK0 ?) -L /2 width=4 by crx_delta/
]
-| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
+| #L #W #U #_ #IHW #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
-| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
+| #L #W #U #_ #IHU #K #c #l #k #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
-| #I #L #W #U #HI #K #s #l #m #_ #X #H
+| #I #L #W #U #HI #K #c #l #k #_ #X #H
elim (lift_fwd_pair2 … H) -H #V #T #_ #H destruct /2 width=1 by crx_ri2/
-| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHW #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
-| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHU #K #c #l #k #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
-| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+| #a #L #W #W0 #U #K #c #l #k #_ #X #H
elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_beta/
-| #a #L #W #W0 #U #K #s #l #m #_ #X #H
+| #a #L #W #W0 #U #K #c #l #k #_ #X #H
elim (lift_inv_flat2 … H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 … H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_theta/
]
(* "RST" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************************)
-inductive fpb (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h g G1 L1 T1 G2 L2 T2
-| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpb h g G1 L1 T1 G1 L1 T2
-| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpb h g G1 L1 T1 G1 L2 T1
+inductive fpb (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpb_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → fpb h o G1 L1 T1 G2 L2 T2
+| fpb_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T2 → (T1 = T2 → ⊥) → fpb h o G1 L1 T1 G1 L1 T2
+| fpb_lpx: ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T1, 0] L2 → ⊥) → fpb h o G1 L1 T1 G1 L2 T1
.
interpretation
"'rst' proper parallel reduction (closure)"
- 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpb h g G1 L1 T1 G2 L2 T2).
+ 'BTPRedProper h o G1 L1 T1 G2 L2 T2 = (fpb h o G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma cpr_fpb: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+lemma cpr_fpb: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄.
/3 width=1 by fpb_cpx, cpr_cpx/ qed.
-lemma lpr_fpb: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) →
- ⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
+lemma lpr_fpb: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → (L1 ≡[T, 0] L2 → ⊥) →
+ ⦃G, L1, T⦄ ≻[h, o] ⦃G, L2, T⦄.
/3 width=1 by fpb_lpx, lpr_lpx/ qed.
(* Properties on lazy equivalence for closures ******************************)
-lemma fleq_fpb_trans: ∀h,g,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, g] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, g] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
-#h #g #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+lemma fleq_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≡[0] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≡[0] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
#K2 #HK12 #G2 #L2 #U2 #H12 elim (lleq_fpb_trans … HK12 … H12) -K2
/3 width=5 by fleq_intro, ex2_3_intro/
qed-.
(* Inversion lemmas on lazy equivalence for closures ************************)
-lemma fpb_inv_fleq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+lemma fpb_inv_fleq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 #H12 #H elim (fleq_inv_gen … H) -H
/3 width=8 by lleq_fwd_length, fqu_inv_eq/
| #T2 #_ #HnT #H elim (fleq_inv_gen … H) -H /2 width=1 by/
(* Advanced properties ******************************************************)
-lemma sta_fpb: ∀h,g,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 →
- ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2)
+lemma sta_fpb: ∀h,o,G,L,T1,T2,d. ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 →
+ ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 → ⦃G, L, T1⦄ ≻[h, o] ⦃G, L, T2⦄.
+#h #o #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2)
/3 width=2 by fpb_cpx, sta_cpx/ #H destruct
elim (lstas_inv_refl_pos h G L T2 0) //
qed.
(* Properties on lazy equivalence for local environments ********************)
-lemma lleq_fpb_trans: ∀h,g,F,K1,K2,T. K1 ≡[T, 0] K2 →
- ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, g] ⦃G, L2, U⦄ →
- ∃∃L1. ⦃F, K1, T⦄ ≻[h, g] ⦃G, L1, U⦄ & L1 ≡[U, 0] L2.
-#h #g #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+lemma lleq_fpb_trans: ∀h,o,F,K1,K2,T. K1 ≡[T, 0] K2 →
+ ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, o] ⦃G, L2, U⦄ →
+ ∃∃L1. ⦃F, K1, T⦄ ≻[h, o] ⦃G, L1, U⦄ & L1 ≡[U, 0] L2.
+#h #o #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
[ #G #L2 #U #H2 elim (lleq_fqu_trans … H2 … HT) -K2
/3 width=3 by fpb_fqu, ex2_intro/
| /4 width=10 by fpb_cpx, cpx_lleq_conf_sn, lleq_cpx_trans, ex2_intro/
(* "QRST" PARALLEL REDUCTION FOR CLOSURES ***********************************)
-inductive fpbq (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h g G1 L1 T1 G2 L2 T2
-| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → fpbq h g G1 L1 T1 G1 L1 T2
-| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, g] L2 → fpbq h g G1 L1 T1 G1 L2 T1
-| fpbq_lleq: ∀L2. L1 ≡[T1, 0] L2 → fpbq h g G1 L1 T1 G1 L2 T1
+inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T2 → fpbq h o G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ➡[h, o] L2 → fpbq h o G1 L1 T1 G1 L2 T1
+| fpbq_lleq: ∀L2. L1 ≡[T1, 0] L2 → fpbq h o G1 L1 T1 G1 L2 T1
.
interpretation
"'qrst' parallel reduction (closure)"
- 'BTPRed h g G1 L1 T1 G2 L2 T2 = (fpbq h g G1 L1 T1 G2 L2 T2).
+ 'BTPRed h o G1 L1 T1 G2 L2 T2 = (fpbq h o G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fpbq_refl: ∀h,g. tri_reflexive … (fpbq h g).
+lemma fpbq_refl: ∀h,o. tri_reflexive … (fpbq h o).
/2 width=1 by fpbq_cpx/ qed.
-lemma cpr_fpbq: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+lemma cpr_fpbq: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
/3 width=1 by fpbq_cpx, cpr_cpx/ qed.
-lemma lpr_fpbq: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, g] ⦃G, L2, T⦄.
+lemma lpr_fpbq: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1, T⦄ ≽[h, o] ⦃G, L2, T⦄.
/3 width=1 by fpbq_lpx, lpr_lpx/ qed.
(* Properties on atomic arity assignment for terms **************************)
-lemma fpbq_aaa_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+lemma fpbq_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=6 by aaa_lleq_conf, lpx_aaa_conf, cpx_aaa_conf, aaa_fquq_conf, ex_intro/
qed-.
(* alternative definition of fpbq *)
definition fpbqa: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g,G1,L1,T1,G2,L2,T2.
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
+ λh,o,G1,L1,T1,G2,L2,T2.
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ ∨ ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
interpretation
"'qrst' parallel reduction (closure) alternative"
- 'BTPRedAlt h g G1 L1 T1 G2 L2 T2 = (fpbqa h g G1 L1 T1 G2 L2 T2).
+ 'BTPRedAlt h o G1 L1 T1 G2 L2 T2 = (fpbqa h o G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fleq_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fpbq_lleq/
+lemma fleq_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fpbq_lleq/
qed.
-lemma fpb_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
qed.
(* Main properties **********************************************************)
-theorem fpbq_fpbqa: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+theorem fpbq_fpbqa: ∀h,o,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽≽[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 #H elim (fquq_inv_gen … H) -H
[ /3 width=1 by fpb_fqu, or_intror/
| * #H1 #H2 #H3 destruct /2 width=1 by or_introl/
]
qed.
-theorem fpbqa_inv_fpbq: ∀h,g,G1,G2,L1,L2,T1,T2.
- ⦃G1, L1, T1⦄ ≽≽[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fleq_fpbq, fpb_fpbq/
+theorem fpbqa_inv_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2.
+ ⦃G1, L1, T1⦄ ≽≽[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * /2 width=1 by fleq_fpbq, fpb_fpbq/
qed-.
(* Advanced eliminators *****************************************************)
-lemma fpbq_ind_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ∀R:Prop.
+lemma fpbq_ind_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ∀R:Prop.
(⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → R) →
- (⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R) →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #R #HI1 #HI2 #H elim (fpbq_fpbqa … H) /2 width=1 by/
+ (⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → R) →
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → R.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #R #HI1 #HI2 #H elim (fpbq_fpbqa … H) /2 width=1 by/
qed-.
(* aternative definition of fpb *********************************************)
-lemma fpb_fpbq_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ ∧ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥).
+lemma fpb_fpbq_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ ∧ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥).
/3 width=10 by fpb_fpbq, fpb_inv_fleq, conj/ qed.
-lemma fpbq_inv_fpb_alt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 @(fpbq_ind_alt … H) -H //
+lemma fpbq_inv_fpb_alt: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
+ (⦃G1, L1, T1⦄ ≡[0] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0 @(fpbq_ind_alt … H) -H //
#H elim H0 -H0 //
qed-.
(* Advanced properties ******************************************************)
-lemma sta_fpbq: ∀h,g,G,L,T1,T2,d.
- ⦃G, L⦄ ⊢ T1 ▪[h, g] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
- ⦃G, L, T1⦄ ≽[h, g] ⦃G, L, T2⦄.
+lemma sta_fpbq: ∀h,o,G,L,T1,T2,d.
+ ⦃G, L⦄ ⊢ T1 ▪[h, o] d+1 → ⦃G, L⦄ ⊢ T1 •*[h, 1] T2 →
+ ⦃G, L, T1⦄ ≽[h, o] ⦃G, L, T2⦄.
/3 width=4 by fpbq_cpx, sta_cpx/ 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/supterm_6.ma".
-include "basic_2/grammar/cl_weight.ma".
-include "basic_2/substitution/drop.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* activate genv *)
-inductive fqu: tri_relation genv lenv term ≝
-| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
-| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
-| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
-| fqu_drop : ∀G,L,K,T,U,m.
- ⬇[⫯m] L ≡ K → ⬆[0, ⫯m] T ≡ U → fqu G L U G K T
-.
-
-interpretation
- "structural successor (closure)"
- 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_drop_lt: ∀G,L,K,T,U,m. 0 < m →
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
-#G #L #K #T #U #m #Hm lapply (ylt_inv_O1 … Hm) -Hm
-#Hm <Hm -Hm /2 width=3 by fqu_drop/
-qed.
-
-lemma fqu_lref_S_lt: ∀I,G,L,V,i. yinj 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(⫰i)⦄.
-/4 width=3 by drop_drop, lift_lref_pred, fqu_drop/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-#G #L #K #T #U #m #HLK #HTU
-lapply (drop_fwd_lw_lt … HLK ?) -HLK // #HKL
-lapply (lift_fwd_tw … HTU) -m #H
-normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
-qed-.
-
-fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| < |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[1: /2 width=1 by monotonic_ylt_plus_sn/
-|3: #a
-|5: /2 width=4 by drop_fwd_length_lt4/
-] #I #G #L #V #T #j #H destruct
-qed-.
-
-lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ →
- |L2| < |L1|.
-/2 width=7 by fqu_fwd_length_lref1_aux/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #V #_ #H lapply (ysucc_inv_refl … H) -H
- #H elim (ylt_yle_false (|L|) (∞)) //
-|5: #G #L #K #T #U #m #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length … HLK) in H; -L
- #H elim (discr_yplus_xy_x … H) -H /2 width=2 by ysucc_inv_O_sn/
- #H elim (ylt_yle_false (|K|) (∞)) //
-]
-/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x/
-qed-.
-
-lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
-#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqu_wf_ind: ∀R:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
-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/suptermopt_6.ma".
-include "basic_2/substitution/fqu.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* activate genv *)
-inductive fquq: tri_relation genv lenv term ≝
-| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
-| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
-| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
-| fquq_drop : ∀G,L,K,T,U,m.
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → fquq G L U G K T
-.
-
-interpretation
- "optional structural successor (closure)"
- 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquq_refl: tri_reflexive … fquq.
-/2 width=3 by fquq_drop/ qed.
-
-lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
-#G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1
-lapply (drop_fwd_lw … HLK1) -HLK1
-lapply (lift_fwd_tw … HTU1) -HTU1
-/2 width=1 by le_plus, le_n/
-qed-.
-
-fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| ≤ |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-[ #a #I #G #L #V #T #j #H destruct
-| #G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #i #H destruct
- /2 width=3 by drop_fwd_length_le4/
-]
-qed-.
-
-lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
-/2 width=7 by fquq_fwd_length_lref1_aux/
-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/suptermoptalt_6.ma".
-include "basic_2/substitution/fquq.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* alternative definition of fquq *)
-definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
-
-interpretation
- "optional structural successor (closure) alternative"
- 'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquqa_refl: tri_reflexive … fquqa.
-// qed.
-
-lemma fquqa_drop: ∀G,L,K,T,U,m.
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
-#G #L #K #T #U #m @(ynat_ind … m) -m /3 width=3 by fqu_drop, or_introl/
-#HLK #HTU >(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
-qed.
-
-(* Main properties **********************************************************)
-
-theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
-* #H1 #H2 #H3 destruct //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
-/2 width=1 by or_introl/
-qed-.
∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #m #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (m+1)) #T2 #HUT2
+#G #L #K #U #T #k #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (k+1)) #T2 #HUT2
lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
qed-.
∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡ L & ⦃G1, L1⦄ ⊢ T1 ➡ U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #m #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (m+1)) #T2 #HUT2
+#G #L #K #U #T #k #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (k+1)) #T2 #HUT2
lapply (cpr_lift … HU2 … HLK … HUT … HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
qed-.
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 … H) -H
#K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
+| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12
elim (drop_lpr_trans … HLK1 … HK12) -HK12
/3 width=7 by fqu_drop, ex3_2_intro/
]
(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
definition lpx: ∀h. sd h → relation3 genv lenv lenv ≝
- λh,g,G. lpx_sn (cpx h g G).
+ λh,o,G. lpx_sn (cpx h o G).
interpretation "extended parallel reduction (local environment, sn variant)"
- 'PRedSn h g G L1 L2 = (lpx h g G L1 L2).
+ 'PRedSn h o G L1 L2 = (lpx h o G L1 L2).
(* Basic inversion lemmas ***************************************************)
-lemma lpx_inv_atom1: ∀h,g,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, g] L2 → L2 = ⋆.
+lemma lpx_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡[h, o] L2 → L2 = ⋆.
/2 width=4 by lpx_sn_inv_atom1_aux/ qed-.
-lemma lpx_inv_pair1: ∀h,g,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+lemma lpx_inv_pair1: ∀h,o,I,G,K1,V1,L2. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, o] L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 &
L2 = K2. ⓑ{I} V2.
/2 width=3 by lpx_sn_inv_pair1_aux/ qed-.
-lemma lpx_inv_atom2: ∀h,g,G,L1. ⦃G, L1⦄ ⊢ ➡[h, g] ⋆ → L1 = ⋆.
+lemma lpx_inv_atom2: ∀h,o,G,L1. ⦃G, L1⦄ ⊢ ➡[h, o] ⋆ → L1 = ⋆.
/2 width=4 by lpx_sn_inv_atom2_aux/ qed-.
-lemma lpx_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 &
+lemma lpx_inv_pair2: ∀h,o,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡[h, o] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 &
L1 = K1. ⓑ{I} V1.
/2 width=3 by lpx_sn_inv_pair2_aux/ qed-.
-lemma lpx_inv_pair: ∀h,g,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, g] L2.ⓑ{I2}V2 →
- ∧∧ ⦃G, L1⦄ ⊢ ➡[h, g] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, g] V2 & I1 = I2.
+lemma lpx_inv_pair: ∀h,o,I1,I2,G,L1,L2,V1,V2. ⦃G, L1.ⓑ{I1}V1⦄ ⊢ ➡[h, o] L2.ⓑ{I2}V2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ➡[h, o] L2 & ⦃G, L1⦄ ⊢ V1 ➡[h, o] V2 & I1 = I2.
/2 width=1 by lpx_sn_inv_pair/ qed-.
(* Basic properties *********************************************************)
-lemma lpx_refl: ∀h,g,G,L. ⦃G, L⦄ ⊢ ➡[h, g] L.
+lemma lpx_refl: ∀h,o,G,L. ⦃G, L⦄ ⊢ ➡[h, o] L.
/2 width=1 by lpx_sn_refl/ qed.
-lemma lpx_pair: ∀h,g,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, g] V2 →
- ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, g] K2.ⓑ{I}V2.
+lemma lpx_pair: ∀h,o,I,G,K1,K2,V1,V2. ⦃G, K1⦄ ⊢ ➡[h, o] K2 → ⦃G, K1⦄ ⊢ V1 ➡[h, o] V2 →
+ ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡[h, o] K2.ⓑ{I}V2.
/2 width=1 by lpx_sn_pair/ qed.
-lemma lpr_lpx: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, g] L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
+lemma lpr_lpx: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡[h, o] L2.
+#h #o #G #L1 #L2 #H elim H -L1 -L2 /3 width=1 by lpx_pair, cpr_cpx/
qed.
(* Basic forward lemmas *****************************************************)
-lemma lpx_fwd_length: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → |L1| = |L2|.
+lemma lpx_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → |L1| = |L2|.
/2 width=2 by lpx_sn_fwd_length/ qed-.
(* Properties on atomic arity assignment for terms **************************)
(* Note: lemma 500 *)
-lemma cpx_lpx_aaa_conf: ∀h,g,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #g #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #g #L1 #k #X #H
+lemma cpx_lpx_aaa_conf: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #o #L1 #s #X #H
elim (cpx_inv_sort1 … H) -H // * //
| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
elim (cpx_inv_lref1 … H) -H
]
qed-.
-lemma cpx_aaa_conf: ∀h,g,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
+lemma cpx_aaa_conf: ∀h,o,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
/2 width=7 by cpx_lpx_aaa_conf/ qed-.
-lemma lpx_aaa_conf: ∀h,g,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
+lemma lpx_aaa_conf: ∀h,o,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
/2 width=7 by cpx_lpx_aaa_conf/ qed-.
lemma cpr_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
(* Properties on local environment slicing ***********************************)
-lemma lpx_drop_conf: ∀h,g,G. dropable_sn (lpx h g G).
+lemma lpx_drop_conf: ∀h,o,G. dropable_sn (lpx h o G).
/3 width=6 by lpx_sn_deliftable_dropable, cpx_inv_lift1/ qed-.
-lemma drop_lpx_trans: ∀h,g,G. dedropable_sn (lpx h g G).
+lemma drop_lpx_trans: ∀h,o,G. dedropable_sn (lpx h o G).
/3 width=10 by lpx_sn_liftable_dedropable, cpx_lift/ qed-.
-lemma lpx_drop_trans_O1: ∀h,g,G. dropable_dx (lpx h g G).
+lemma lpx_drop_trans_O1: ∀h,o,G. dropable_dx (lpx h o G).
/2 width=3 by lpx_sn_dropable/ qed-.
(* Properties on supclosure *************************************************)
-lemma fqu_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma fqu_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_flat_dx, lpx_pair, ex3_2_intro/
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 … H) -H
#K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
+| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #K2 #HK12
elim (drop_lpx_trans … HLK1 … HK12) -HK12
/3 width=7 by fqu_drop, ex3_2_intro/
]
qed-.
-lemma fquq_lpx_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, g] K2 →
- ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, g] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
+lemma fquq_lpx_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K2. ⦃G2, L2⦄ ⊢ ➡[h, o] K2 →
+ ∃∃K1,T. ⦃G1, L1⦄ ⊢ ➡[h, o] K1 & ⦃G1, L1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K2 #HLK2 elim (fquq_inv_gen … H) -H
[ #HT12 elim (fqu_lpx_trans … HT12 … HLK2) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lpx_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma lpx_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=7 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpx_pair, ex3_2_intro/
[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 … H) -H
#K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
/4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #L0 #HL01
+| #G #L1 #K1 #T1 #U1 #k #HLK1 #HTU1 #L0 #HL01
elim (lpx_drop_trans_O1 … HL01 … HLK1) -L1
/3 width=5 by fqu_drop, ex3_2_intro/
]
qed-.
-lemma lpx_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
- ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, g] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
+lemma lpx_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #HKL1 elim (fquq_inv_gen … H) -H
[ #HT12 elim (lpx_fqu_trans … HT12 … HKL1) /3 width=5 by fqu_fquq, ex3_2_intro/
| * #H1 #H2 #H3 destruct /2 width=5 by ex3_2_intro/
]
(* Properties on context-sensitive free variables ***************************)
-lemma lpx_cpx_frees_trans: ∀h,g,G,L1,U1,U2. ⦃G, L1⦄ ⊢ U1 ➡[h, g] U2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+lemma lpx_cpx_frees_trans: ∀h,o,G,L1,U1,U2. ⦃G, L1⦄ ⊢ U1 ➡[h, o] U2 →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 →
∀i. L2 ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U1⦄.
-#h #g #G #L1 #U1 @(fqup_wf_ind_eq … G L1 U1) -G -L1 -U1
+#h #o #G #L1 #U1 @(fqup_wf_ind_eq … G L1 U1) -G -L1 -U1
#G0 #L0 #U0 #IH #G #L1 * *
-[ -IH #k #HG #HL #HU #U2 #H1 #L2 #_ #i #H2 elim (cpx_inv_sort1 … H1) -H1
+[ -IH #s #HG #HL #HU #U2 #H1 #L2 #_ #i #H2 elim (cpx_inv_sort1 … H1) -H1
[| * #d #_ ] #H destruct elim (frees_inv_sort … H2)
| #j #HG #HL #HU #U2 #H1 #L2 #HL12 #i #H2 elim (cpx_inv_lref1 … H1) -H1
[ #H destruct elim (frees_inv_lref … H2) -H2 //
]
qed-.
-lemma cpx_frees_trans: ∀h,g,G. frees_trans (cpx h g G).
+lemma cpx_frees_trans: ∀h,o,G. frees_trans (cpx h o G).
/2 width=8 by lpx_cpx_frees_trans/ qed-.
-lemma lpx_frees_trans: ∀h,g,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, g] L2 →
+lemma lpx_frees_trans: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 →
∀U,i. L2 ⊢ i ϵ 𝐅*[0]⦃U⦄ → L1 ⊢ i ϵ 𝐅*[0]⦃U⦄.
/2 width=8 by lpx_cpx_frees_trans/ qed-.
(* Properties on lazy equivalence for local environments ********************)
(* Note: contains a proof of llpx_cpx_conf *)
-lemma lleq_lpx_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, g] K2 →
+lemma lleq_lpx_trans: ∀h,o,G,L2,K2. ⦃G, L2⦄ ⊢ ➡[h, o] K2 →
∀L1,T,l. L1 ≡[T, l] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, g] K1 & K1 ≡[T, l] K2.
-#h #g #G #L2 #K2 #HLK2 #L1 #T #l #HL12
+ ∃∃K1. ⦃G, L1⦄ ⊢ ➡[h, o] K1 & K1 ≡[T, l] K2.
+#h #o #G #L2 #K2 #HLK2 #L1 #T #l #HL12
lapply (lpx_fwd_length … HLK2) #H1
lapply (lleq_fwd_length … HL12) #H2
lapply (lpx_sn_llpx_sn … T … l HLK2) // -HLK2 #H
]
qed-.
-lemma lpx_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+lemma lpx_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 … H1) -H1
#K0 #V0 #H1KL1 #_ #H destruct
elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
/3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
/2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (drop_O1_le (Ⓕ) (m+1) K1)
+| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+ elim (drop_O1_le (Ⓕ) (k+1) K1)
[ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
#H2KL elim (lpx_drop_trans_O1 … H1KL1 … HL1) -L1
#K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
/3 width=4 by fqu_drop, ex3_intro/
- | lapply (drop_fwd_length_le2 … HL1) -L -T1 -g
+ | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
lapply (lleq_fwd_length … H2KL1) //
]
]
qed-.
-lemma lpx_lleq_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+lemma lpx_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
elim (fquq_inv_gen … H) -H
[ #H elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
/3 width=4 by fqu_fquq, ex3_intro/
]
qed-.
-lemma lpx_lleq_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+lemma lpx_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpx_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
/3 width=4 by fqu_fqup, ex3_intro/
| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
]
qed-.
-lemma lpx_lleq_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2 & K2 ≡[T2, 0] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+lemma lpx_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
elim (fqus_inv_gen … H) -H
[ #H elim (lpx_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
/3 width=4 by fqup_fqus, ex3_intro/
]
qed-.
-fact lreq_lpx_trans_lleq_aux: ∀h,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 → m = ∞ →
- ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
- ∃∃L. L ⩬[l, m] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+fact lreq_lpx_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
+ ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] L &
(∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
-#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
-[ #l #m #_ #L2 #H >(lpx_inv_atom1 … H) -H
+#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
+[ #l #k #_ #L2 #H >(lpx_inv_atom1 … H) -H
/3 width=5 by ex3_intro, conj/
| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
-| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
+| #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
@(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpx_pair, lreq_cpx_trans, lreq_pair/
#T elim (IH T) #HL0dx #HL0sn
@conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
-| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #Y #H
elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
@(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpx_pair, lreq_succ/
]
qed-.
-lemma lreq_lpx_trans_lleq: ∀h,g,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
- ∀L2. ⦃G, L0⦄ ⊢ ➡[h, g] L2 →
- ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, g] L &
+lemma lreq_lpx_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡[h, o] L2 →
+ ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡[h, o] L &
(∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
/2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.
(**************************************************************************)
include "ground_2/relocation/rtmap_isfin.ma".
+include "basic_2/notation/relations/rdropstar_3.ma".
include "basic_2/notation/relations/rdropstar_4.ma".
include "basic_2/relocation/lreq.ma".
include "basic_2/relocation/lifts.ma".
drops c (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
+interpretation "uniform slicing (local environment)"
+ 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
+
interpretation "general slicing (local environment)"
'RDropStar c f L1 L2 = (drops c f L1 L2).
∀f2. f ⊚ f1 ≡ f2 →
∃∃L2. R f2 L1 L2 & ⬇*[c, f] L2 ≡ K2 & L1 ≡[f] L2.
-(* Basic inversion lemmas ***************************************************)
-
-fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ →
- Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
-#X #Y #c #f * -X -Y -f
-[ /3 width=1 by conj/
-| #I #L1 #L2 #V #f #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_sort *)
-(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀Y,c,f. ⬇*[c, f] ⋆ ≡ Y → Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
-/2 width=3 by drops_inv_atom1_aux/ qed-.
-
-fact drops_inv_drop1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K,V,g. X = K.ⓑ{I}V → f = ⫯g →
- ⬇*[c, g] K ≡ Y.
-#X #Y #c #f * -X -Y -f
-[ #f #Ht #J #K #W #g #H destruct
-| #I #L1 #L2 #V #f #HL #J #K #W #g #H1 #H2 <(injective_next … H2) -g destruct //
-| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K #W #g #_ #H2 elim (discr_push_next … H2)
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_drop *)
-(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀I,K,Y,V,c,f. ⬇*[c, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[c, f] K ≡ Y.
-/2 width=7 by drops_inv_drop1_aux/ qed-.
-
-
-fact drops_inv_skip1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K1,V1,g. X = K1.ⓑ{I}V1 → f = ↑g →
- ∃∃K2,V2. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
-#X #Y #c #f * -X -Y -f
-[ #f #Ht #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V #f #_ #J #K1 #W1 #g #_ #H2 elim (discr_next_push … H2)
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_skip_l *)
-(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀I,K1,V1,Y,c,f. ⬇*[c, ↑f] K1.ⓑ{I}V1 ≡ Y →
- ∃∃K2,V2. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
-/2 width=5 by drops_inv_skip1_aux/ qed-.
-
-fact drops_inv_skip2_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K2,V2,g. Y = K2.ⓑ{I}V2 → f = ↑g →
- ∃∃K1,V1. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1.
-#X #Y #c #f * -X -Y -f
-[ #f #Ht #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V #f #_ #J #K2 #W2 #g #_ #H2 elim (discr_next_push … H2)
-| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_skip_r *)
-(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1.
-/2 width=5 by drops_inv_skip2_aux/ qed-.
-
(* Basic properties *********************************************************)
lemma drops_eq_repl_back: ∀L1,L2,c. eq_repl_back … (λf. ⬇*[c, f] L1 ≡ L2).
/3 width=1 by drops_skip, lifts_refl/
qed.
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀L1,L2,f,c. ⬇*[c, f] L1 ≡ L2 → ∀f1,f2. f1 ⊚ f2 ≡ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[c, f1] L1 ≡ L & ⬇*[c, f2] L ≡ L2.
+#L1 #L2 #f #c #H elim H -L1 -L2 -f
+[ #f #Hc #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
+ #H lapply (Hc H) -c
+ #H elim (after_inv_isid3 … Hf H) -f //
+| #I #L1 #L2 #V #f #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ [ #g1 #g2 #Hf #H1 #H2 destruct
+ lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
+ elim (IHL12 … Hf) -f
+ /4 width=5 by drops_drop, drops_skip, lifts_refl, isuni_isid, ex2_intro/
+ | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
+ /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
+ ]
+| #I #L1 #L2 #V1 #V2 #f #_ #HV21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
+ #g1 #g2 #Hf #H1 #H2 destruct elim (lifts_split_trans … HV21 … Hf) -HV21
+ elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
+]
+qed-.
+
(* Basic_2A1: includes: drop_FT *)
lemma drops_TF: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → ⬇*[Ⓕ, f] L1 ≡ L2.
#L1 #L2 #f #H elim H -L1 -L2 -f
lemma drops_isuni_fwd_drop2: ∀I,X,K,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] X ≡ K.ⓑ{I}V → ⬇*[c, ⫯f] X ≡ K.
/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
-(* forward lemmas with test for finite colength *****************************)
+(* Forward lemmas with test for finite colength *****************************)
lemma drops_fwd_isfin: ∀L1,L2,f. ⬇*[Ⓣ, f] L1 ≡ L2 → 𝐅⦃f⦄.
#L1 #L2 #f #H elim H -L1 -L2 -f
/3 width=1 by isfin_next, isfin_push, isfin_isid/
qed-.
-(* Basic_2A1: removed theorems 14:
+(* Basic inversion lemmas ***************************************************)
+
+fact drops_inv_atom1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → X = ⋆ →
+ Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
+#X #Y #c #f * -X -Y -f
+[ /3 width=1 by conj/
+| #I #L1 #L2 #V #f #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_sort *)
+(* Basic_2A1: includes: drop_inv_atom1 *)
+lemma drops_inv_atom1: ∀Y,c,f. ⬇*[c, f] ⋆ ≡ Y → Y = ⋆ ∧ (c = Ⓣ → 𝐈⦃f⦄).
+/2 width=3 by drops_inv_atom1_aux/ qed-.
+
+fact drops_inv_drop1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K,V,g. X = K.ⓑ{I}V → f = ⫯g →
+ ⬇*[c, g] K ≡ Y.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K #W #g #H destruct
+| #I #L1 #L2 #V #f #HL #J #K #W #g #H1 #H2 <(injective_next … H2) -g destruct //
+| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K #W #g #_ #H2 elim (discr_push_next … H2)
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_drop *)
+(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
+lemma drops_inv_drop1: ∀I,K,Y,V,c,f. ⬇*[c, ⫯f] K.ⓑ{I}V ≡ Y → ⬇*[c, f] K ≡ Y.
+/2 width=7 by drops_inv_drop1_aux/ qed-.
+
+
+fact drops_inv_skip1_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K1,V1,g. X = K1.ⓑ{I}V1 → f = ↑g →
+ ∃∃K2,V2. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K1 #W1 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K1 #W1 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_skip_l *)
+(* Basic_2A1: includes: drop_inv_skip1 *)
+lemma drops_inv_skip1: ∀I,K1,V1,Y,c,f. ⬇*[c, ↑f] K1.ⓑ{I}V1 ≡ Y →
+ ∃∃K2,V2. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+/2 width=5 by drops_inv_skip1_aux/ qed-.
+
+fact drops_inv_skip2_aux: ∀X,Y,c,f. ⬇*[c, f] X ≡ Y → ∀I,K2,V2,g. Y = K2.ⓑ{I}V2 → f = ↑g →
+ ∃∃K1,V1. ⬇*[c, g] K1 ≡ K2 & ⬆*[g] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+#X #Y #c #f * -X -Y -f
+[ #f #Ht #J #K2 #W2 #g #H destruct
+| #I #L1 #L2 #V #f #_ #J #K2 #W2 #g #_ #H2 elim (discr_next_push … H2)
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_skip_r *)
+(* Basic_2A1: includes: drop_inv_skip2 *)
+lemma drops_inv_skip2: ∀I,X,K2,V2,c,f. ⬇*[c, ↑f] X ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[c, f] K1 ≡ K2 & ⬆*[f] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+/2 width=5 by drops_inv_skip2_aux/ qed-.
+
+(* Inversion lemmas with test for uniformity ********************************)
+
+(* Basic_2A1: was: drop_inv_O1_pair1 *)
+lemma drops_inv_pair1_isuni: ∀I,K,L2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] K.ⓑ{I}V ≡ L2 →
+ (𝐈⦃f⦄ ∧ L2 = K.ⓑ{I}V) ∨
+ ∃∃g. ⬇*[c, g] K ≡ L2 & f = ⫯g.
+#I #K #L2 #V #c #f #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
+[ lapply (drops_inv_skip1 … H) -H * #Y #X #HY #HX #H destruct
+ <(drops_fwd_isid … HY Hg) -Y >(lifts_fwd_isid … HX Hg) -X
+ /4 width=3 by isid_push, or_introl, conj/
+| lapply (drops_inv_drop1 … H) -H /3 width=3 by ex2_intro, or_intror/
+]
+qed-.
+
+(* Basic_2A1: was: drop_inv_O1_pair2 *)
+lemma drops_inv_pair2_isuni: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, f] L1 ≡ K.ⓑ{I}V →
+ (𝐈⦃f⦄ ∧ L1 = K.ⓑ{I}V) ∨
+ ∃∃I1,K1,V1,g. ⬇*[c, g] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & f = ⫯g.
+#I #K #V #c #f *
+[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #V1 #Hf #H elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct /3 width=1 by or_introl, conj/
+ | /3 width=7 by ex3_4_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma drops_inv_pair2_isuni_next: ∀I,K,V,c,f,L1. 𝐔⦃f⦄ → ⬇*[c, ⫯f] L1 ≡ K.ⓑ{I}V →
+ ∃∃I1,K1,V1. ⬇*[c, f] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1.
+#I #K #V #c #f #L1 #Hf #H elim (drops_inv_pair2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 12:
drops_inv_nil drops_inv_cons d1_liftable_liftables
- drop_refl_atom_O2
- drop_inv_O1_pair1 drop_inv_pair1 drop_inv_O1_pair2
+ drop_refl_atom_O2 drop_inv_pair1
drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
drop_fwd_length_minus2 drop_fwd_length_minus4
*)
lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
.
+interpretation "uniform relocation (term)"
+ 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
+
interpretation "generic relocation (term)"
- 'RLiftStar cs T1 T2 = (lifts cs T1 T2).
+ 'RLiftStar f T1 T2 = (lifts f T1 T2).
+
(* Basic inversion lemmas ***************************************************)
Y = ⓑ{p,I}V2.T2.
/2 width=3 by lifts_inv_bind1_aux/ qed-.
-fact lifts_inv_flat1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
+fact lifts_inv_flat1_aux: ∀X,Y. ∀f:rtmap. ⬆*[f] X ≡ Y →
∀I,V1,T1. X = ⓕ{I}V1.T1 →
∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
Y = ⓕ{I}V2.T2.
(* Basic_1: was: lift1_flat *)
(* Basic_2A1: includes: lift_inv_flat1 *)
-lemma lifts_inv_flat1: ∀I,V1,T1,Y,f. ⬆*[f] ⓕ{I}V1.T1 ≡ Y →
+lemma lifts_inv_flat1: ∀I,V1,T1,Y. ∀f:rtmap. ⬆*[f] ⓕ{I}V1.T1 ≡ Y →
∃∃V2,T2. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
Y = ⓕ{I}V2.T2.
/2 width=3 by lifts_inv_flat1_aux/ qed-.
X = ⓑ{p,I}V1.T1.
/2 width=3 by lifts_inv_bind2_aux/ qed-.
-fact lifts_inv_flat2_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
+fact lifts_inv_flat2_aux: ∀X,Y. ∀f:rtmap. ⬆*[f] X ≡ Y →
∀I,V2,T2. Y = ⓕ{I}V2.T2 →
∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
X = ⓕ{I}V1.T1.
(* Basic_1: includes: lift_gen_flat *)
(* Basic_2A1: includes: lift_inv_flat2 *)
-lemma lifts_inv_flat2: ∀I,V2,T2,X,f. ⬆*[f] X ≡ ⓕ{I}V2.T2 →
+lemma lifts_inv_flat2: ∀I,V2,T2,X. ∀f:rtmap. ⬆*[f] X ≡ ⓕ{I}V2.T2 →
∃∃V1,T1. ⬆*[f] V1 ≡ V2 & ⬆*[f] T1 ≡ T2 &
X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
qed-.
(* Basic_2A1: includes: lift_fwd_pair1 *)
-lemma lifts_fwd_pair1: ∀I,V1,T1,Y,f. ⬆*[f] ②{I}V1.T1 ≡ Y →
+lemma lifts_fwd_pair1: ∀I,V1,T1,Y. ∀f:rtmap. ⬆*[f] ②{I}V1.T1 ≡ Y →
∃∃V2,T2. ⬆*[f] V1 ≡ V2 & Y = ②{I}V2.T2.
* [ #p ] #I #V1 #T1 #Y #f #H
[ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
qed-.
(* Basic_2A1: includes: lift_fwd_pair2 *)
-lemma lifts_fwd_pair2: ∀I,V2,T2,X,f. ⬆*[f] X ≡ ②{I}V2.T2 →
+lemma lifts_fwd_pair2: ∀I,V2,T2,X. ∀f:rtmap. ⬆*[f] X ≡ ②{I}V2.T2 →
∃∃V1,T1. ⬆*[f] V1 ≡ V2 & X = ②{I}V1.T1.
* [ #p ] #I #V2 #T2 #X #f #H
[ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
]
qed-.
+lemma lift_SO: ∀i. ⬆*[1] #i ≡ #(⫯i).
+/2 width=1 by lifts_lref/ qed.
+
(* Basic_1: includes: lift_free (right to left) *)
(* Basic_2A1: includes: lift_split *)
lemma lifts_split_trans: ∀T1,T2,f. ⬆*[f] T1 ≡ T2 →
(* Main properties **********************************************************)
(* Basic_1: includes: lifts_inj *)
-theorem liftsv_inj: ∀T1c,Us,f. ⬆*[f] T1c ≡ Us →
- ∀T2c. ⬆*[f] T2c ≡ Us → T1c = T2c.
-#T1c #Us #f #H elim H -T1c -Us
-[ #T2c #H >(liftsv_inv_nil2 … H) -H //
-| #T1c #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
- elim (liftsv_inv_cons2 … H) -H #T2 #T2c #HT2U #HT2Us #H destruct
+theorem liftsv_inj: ∀T1s,Us,f. ⬆*[f] T1s ≡ Us →
+ ∀T2s. ⬆*[f] T2s ≡ Us → T1s = T2s.
+#T1s #Us #f #H elim H -T1s -Us
+[ #T2s #H >(liftsv_inv_nil2 … H) -H //
+| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
+ elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct
>(lifts_inj … HT1U … HT2U) -U /3 width=1 by eq_f/
]
qed-.
(* Basic_2A1: includes: liftv_mono *)
-theorem liftsv_mono: ∀Ts,U1c,f. ⬆*[f] Ts ≡ U1c →
- ∀U2c. ⬆*[f] Ts ≡ U2c → U1c = U2c.
-#Ts #U1c #f #H elim H -Ts -U1c
-[ #U2c #H >(liftsv_inv_nil1 … H) -H //
-| #Ts #U1c #T #U1 #HTU1 #_ #IHTU1c #X #H destruct
- elim (liftsv_inv_cons1 … H) -H #U2 #U2c #HTU2 #HTU2c #H destruct
+theorem liftsv_mono: ∀Ts,U1s,f. ⬆*[f] Ts ≡ U1s →
+ ∀U2s. ⬆*[f] Ts ≡ U2s → U1s = U2s.
+#Ts #U1s #f #H elim H -Ts -U1s
+[ #U2s #H >(liftsv_inv_nil1 … H) -H //
+| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
+ elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
>(lifts_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
]
qed-.
(* Basic_1: includes: lifts1_xhg (right to left) *)
(* Basic_2A1: includes: liftsv_liftv_trans_le *)
-theorem liftsv_trans: ∀T1c,Ts,f1. ⬆*[f1] T1c ≡ Ts → ∀T2c,f2. ⬆*[f2] Ts ≡ T2c →
- ∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1c ≡ T2c.
-#T1c #Ts #f1 #H elim H -T1c -Ts
-[ #T2c #f2 #H >(liftsv_inv_nil1 … H) -T2c /2 width=3 by liftsv_nil/
-| #T1c #Ts #T1 #T #HT1 #_ #IHT1c #X #f2 #H elim (liftsv_inv_cons1 … H) -H
- #T2 #T2c #HT2 #HT2c #H destruct /3 width=6 by lifts_trans, liftsv_cons/
+theorem liftsv_trans: ∀T1s,Ts,f1. ⬆*[f1] T1s ≡ Ts → ∀T2s,f2. ⬆*[f2] Ts ≡ T2s →
+ ∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1s ≡ T2s.
+#T1s #Ts #f1 #H elim H -T1s -Ts
+[ #T2s #f2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
+| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #f2 #H elim (liftsv_inv_cons1 … H) -H
+ #T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/
]
qed-.
(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
(* Basic_2A1: includes: liftv_nil liftv_cons *)
-inductive liftsv (f): relation (list term) ≝
+inductive liftsv (f:rtmap): relation (list term) ≝
| liftsv_nil : liftsv f (◊) (◊)
-| liftsv_cons: ∀T1c,T2c,T1,T2.
- ⬆*[f] T1 ≡ T2 → liftsv f T1c T2c →
- liftsv f (T1 @ T1c) (T2 @ T2c)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+ ⬆*[f] T1 ≡ T2 → liftsv f T1s T2s →
+ liftsv f (T1 @ T1s) (T2 @ T2s)
.
+interpretation "uniform relocation (vector)"
+ 'RLiftStar i T1s T2s = (liftsv (uni i) T1s T2s).
+
interpretation "generic relocation (vector)"
- 'RLiftStar f T1c T2c = (liftsv f T1c T2c).
+ 'RLiftStar f T1s T2s = (liftsv f T1s T2s).
(* Basic inversion lemmas ***************************************************)
fact liftsv_inv_nil1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y → X = ◊ → Y = ◊.
#X #Y #f * -X -Y //
-#T1c #T2c #T1 #T2 #_ #_ #H destruct
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes: liftv_inv_nil1 *)
lemma liftsv_inv_nil1: ∀Y,f. ⬆*[f] ◊ ≡ Y → Y = ◊.
/2 width=5 by liftsv_inv_nil1_aux/ qed-.
-fact liftsv_inv_cons1_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
- ∀T1,T1c. X = T1 @ T1c →
- ∃∃T2,T2c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- Y = T2 @ T2c.
+fact liftsv_inv_cons1_aux: ∀X,Y. ∀f:rtmap. ⬆*[f] X ≡ Y →
+ ∀T1,T1s. X = T1 @ T1s →
+ ∃∃T2,T2s. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1s ≡ T2s &
+ Y = T2 @ T2s.
#X #Y #f * -X -Y
-[ #U1 #U1c #H destruct
-| #T1c #T2c #T1 #T2 #HT12 #HT12c #U1 #U1c #H destruct /2 width=5 by ex3_2_intro/
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_2A1: includes: liftv_inv_cons1 *)
-lemma liftsv_inv_cons1: ∀T1,T1c,Y,f. ⬆*[f] T1 @ T1c ≡ Y →
- ∃∃T2,T2c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- Y = T2 @ T2c.
+lemma liftsv_inv_cons1: ∀T1,T1s,Y. ∀f:rtmap. ⬆*[f] T1 @ T1s ≡ Y →
+ ∃∃T2,T2s. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1s ≡ T2s &
+ Y = T2 @ T2s.
/2 width=3 by liftsv_inv_cons1_aux/ qed-.
fact liftsv_inv_nil2_aux: ∀X,Y,f. ⬆*[f] X ≡ Y → Y = ◊ → X = ◊.
#X #Y #f * -X -Y //
-#T1c #T2c #T1 #T2 #_ #_ #H destruct
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
qed-.
lemma liftsv_inv_nil2: ∀X,f. ⬆*[f] X ≡ ◊ → X = ◊.
/2 width=5 by liftsv_inv_nil2_aux/ qed-.
-fact liftsv_inv_cons2_aux: ∀X,Y,f. ⬆*[f] X ≡ Y →
- ∀T2,T2c. Y = T2 @ T2c →
- ∃∃T1,T1c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- X = T1 @ T1c.
+fact liftsv_inv_cons2_aux: ∀X,Y. ∀f:rtmap. ⬆*[f] X ≡ Y →
+ ∀T2,T2s. Y = T2 @ T2s →
+ ∃∃T1,T1s. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1s ≡ T2s &
+ X = T1 @ T1s.
#X #Y #f * -X -Y
-[ #U2 #U2c #H destruct
-| #T1c #T2c #T1 #T2 #HT12 #HT12c #U2 #U2c #H destruct /2 width=5 by ex3_2_intro/
+[ #U2 #U2s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma liftsv_inv_cons2: ∀X,T2,T2c,f. ⬆*[f] X ≡ T2 @ T2c →
- ∃∃T1,T1c. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1c ≡ T2c &
- X = T1 @ T1c.
+lemma liftsv_inv_cons2: ∀X,T2,T2s. ∀f:rtmap. ⬆*[f] X ≡ T2 @ T2s →
+ ∃∃T1,T1s. ⬆*[f] T1 ≡ T2 & ⬆*[f] T1s ≡ T2s &
+ X = T1 @ T1s.
/2 width=3 by liftsv_inv_cons2_aux/ qed-.
(* Basic_1: was: lifts1_flat (left to right) *)
-lemma lifts_inv_applv1: ∀V1c,U1,T2,f. ⬆*[f] Ⓐ V1c.U1 ≡ T2 →
- ∃∃V2c,U2. ⬆*[f] V1c ≡ V2c & ⬆*[f] U1 ≡ U2 &
- T2 = Ⓐ V2c.U2.
-#V1c elim V1c -V1c
+lemma lifts_inv_applv1: ∀V1s,U1,T2. ∀f:rtmap. ⬆*[f] Ⓐ V1s.U1 ≡ T2 →
+ ∃∃V2s,U2. ⬆*[f] V1s ≡ V2s & ⬆*[f] U1 ≡ U2 &
+ T2 = Ⓐ V2s.U2.
+#V1s elim V1s -V1s
[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V1 #V1c #IHV1c #T1 #X #f #H elim (lifts_inv_flat1 … H) -H
- #V2 #Y #HV12 #HY #H destruct elim (IHV1c … HY) -IHV1c -HY
- #V2c #T2 #HV12c #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+| #V1 #V1s #IHV1s #T1 #X #f #H elim (lifts_inv_flat1 … H) -H
+ #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY
+ #V2c #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
]
qed-.
-lemma lifts_inv_applv2: ∀V2c,U2,T1,f. ⬆*[f] T1 ≡ Ⓐ V2c.U2 →
- ∃∃V1c,U1. ⬆*[f] V1c ≡ V2c & ⬆*[f] U1 ≡ U2 &
- T1 = Ⓐ V1c.U1.
-#V2c elim V2c -V2c
+lemma lifts_inv_applv2: ∀V2s,U2,T1. ∀f:rtmap. ⬆*[f] T1 ≡ Ⓐ V2s.U2 →
+ ∃∃V1s,U1. ⬆*[f] V1s ≡ V2s & ⬆*[f] U1 ≡ U2 &
+ T1 = Ⓐ V1s.U1.
+#V2s elim V2s -V2s
[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V2 #V2c #IHV2c #T2 #X #f #H elim (lifts_inv_flat2 … H) -H
- #V1 #Y #HV12 #HY #H destruct elim (IHV2c … HY) -IHV2c -HY
- #V1c #T1 #HV12c #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+| #V2 #V2s #IHV2s #T2 #X #f #H elim (lifts_inv_flat2 … H) -H
+ #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY
+ #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
]
qed-.
(* Basic properties *********************************************************)
(* Basic_2A1: includes: liftv_total *)
-lemma liftsv_total: ∀f. ∀T1c:list term. ∃T2c. ⬆*[f] T1c ≡ T2c.
-#f #T1c elim T1c -T1c
+lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≡ T2s.
+#f #T1s elim T1s -T1s
[ /2 width=2 by liftsv_nil, ex_intro/
-| #T1 #T1c * #T2c #HT12c
+| #T1 #T1s * #T2s #HT12s
elim (lifts_total T1 f) /3 width=2 by liftsv_cons, ex_intro/
]
qed-.
(* Basic_1: was: lifts1_flat (right to left) *)
-lemma lifts_applv: ∀V1c,V2c,f. ⬆*[f] V1c ≡ V2c →
+lemma lifts_applv: ∀V1s,V2s. ∀f:rtmap. ⬆*[f] V1s ≡ V2s →
∀T1,T2. ⬆*[f] T1 ≡ T2 →
- ⬆*[f] Ⓐ V1c. T1 ≡ Ⓐ V2c. T2.
-#V1c #V2c #f #H elim H -V1c -V2c /3 width=1 by lifts_flat/
+ ⬆*[f] Ⓐ V1s.T1 ≡ Ⓐ V2s.T2.
+#V1s #V2s #f #H elim H -V1s -V2s /3 width=1 by lifts_flat/
qed.
(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)
--- /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/supterm_6.ma".
+include "basic_2/grammar/lenv.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/relocation/lifts.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* activate genv *)
+inductive fqu: tri_relation genv lenv term ≝
+| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
+| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
+| fqu_bind_dx: ∀p,I,G,L,V,T. fqu G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
+| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
+| fqu_drop : ∀I,G,L,V,T,U. ⬆*[1] T ≡ U → fqu G (L.ⓑ{I}V) U G L T
+.
+
+interpretation
+ "structural successor (closure)"
+ 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_lref_S: ∀I,G,L,V,i. ⦃G, L.ⓑ{I}V, #(⫯i)⦄ ⊐ ⦃G, L, #(i)⦄.
+/2 width=1 by fqu_drop/ qed.
+
+(* Basic_2A1: removed theorems 3:
+ fqu_drop fqu_drop_lt fqu_lref_S_lt
+*)
--- /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/grammar/lenv_length.ma".
+include "basic_2/s_transition/fqu.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| < |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2: #p]
+#I #G #L #V #T #j #H destruct
+qed-.
+
+lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ →
+ |L2| < |L1|.
+/2 width=7 by fqu_fwd_length_lref1_aux/
+qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H)
+|5: #I #G #L #V #T #U #_ #_ #H elim (succ_inv_refl_sn … H)
+]
+/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x/
+qed-.
+
+lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
+#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
+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/grammar/cl_weight.ma".
+include "basic_2/relocation/lifts_weight.ma".
+include "basic_2/s_transition/fqu.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+#I #G #L #V #T #U #HTU normalize in ⊢ (?%%); -I
+<(lifts_fwd_tw … HTU) -U /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqu_wf_ind: ∀R:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
+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/suptermopt_6.ma".
+include "basic_2/s_transition/fqu.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Basic_2A1: was: fquqa *)
+(* Basic_2A1: includes: fquq_inv_gen *)
+definition fquq: tri_relation genv lenv term ≝ tri_RC … fqu.
+
+interpretation
+ "optional structural successor (closure)"
+ 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: includes: fquqa_refl *)
+lemma fquq_refl: tri_reflexive … fquq.
+// qed.
+
+lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+/2 width=1 by or_introl/ qed.
+
+(* Basic_2A1: removed theorems 8:
+ fquq_lref_O fquq_pair_sn fquq_bind_dx fquq_flat_dx fquq_drop
+ fquqa_drop fquq_fquqa fquqa_inv_fquq
+*)
--- /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/s_transition/fqu_length.ma".
+include "basic_2/s_transition/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
+#G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ]
+/3 width=5 by fqu_fwd_length_lref1, lt_to_le/
+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/s_transition/fqu_weight.ma".
+include "basic_2/s_transition/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ]
+/3 width=1 by fqu_fwd_fw, lt_to_le/
+qed-.
include "basic_2/notation/relations/atomicarity_4.ma".
include "basic_2/grammar/aarity.ma".
+include "basic_2/grammar/lenv.ma".
include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/drop.ma".
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+(* ATONIC ARITY ASSIGNMENT FOR TERMS ****************************************)
(* activate genv *)
inductive aaa: relation4 genv lenv term aarity ≝
-| aaa_sort: ∀G,L,k. aaa G L (⋆k) (⓪)
-| aaa_lref: ∀I,G,L,K,V,B,i. ⬇[i] L ≡ K. ⓑ{I}V → aaa G K V B → aaa G L (#i) B
-| aaa_abbr: ∀a,G,L,V,T,B,A.
- aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{a}V.T) A
-| aaa_abst: ∀a,G,L,V,T,B,A.
- aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{a}V.T) (②B.A)
+| aaa_sort: ∀G,L,s. aaa G L (⋆s) (⓪)
+| aaa_zero: ∀I,G,L,V,B. aaa G L V B → aaa G (L.ⓑ{I}V) (#0) B
+| aaa_lref: ∀I,G,L,V,A,i. aaa G L (#i) A → aaa G (L.ⓑ{I}V) (#⫯i) A
+| aaa_abbr: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{p}V.T) A
+| aaa_abst: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{p}V.T) (②B.A)
| aaa_appl: ∀G,L,V,T,B,A. aaa G L V B → aaa G L T (②B.A) → aaa G L (ⓐV.T) A
| aaa_cast: ∀G,L,V,T,A. aaa G L V A → aaa G L T A → aaa G L (ⓝV.T) A
.
(* Basic inversion lemmas ***************************************************)
-fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀k. T = ⋆k → A = ⓪.
-#G #L #T #A * -G -L -T -A
-[ //
-| #I #G #L #K #V #B #i #_ #_ #k #H destruct
-| #a #G #L #V #T #B #A #_ #_ #k #H destruct
-| #a #G #L #V #T #B #A #_ #_ #k #H destruct
-| #G #L #V #T #B #A #_ #_ #k #H destruct
-| #G #L #V #T #A #_ #_ #k #H destruct
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
+#G #L #T #A * -G -L -T -A //
+[ #I #G #L #V #B #_ #s #H destruct
+| #I #G #L #V #A #i #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #A #_ #_ #s #H destruct
]
qed-.
-lemma aaa_inv_sort: ∀G,L,A,k. ⦃G, L⦄ ⊢ ⋆k ⁝ A → A = ⓪.
+lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
/2 width=6 by aaa_inv_sort_aux/ qed-.
-fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #i →
- ∃∃I,K,V. ⬇[i] L ≡ K.ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
+[ #G #L #s #H destruct
+| #I #G #L #V #A #i #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #A #_ #_ #H destruct
+]
+qed-.
+
+lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+/2 width=3 by aaa_inv_zero_aux/ qed-.
+
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(⫯i) →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ #i ⁝ A.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #i #H destruct
-| #I #G #L #K #V #B #j #HLK #HB #i #H destruct /2 width=5 by ex2_3_intro/
-| #a #G #L #V #T #B #A #_ #_ #i #H destruct
-| #a #G #L #V #T #B #A #_ #_ #i #H destruct
-| #G #L #V #T #B #A #_ #_ #i #H destruct
-| #G #L #V #T #A #_ #_ #i #H destruct
+[ #G #L #s #j #H destruct
+| #I #G #L #V #B #_ #j #H destruct
+| #I #G #L #V #A #i #HA #j #H destruct /2 width=5 by ex2_3_intro/
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #A #_ #_ #j #H destruct
]
qed-.
-lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇[i] L ≡ K. ⓑ{I} V & ⦃G, K⦄ ⊢ V ⁝ A.
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #⫯i ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ #i ⁝ A.
/2 width=3 by aaa_inv_lref_aux/ qed-.
-fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p. T = §p → ⊥.
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #q #H destruct
-| #I #G #L #K #V #B #i #HLK #HB #q #H destruct
-| #a #G #L #V #T #B #A #_ #_ #q #H destruct
-| #a #G #L #V #T #B #A #_ #_ #q #H destruct
-| #G #L #V #T #B #A #_ #_ #q #H destruct
-| #G #L #V #T #A #_ #_ #q #H destruct
+[ #G #L #s #k #H destruct
+| #I #G #L #V #B #_ #k #H destruct
+| #I #G #L #V #A #i #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #A #_ #_ #k #H destruct
]
qed-.
-lemma aaa_inv_gref: ∀G,L,A,p. ⦃G, L⦄ ⊢ §p ⁝ A → ⊥.
+lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
/2 width=7 by aaa_inv_gref_aux/ qed-.
-fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓓ{a}W. U →
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #a #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=2 by ex2_intro/
-| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #V #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=2 by ex2_intro/
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abbr: ∀a,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{a}V. T ⁝ A →
+lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
/2 width=4 by aaa_inv_abbr_aux/ qed-.
-fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀a,W,U. T = ⓛ{a}W. U →
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #a #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #b #G #L #V #T #B #A #HV #HT #a #W #U #H destruct /2 width=5 by ex3_2_intro/
-| #G #L #V #T #B #A #_ #_ #a #W #U #H destruct
-| #G #L #V #T #A #_ #_ #a #W #U #H destruct
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #V #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
]
qed-.
-lemma aaa_inv_abst: ∀a,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{a}W. T ⁝ A →
+lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
/2 width=4 by aaa_inv_abst_aux/ qed-.
fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #V #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
| #G #L #V #T #A #_ #_ #W #U #H destruct
]
fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW. U →
⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
#G #L #T #A * -G -L -T -A
-[ #G #L #k #W #U #H destruct
-| #I #G #L #K #V #B #i #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #a #G #L #V #T #B #A #_ #_ #W #U #H destruct
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #V #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #G #L #V #T #B #A #_ #_ #W #U #H destruct
| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
]
theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
#G #L #T #A1 #H elim H -G -L -T -A1
-[ #G #L #k #A2 #H
+[ #G #L #s #A2 #H
>(aaa_inv_sort … H) -H //
| #I1 #G #L #K1 #V1 #B #i #HLK1 #_ #IHA1 #A2 #H
elim (aaa_inv_lref … H) -H #I2 #K2 #V2 #HLK2 #HA2
--- /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/drops.ma".
+include "basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: was: aaa_lref *)
+lemma aaa_lref_gen: ∀I,G,K,V,B,i,L. ⬇*[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+#I #G #K #V #B #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+ #H destruct /2 width=1 by aaa_zero/
+| #i #IH #L <uni_succ #H #HB lapply (drops_inv_pair2_isuni_next … H) -H // *
+ #Z #Y #X #HY #H destruct /3 width=1 by aaa_lref/
+]
+qed.
+
+(* Basic_2A1: includes: aaa_lift *)
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,c,f. ⬇*[c, f] L2 ≡ L1 →
+ ∀T2. ⬆*[f] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #s #L2 #c #f #_ #T2 #H
+ >(lifts_inv_sort1 … H) -H //
+| #I #G #L1 #V1 #B #_ #IHB #L2 #c #f #HL21 #T2 #H
+ elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
+ @aaa_lref_gen [5: @IHB ]
+| #I #G #L1 #V1 #B #i1 #_ #IHB #L2 #c #f #HL21 #T2 #H
+ elim (lifts_inv_lref1 … H) -H #i2 #Hi #H destruct
+ lapply (at_inv_nxx … H)
+
+
+
+
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #c #l #k #HL21 #T2 #H
+ elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+ [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
+ /3 width=9 by aaa_lref/
+ | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
+ /3 width=9 by aaa_lref, drop_inv_gen/
+ ]
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=5 by aaa_abbr, drop_skip/
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /4 width=5 by aaa_abst, drop_skip/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=5 by aaa_appl/
+| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=5 by aaa_cast/
+]
+qed.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: was: aaa_inv_lref *)
+lemma aaa_inv_lref_gen: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #A #i elim i -i
+[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
+| #i #IH #L #H elim (aaa_inv_lref … H) -H
+ #I #K #V #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
+]
+qed-.
+
+lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,c,l,k. ⬇[c, l, k] L2 ≡ L1 →
+ ∀T1. ⬆[l, k] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
+[ #G #L2 #s #L1 #c #l #k #_ #T1 #H
+ >(lift_inv_sort2 … H) -H //
+| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #c #l #k #HL21 #T1 #H
+ elim (lift_inv_lref2 … H) -H * #Hil #H destruct
+ [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
+ | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
+ ]
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=5 by aaa_abbr, drop_skip/
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /4 width=5 by aaa_abst, drop_skip/
+| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=5 by aaa_appl/
+| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #c #l #k #HL21 #X #H
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=5 by aaa_cast/
+]
+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/drop_drop.ma".
-include "basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties on basic relocation *******************************************)
-
-lemma aaa_lift: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀L2,s,l,m. ⬇[s, l, m] L2 ≡ L1 →
- ∀T2. ⬆[l, m] T1 ≡ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #k #L2 #s #l #m #_ #T2 #H
- >(lift_inv_sort1 … H) -H //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #l #m #HL21 #T2 #H
- elim (lift_inv_lref1 … H) -H * #Hil #H destruct
- [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=9 by aaa_lref/
- | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
- /3 width=9 by aaa_lref, drop_inv_gen/
- ]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abst, drop_skip/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_cast/
-]
-qed.
-
-lemma aaa_inv_lift: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀L1,s,l,m. ⬇[s, l, m] L2 ≡ L1 →
- ∀T1. ⬆[l, m] T1 ≡ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
-#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
-[ #G #L2 #k #L1 #s #l #m #_ #T1 #H
- >(lift_inv_sort2 … H) -H //
-| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #l #m #HL21 #T1 #H
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=9 by aaa_lref/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /3 width=9 by aaa_lref/
- ]
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=5 by aaa_abst, drop_skip/
-| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_appl/
-| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=5 by aaa_cast/
-]
-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/multiple/drops.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties concerning generic relocation *********************************)
-
-lemma aaa_lifts: ∀G,L1,L2,T2,A,s,cs. ⬇*[s, cs] L2 ≡ L1 → ∀T1. ⬆*[cs] T1 ≡ T2 →
- ⦃G, L1⦄ ⊢ T1 ⁝ A → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#G #L1 #L2 #T2 #A #s #cs #H elim H -L1 -L2 -cs
-[ #L #T1 #H #HT1
- <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL1 #T1 #H #HT1
- elim (lifts_inv_cons … H) -H /3 width=10 by aaa_lift/
-]
-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/degree_6.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/drop.ma".
-include "basic_2/static/sd.ma".
-
-(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
-
-(* activate genv *)
-inductive da (h:sh) (g:sd h): relation4 genv lenv term nat ≝
-| da_sort: ∀G,L,k,d. deg h g k d → da h g G L (⋆k) d
-| da_ldef: ∀G,L,K,V,i,d. ⬇[i] L ≡ K.ⓓV → da h g G K V d → da h g G L (#i) d
-| da_ldec: ∀G,L,K,W,i,d. ⬇[i] L ≡ K.ⓛW → da h g G K W d → da h g G L (#i) (d+1)
-| da_bind: ∀a,I,G,L,V,T,d. da h g G (L.ⓑ{I}V) T d → da h g G L (ⓑ{a,I}V.T) d
-| da_flat: ∀I,G,L,V,T,d. da h g G L T d → da h g G L (ⓕ{I}V.T) d
-.
-
-interpretation "degree assignment (term)"
- 'Degree h g G L T d = (da h g G L T d).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact da_inv_sort_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
- ∀k0. T = ⋆k0 → deg h g k0 d.
-#h #g #G #L #T #d * -G -L -T -d
-[ #G #L #k #d #Hkd #k0 #H destruct //
-| #G #L #K #V #i #d #_ #_ #k0 #H destruct
-| #G #L #K #W #i #d #_ #_ #k0 #H destruct
-| #a #I #G #L #V #T #d #_ #k0 #H destruct
-| #I #G #L #V #T #d #_ #k0 #H destruct
-]
-qed-.
-
-lemma da_inv_sort: ∀h,g,G,L,k,d. ⦃G, L⦄ ⊢ ⋆k ▪[h, g] d → deg h g k d.
-/2 width=5 by da_inv_sort_aux/ qed-.
-
-fact da_inv_lref_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀j. T = #j →
- (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] d) ∨
- (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] d0 &
- d = d0 + 1
- ).
-#h #g #G #L #T #d * -G -L -T -d
-[ #G #L #k #d #_ #j #H destruct
-| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
-| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
-| #a #I #G #L #V #T #d #_ #j #H destruct
-| #I #G #L #V #T #d #_ #j #H destruct
-]
-qed-.
-
-lemma da_inv_lref: ∀h,g,G,L,j,d. ⦃G, L⦄ ⊢ #j ▪[h, g] d →
- (∃∃K,V. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ▪[h, g] d) ∨
- (∃∃K,W,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W ▪[h, g] d0 & d = d0+1).
-/2 width=3 by da_inv_lref_aux/ qed-.
-
-fact da_inv_gref_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d → ∀p0. T = §p0 → ⊥.
-#h #g #G #L #T #d * -G -L -T -d
-[ #G #L #k #d #_ #p0 #H destruct
-| #G #L #K #V #i #d #_ #_ #p0 #H destruct
-| #G #L #K #W #i #d #_ #_ #p0 #H destruct
-| #a #I #G #L #V #T #d #_ #p0 #H destruct
-| #I #G #L #V #T #d #_ #p0 #H destruct
-]
-qed-.
-
-lemma da_inv_gref: ∀h,g,G,L,p,d. ⦃G, L⦄ ⊢ §p ▪[h, g] d → ⊥.
-/2 width=9 by da_inv_gref_aux/ qed-.
-
-fact da_inv_bind_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
- ∀b,J,X,Y. T = ⓑ{b,J}Y.X → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, g] d.
-#h #g #G #L #T #d * -G -L -T -d
-[ #G #L #k #d #_ #b #J #X #Y #H destruct
-| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct
-| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct //
-| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct
-]
-qed-.
-
-lemma da_inv_bind: ∀h,g,b,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓑ{b,J}Y.X ▪[h, g] d → ⦃G, L.ⓑ{J}Y⦄ ⊢ X ▪[h, g] d.
-/2 width=4 by da_inv_bind_aux/ qed-.
-
-fact da_inv_flat_aux: ∀h,g,G,L,T,d. ⦃G, L⦄ ⊢ T ▪[h, g] d →
- ∀J,X,Y. T = ⓕ{J}Y.X → ⦃G, L⦄ ⊢ X ▪[h, g] d.
-#h #g #G #L #T #d * -G -L -T -d
-[ #G #L #k #d #_ #J #X #Y #H destruct
-| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct
-| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct
-| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct
-| #I #G #L #V #T #d #HT #J #X #Y #H destruct //
-]
-qed-.
-
-lemma da_inv_flat: ∀h,g,J,G,L,Y,X,d. ⦃G, L⦄ ⊢ ⓕ{J}Y.X ▪[h, g] d → ⦃G, L⦄ ⊢ X ▪[h, g] d.
-/2 width=5 by da_inv_flat_aux/ 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/static/aaa_lift.ma".
-include "basic_2/static/da.ma".
-
-(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
-
-(* Properties on atomic arity assignment for terms **************************)
-
-lemma aaa_da: ∀h,g,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∃d. ⦃G, L⦄ ⊢ T ▪[h, g] d.
-#h #g #G #L #T #A #H elim H -G -L -T -A
-[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
-| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
-| #a #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_bind, ex_intro/
-| #G #L #V #T #B #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
-| #G #L #W #T #A #_ #_ #_ * /3 width=2 by da_flat, ex_intro/
-]
-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/static/da_lift.ma".
-
-(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
-
-(* Main properties **********************************************************)
-
-theorem da_mono: ∀h,g,G,L,T,d1. ⦃G, L⦄ ⊢ T ▪[h, g] d1 →
- ∀d2. ⦃G, L⦄ ⊢ T ▪[h, g] d2 → d1 = d2.
-#h #g #G #L #T #d1 #H elim H -G -L -T -d1
-[ #G #L #k #d1 #Hkd1 #d2 #H
- lapply (da_inv_sort … H) -G -L #Hkd2
- >(deg_mono … Hkd2 … Hkd1) -h -k -d2 //
-| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H
- elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ]
- lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /2 width=1 by/
-| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H
- elim (da_inv_lref … H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ]
- lapply (drop_mono … HLK0 … HLK) -HLK -HLK0 #H destruct /3 width=1 by eq_f/
-| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H
- lapply (da_inv_bind … H) -H /2 width=1 by/
-| #I #G #L #V #T #d1 #_ #IHT #d2 #H
- lapply (da_inv_flat … H) -H /2 width=1 by/
-]
-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/drop_drop.ma".
-include "basic_2/static/da.ma".
-
-(* DEGREE ASSIGNMENT FOR TERMS **********************************************)
-
-(* Properties on relocation *************************************************)
-
-lemma da_lift: ∀h,g,G,L1,T1,d. ⦃G, L1⦄ ⊢ T1 ▪[h, g] d →
- ∀L2,s,l,m. ⬇[s, l, m] L2 ≡ L1 → ∀T2. ⬆[l, m] T1 ≡ T2 →
- ⦃G, L2⦄ ⊢ T2 ▪[h, g] d.
-#h #g #G #L1 #T1 #d #H elim H -G -L1 -T1 -d
-[ #G #L1 #k #d #Hkd #L2 #s #l #m #_ #X #H
- >(lift_inv_sort1 … H) -X /2 width=1 by da_sort/
-| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_lref1 … H) * #Hil #H destruct
- [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
- /3 width=9 by da_ldef/
- | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
- /3 width=8 by da_ldef, drop_inv_gen/
- ]
-| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_lref1 … H) * #Hil #H destruct
- [ elim (drop_trans_le … HL21 … HLK1) -L1 /2 width=2 by ylt_fwd_le/ #X #HLK2 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
- /3 width=8 by da_ldec/
- | lapply (drop_trans_ge … HL21 … HLK1 ?) -L1
- /3 width=8 by da_ldec, drop_inv_gen/
- ]
-| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_bind1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
- /4 width=5 by da_bind, drop_skip/
-| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
- elim (lift_inv_flat1 … H) -H #V2 #T2 #HV12 #HU12 #H destruct
- /3 width=5 by da_flat/
-]
-qed.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma da_inv_lift: ∀h,g,G,L2,T2,d. ⦃G, L2⦄ ⊢ T2 ▪[h, g] d →
- ∀L1,s,l,m. ⬇[s, l, m] L2 ≡ L1 → ∀T1. ⬆[l, m] T1 ≡ T2 →
- ⦃G, L1⦄ ⊢ T1 ▪[h, g] d.
-#h #g #G #L2 #T2 #d #H elim H -G -L2 -T2 -d
-[ #G #L2 #k #d #Hkd #L1 #s #l #m #_ #X #H
- >(lift_inv_sort2 … H) -X /2 width=1 by da_sort/
-| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HV2 | -IHV2 ]
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldef/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldef/
- ]
-| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_lref2 … H) * #Hil #H destruct [ -HW2 | -IHW2 ]
- [ elim (drop_conf_lt … HL21 … HLK2) -L2 /3 width=8 by da_ldec/
- | lapply (drop_conf_ge … HL21 … HLK2 ?) -L2 /2 width=4 by da_ldec/
- ]
-| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /4 width=5 by da_bind, drop_skip/
-| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=5 by da_flat/
-]
-qed-.
qed.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L2 ≡ K2.
+lemma lsuba_drop_O1_conf: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K1,c,k. ⬇[c, 0, k] L1 ≡ K1 →
+ ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇[c, 0, k] L2 ≡ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+| #I #L1 #L2 #V #_ #IHL12 #K1 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
| elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #m #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
[ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L1 c 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
| elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[s, 0, m] L1 ≡ K1.
+lemma lsuba_drop_O1_trans: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
+ ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇[c, 0, k] L1 ≡ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+| #I #L1 #L2 #V #_ #IHL12 #K2 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
| elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #m #H
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #c #k #H
elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
[ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
+ elim (IHL12 L2 c 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
| elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
+++ /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/lrsubeqd_5.ma".
-include "basic_2/static/lsubr.ma".
-include "basic_2/static/da.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
-
-inductive lsubd (h) (g) (G): relation lenv ≝
-| lsubd_atom: lsubd h g G (⋆) (⋆)
-| lsubd_pair: ∀I,L1,L2,V. lsubd h g G L1 L2 →
- lsubd h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubd_beta: ∀L1,L2,W,V,d. ⦃G, L1⦄ ⊢ V ▪[h, g] d+1 → ⦃G, L2⦄ ⊢ W ▪[h, g] d →
- lsubd h g G L1 L2 → lsubd h g G (L1.ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (degree assignment)"
- 'LRSubEqD h g G L1 L2 = (lsubd h g G L1 L2).
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubd_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L1 ⫃ L2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubd_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #g #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubd_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ⫃▪[h, g] L2 → L2 = ⋆.
-/2 width=6 by lsubd_inv_atom1_aux/ qed-.
-
-fact lsubd_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
- G ⊢ K1 ⫃▪[h, g] K2 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-#h #g #G #L1 #L2 * -L1 -L2
-[ #J #K1 #X #H destruct
-| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
-]
-qed-.
-
-lemma lsubd_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃▪[h, g] L2 →
- (∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
- G ⊢ K1 ⫃▪[h, g] K2 &
- I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-/2 width=3 by lsubd_inv_pair1_aux/ qed-.
-
-fact lsubd_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 → L2 = ⋆ → L1 = ⋆.
-#h #g #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubd_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ⫃▪[h, g] ⋆ → L1 = ⋆.
-/2 width=6 by lsubd_inv_atom2_aux/ qed-.
-
-fact lsubd_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀I,K2,W. L2 = K2.ⓑ{I}W →
- (∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
- G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
-#h #g #G #L1 #L2 * -L1 -L2
-[ #J #K2 #U #H destruct
-| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
-]
-qed-.
-
-lemma lsubd_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ⫃▪[h, g] K2.ⓑ{I}W →
- (∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,d. ⦃G, K1⦄ ⊢ V ▪[h, g] d+1 & ⦃G, K2⦄ ⊢ W ▪[h, g] d &
- G ⊢ K1 ⫃▪[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
-/2 width=3 by lsubd_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsubd_refl: ∀h,g,G,L. G ⊢ L ⫃▪[h, g] L.
-#h #g #G #L elim L -L /2 width=1 by lsubd_pair/
-qed.
-
-(* Note: the constant 0 cannot be generalized *)
-lemma lsubd_drop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀K1,s,m. ⬇[s, 0, m] L1 ≡ K1 →
- ∃∃K2. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L2 ≡ K2.
-#h #g #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
- [ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #s #m #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK1
- [ destruct
- elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
- | elim (IHL12 … HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-]
-qed-.
-
-(* Note: the constant 0 cannot be generalized *)
-lemma lsubd_drop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ⫃▪[h, g] L2 →
- ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
- ∃∃K1. G ⊢ K1 ⫃▪[h, g] K2 & ⬇[s, 0, m] L1 ≡ K1.
-#h #g #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
- [ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #s #m #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK2
- [ destruct
- elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
- <(drop_inv_O2 … H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
- | elim (IHL12 … HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-]
-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/static/da_da.ma".
-include "basic_2/static/lsubd.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
-
-(* Properties on degree assignment ******************************************)
-
-lemma lsubd_da_trans: ∀h,g,G,L2,T,d. ⦃G, L2⦄ ⊢ T ▪[h, g] d →
- ∀L1. G ⊢ L1 ⫃▪[h, g] L2 → ⦃G, L1⦄ ⊢ T ▪[h, g] d.
-#h #g #G #L2 #T #d #H elim H -G -L2 -T -d
-[ /2 width=1 by da_sort/
-| #G #L2 #K2 #V #i #d #HLK2 #_ #IHV #L1 #HL12
- elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubd_inv_pair2 … H) -H * #K1 [ | -IHV -HLK1 ]
- [ #HK12 #H destruct /3 width=4 by da_ldef/
- | #W #d0 #_ #_ #_ #H destruct
- ]
-| #G #L2 #K2 #W #i #d #HLK2 #HW #IHW #L1 #HL12
- elim (lsubd_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
- elim (lsubd_inv_pair2 … H) -H * #K1 [ -HW | -IHW ]
- [ #HK12 #H destruct /3 width=4 by da_ldec/
- | #V #d0 #HV #H0W #_ #_ #H destruct
- lapply (da_mono … H0W … HW) -H0W -HW #H destruct /3 width=7 by da_ldef, da_flat/
- ]
-| /4 width=1 by lsubd_pair, da_bind/
-| /3 width=1 by da_flat/
-]
-qed-.
-
-lemma lsubd_da_conf: ∀h,g,G,L1,T,d. ⦃G, L1⦄ ⊢ T ▪[h, g] d →
- ∀L2. G ⊢ L1 ⫃▪[h, g] L2 → ⦃G, L2⦄ ⊢ T ▪[h, g] d.
-#h #g #G #L1 #T #d #H elim H -G -L1 -T -d
-[ /2 width=1 by da_sort/
-| #G #L1 #K1 #V #i #d #HLK1 #HV #IHV #L2 #HL12
- elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
- elim (lsubd_inv_pair1 … H) -H * #K2 [ -HV | -IHV ]
- [ #HK12 #H destruct /3 width=4 by da_ldef/
- | #W0 #V0 #d0 #HV0 #HW0 #_ #_ #H1 #H2 destruct
- lapply (da_inv_flat … HV) -HV #H0V0
- lapply (da_mono … H0V0 … HV0) -H0V0 -HV0 #H destruct /2 width=4 by da_ldec/
- ]
-| #G #L1 #K1 #W #i #d #HLK1 #HW #IHW #L2 #HL12
- elim (lsubd_drop_O1_conf … HL12 … HLK1) -L1 #X #H #HLK2
- elim (lsubd_inv_pair1 … H) -H * #K2 [ -HW | -IHW ]
- [ #HK12 #H destruct /3 width=4 by da_ldec/
- | #W0 #V0 #d0 #HV0 #HW0 #_ #H destruct
- ]
-| /4 width=1 by lsubd_pair, da_bind/
-| /3 width=1 by da_flat/
-]
-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/static/lsubd_da.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR DEGREE ASSIGNMENT ***********************)
-
-(* Main properties **********************************************************)
-
-theorem lsubd_trans: ∀h,g,G. Transitive … (lsubd h g G).
-#h #g #G #L1 #L #H elim H -L1 -L
-[ #X #H >(lsubd_inv_atom1 … H) -H //
-| #I #L1 #L #Y #HL1 #IHL1 #X #H
- elim (lsubd_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=1 by lsubd_pair/
- | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct
- /3 width=3 by lsubd_beta, lsubd_da_trans/
- ]
-| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H
- elim (lsubd_inv_pair1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/
- | #W0 #V0 #d0 #_ #_ #_ #H destruct
- ]
-]
-qed-.
(**************************************************************************)
include "basic_2/notation/relations/lrsubeqc_2.ma".
-include "basic_2/substitution/drop.ma".
+include "basic_2/grammar/lenv.ma".
-(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
inductive lsubr: relation lenv ≝
| lsubr_atom: ∀L. lsubr L (⋆)
lemma lsubr_inv_abbr2: ∀L1,K2,W. L1 ⫃ K2.ⓓW →
∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓW.
/2 width=3 by lsubr_inv_abbr2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubr_fwd_length: ∀L1,L2. L1 ⫃ L2 → |L2| ≤ |L1|.
-#L1 #L2 #H elim H -L1 -L2 /2 width=1 by monotonic_le_plus_l/
-qed-.
-
-lemma lsubr_fwd_drop2_pair: ∀L1,L2. L1 ⫃ L2 →
- ∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
- (∃∃K1. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W) ∨
- ∃∃K1,V. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓓⓝW.V & I = Abst.
-#L1 #L2 #H elim H -L1 -L2
-[ #L #I #K2 #W #s #i #H
- elim (drop_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #s #i #H
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=3 by drop_pair, ex2_intro, or_introl/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 *
- /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
-| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #s #i #H
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK2 destruct [ -IHL12 | -HL12 ]
- [ /3 width=4 by drop_pair, ex3_2_intro, or_intror/
- | elim (IHL12 … HLK2) -IHL12 -HLK2 *
- /4 width=4 by drop_drop_lt, ex3_2_intro, ex2_intro, or_introl, or_intror/
- ]
-]
-qed-.
-
-lemma lsubr_fwd_drop2_abbr: ∀L1,L2. L1 ⫃ L2 →
- ∀K2,V,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & ⬇[s, 0, i] L1 ≡ K1.ⓓV.
-#L1 #L2 #HL12 #K2 #V #s #i #HLK2 elim (lsubr_fwd_drop2_pair … HL12 … HLK2) -L2 // *
-#K1 #W #_ #_ #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/relocation/drops.ma".
+include "basic_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Forward lemmas with generic slicing for local environments ***************)
+
+(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
+lemma lsubr_fwd_drops2_pair: ∀L1,L2. L1 ⫃ L2 →
+ ∀I,K2,W,c,f. 𝐔⦃f⦄ → ⬇*[c, f] L2 ≡ K2.ⓑ{I}W →
+ (∃∃K1. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓑ{I}W) ∨
+ ∃∃K1,V. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓓⓝW.V & I = Abst.
+#L1 #L2 #H elim H -L1 -L2
+[ #L #I #K2 #W #c #f #_ #H
+ elim (drops_inv_atom1 … H) -H #H destruct
+| #J #L1 #L2 #V #HL12 #IHL12 #I #K2 #W #c #f #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -H * #g #HLK2 try #H destruct [ -IHL12 | -HL12 ]
+ [ /4 width=4 by drops_refl, ex2_intro, or_introl/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 /2 width=3 by isuni_inv_next/ *
+ /4 width=4 by drops_drop, ex3_2_intro, ex2_intro, or_introl, or_intror/
+ ]
+| #L1 #L2 #V1 #V2 #HL12 #IHL12 #I #K2 #W #c #f #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -H * #g #HLK2 try #H destruct [ -IHL12 | -HL12 ]
+ [ /4 width=4 by drops_refl, ex3_2_intro, or_intror/
+ | elim (IHL12 … HLK2) -IHL12 -HLK2 /2 width=3 by isuni_inv_next/ *
+ /4 width=4 by drops_drop, ex3_2_intro, ex2_intro, or_introl, or_intror/
+ ]
+]
+qed-.
+
+(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
+lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
+ ∀K2,V,c,f. 𝐔⦃f⦄ → ⬇*[c, f] L2 ≡ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & ⬇*[c, f] L1 ≡ K1.ⓓV.
+#L1 #L2 #HL12 #K2 #V #c #f #Hf #HLK2
+elim (lsubr_fwd_drops2_pair … HL12 … Hf HLK2) -L2 -Hf // *
+#K1 #W #_ #_ #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/grammar/lenv_length.ma".
+include "basic_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⫃ L2 → |L2| ≤ |L1|.
+#L1 #L2 #H elim H -L1 -L2 /2 width=1 by O, le_S_S/
+qed-.
include "basic_2/static/lsubr.ma".
-(* RESTRICTED LOCAL ENVIRONMENT REFINEMENT **********************************)
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
(* Auxiliary inversion lemmas ***********************************************)
(* sort degree specification *)
record sd (h:sh): Type[0] ≝ {
deg : relation nat; (* degree of the sort *)
- deg_total: ∀k. ∃d. deg k d; (* functional relation axioms *)
- deg_mono : ∀k,d1,d2. deg k d1 → deg k d2 → d1 = d2;
- deg_next : ∀k,d. deg k d → deg (next h k) (d - 1) (* compatibility condition *)
+ deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
+ deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
+ deg_next : ∀s,d. deg s d → deg (next h s) (⫰d) (* compatibility condition *)
}.
(* Notable specifications ***************************************************)
-definition deg_O: relation nat ≝ λk,d. d = 0.
+definition deg_O: relation nat ≝ λs,d. d = 0.
definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
-inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat ≝
-| deg_SO_pos : ∀d0. (next h)^d0 k0 = k → deg_SO h k k0 (d0 + 1)
-| deg_SO_zero: ((∃d0. (next h)^d0 k0 = k) → ⊥) → deg_SO h k k0 0
+(* Basic_2A1: includes: deg_SO_pos *)
+inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
+| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (⫯n)
+| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
.
-fact deg_SO_inv_pos_aux: ∀h,k,k0,d0. deg_SO h k k0 d0 → ∀d. d0 = d + 1 →
- (next h)^d k0 = k.
-#h #k #k0 #d0 * -d0
-[ #d0 #Hd0 #d #H
- lapply (injective_plus_l … H) -H #H destruct //
-| #_ #d0 <plus_n_Sm #H destruct
+fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ⫯n →
+ (next h)^n s0 = s.
+#h #s #s0 #n0 * -n0
+[ #n #Hn #x #H destruct //
+| #_ #x #H destruct
]
-qed.
+qed-.
-lemma deg_SO_inv_pos: ∀h,k,k0,d0. deg_SO h k k0 (d0 + 1) → (next h)^d0 k0 = k.
-/2 width=3 by deg_SO_inv_pos_aux/ qed-.
+(* Basic_2A1: was: deg_SO_inv_pos *)
+lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (⫯n) → (next h)^n s0 = s.
+/2 width=3 by deg_SO_inv_succ_aux/ qed-.
-lemma deg_SO_refl: ∀h,k. deg_SO h k k 1.
-#h #k @(deg_SO_pos … 0 ?) //
+lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
+#h #s @(deg_SO_succ … 0 ?) //
qed.
-lemma deg_SO_gt: ∀h,k1,k2. k1 < k2 → deg_SO h k1 k2 0.
-#h #k1 #k2 #HK12 @deg_SO_zero * #d elim d -d normalize
+lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
+#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
[ #H destruct
elim (lt_refl_false … HK12)
-| #d #_ #H
- lapply (next_lt h ((next h)^d k2)) >H -H #H
- lapply (transitive_lt … H HK12) -k1 #H1
- lapply (nexts_le h k2 d) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -d #H
+| #n #_ #H
+ lapply (next_lt h ((next h)^n s2)) >H -H #H
+ lapply (transitive_lt … H HK12) -s1 #H1
+ lapply (nexts_le h s2 n) #H2
+ lapply (le_to_lt_to_lt … H2 H1) -h -n #H
elim (lt_refl_false … H)
]
qed.
-definition sd_SO: ∀h. nat → sd h ≝ λh,k. mk_sd h (deg_SO h k) ….
-[ #k0
- lapply (nexts_dec h k0 k) *
- [ * /3 width=2 by deg_SO_pos, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #d1 #d2 * [ #d01 ] #H1 * [1,3: #d02 ] #H2 //
+definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
+[ #s0
+ lapply (nexts_dec h s0 s) *
+ [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
+| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
[ < H2 in H1; -H2 #H
lapply (nexts_inj … H) -H #H destruct //
| elim H1 /2 width=2 by ex_intro/
| elim H2 /2 width=2 by ex_intro/
]
-| #k0 #d0 *
+| #s0 #n *
[ #d #H destruct elim d -d normalize
- /2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
+ /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
| #H1 @deg_SO_zero * #d #H2 destruct
- @H1 -H1 @(ex_intro … (S d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+ @H1 -H1 @(ex_intro … (⫯d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
]
]
defined.
-rec definition sd_d (h:sh) (k:nat) (d:nat) on d : sd h ≝
+rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
match d with
[ O ⇒ sd_O h
| S d ⇒ match d with
- [ O ⇒ sd_SO h k
- | _ ⇒ sd_d h (next h k) d
+ [ O ⇒ sd_SO h s
+ | _ ⇒ sd_d h (next h s) d
]
].
(* Basic inversion lemmas ***************************************************)
-lemma deg_inv_pred: ∀h,g,k,d. deg h g (next h k) (d+1) → deg h g k (d+2).
-#h #g #k #d #H1
-elim (deg_total h g k) #d0 #H0
+lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (⫯d) → deg h o s (⫯⫯d).
+#h #o #s #d #H1
+elim (deg_total h o s) #n #H0
lapply (deg_next … H0) #H2
-lapply (deg_mono … H1 H2) -H1 -H2 #H
-<(associative_plus d 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
qed-.
-lemma deg_inv_prec: ∀h,g,k,d,d0. deg h g ((next h)^d k) (d0+1) → deg h g k (d+d0+1).
-#h #g #k #d @(nat_ind_plus … d) -d //
-#d #IHd #d0 >iter_SO #H
-lapply (deg_inv_pred … H) -H <(associative_plus d0 1 1) #H
-lapply (IHd … H) -IHd -H //
+lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (⫯d) → deg h o s (⫯(d+n)).
+#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
qed-.
(* Basic properties *********************************************************)
-lemma deg_iter: ∀h,g,k,d1,d2. deg h g k d1 → deg h g ((next h)^d2 k) (d1-d2).
-#h #g #k #d1 #d2 @(nat_ind_plus … d2) -d2 [ <minus_n_O // ]
-#d2 #IHd2 #Hkd1 >iter_SO <minus_plus /3 width=1 by deg_next/
+lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
+#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
qed.
-lemma deg_next_SO: ∀h,g,k,d. deg h g k (d+1) → deg h g (next h k) d.
-#h #g #k #d #Hkd
-lapply (deg_next … Hkd) -Hkd <minus_plus_m_m //
-qed-.
+lemma deg_next_SO: ∀h,o,s,d. deg h o s (⫯d) → deg h o (next h s) d.
+/2 width=1 by deg_next/ qed-.
-lemma sd_d_SS: ∀h,k,d. sd_d h k (d + 2) = sd_d h (next h k) (d + 1).
-#h #k #d <plus_n_Sm <plus_n_Sm //
-qed.
+lemma sd_d_SS: ∀h,s,d. sd_d h s (⫯⫯d) = sd_d h (next h s) (⫯d).
+// qed.
-lemma sd_d_correct: ∀h,d,k. deg h (sd_d h k d) k d.
-#h #d @(nat_ind_plus … d) -d // #d @(nat_ind_plus … d) -d /3 width=1 by deg_inv_pred/
+lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
+#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
qed.
(* sort hierarchy specification *)
record sh: Type[0] ≝ {
next : nat → nat; (* next sort in the hierarchy *)
- next_lt: ∀k. k < next k (* strict monotonicity condition *)
+ next_lt: ∀s. s < next s (* strict monotonicity condition *)
}.
definition sh_N: sh ≝ mk_sh S ….
(* Basic properties *********************************************************)
-lemma nexts_le: ∀h,k,d. k ≤ (next h)^d k.
-#h #k #d elim d -d // normalize #d #IHd
-lapply (next_lt h ((next h)^d k)) #H
-lapply (le_to_lt_to_lt … IHd H) -IHd -H /2 width=2 by lt_to_le/
+lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
+#h #s #n elim n -n // normalize #n #IH
+lapply (next_lt h ((next h)^n s)) #H
+lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
qed.
-lemma nexts_lt: ∀h,k,d. k < (next h)^(d+1) k.
-#h #k #d >iter_SO
-lapply (nexts_le h k d) #H
+lemma nexts_lt: ∀h,s,n. s < (next h)^(⫯n) s.
+#h #s #n normalize
+lapply (nexts_le h s n) #H
@(le_to_lt_to_lt … H) //
qed.
-axiom nexts_dec: ∀h,k1,k2. Decidable (∃d. (next h)^d k1 = k2).
+axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
-axiom nexts_inj: ∀h,k,d1,d2. (next h)^d1 k = (next h)^d2 k → d1 = d2.
+axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
]
}
]
- class "grass"
+*)
+ class "green"
[ { "static typing" * } {
- [ { "local env. ref. for degree assignment" * } {
- [ "lsubd ( ? ⊢ ? ⫃▪[?,?] ? )" "lsubd_da" + "lsubd_lsubd" * ]
- }
- ]
- [ { "degree assignment" * } {
- [ "da ( ⦃?,?⦄ ⊢ ? ▪[?,?] ? )" "da_lift" + "da_aaa" + "da_da" * ]
- }
- ]
[ { "parameters" * } {
[ "sh" "sd" * ]
}
]
+(*
[ { "local env. ref. for atomic arity assignment" * } {
[ "lsuba ( ? ⊢ ? ⫃⁝ ? )" "lsuba_aaa" + "lsuba_lsuba" * ]
}
[ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_lift" + "aaa_lifts" + "aaa_fqus" + "aaa_lleq" + "aaa_aaa" * ]
}
]
- [ { "restricted local env. ref." * } {
- [ "lsubr ( ? ⫃ ? )" "lsubr_lsubr" * ]
- }
- ]
- }
- ]
-
- [ { "context-sensitive multiple rt-substitution" * } {
- [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
- }
- ]
- [ { "iterated structural successor for closures" * } {
- [ "fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_fqus" * ]
- [ "fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_fqup" * ]
+*)
+ [ { "restricted ref. for local env." * } {
+ [ "lsubr ( ? ⫃ ? )" "lsubr_length" + "lsubr_drops" + "lsubr_lsubr" * ]
}
]
- [ { "pointwise union for local environments" * } {
- [ "llor ( ? ⋓[?,?] ? ≡ ? )" "llor_alt" + "llor_drop" * ]
+ [ { "ranged equivalence for closures" * } {
+ [ "freq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )" "freq_freq" * ]
}
]
- [ { "support for multiple relocation" * } {
- [ "mr2 ( @⦃?,?⦄ ≡ ? )" "mr2_plus ( ? + ? )" "mr2_minus ( ? ▭ ? ≡ ? )" "mr2_mr2" * ]
+ [ { "context-sensitive free variables" * } {
+ [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_lreq" + "frees_frees" * ]
}
]
- [ { "lazy pointwise extension of a relation" * } {
- [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt_rec" + "llpx_sn_tc" + "llpx_sn_lreq" + "llpx_sn_drop" + "llpx_sn_lpx_sn" + "llpx_sn_frees" + "llpx_sn_llor" * ]
+ }
+ ]
+ class "grass"
+ [ { "s-computation" * } {
+ [ { "" * } {
+ [ * ]
}
]
+ }
+ ]
+ class "yellow"
+ [ { "s-transition" * } {
[ { "structural successor for closures" * } {
- [ "fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )" "fquq_alt ( ⦃?,?,?⦄ ⊐⊐⸮ ⦃?,?,?⦄ )" * ]
- [ "fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )" * ]
- }
- ]
- [ { "global env. slicing" * } {
- [ "gget ( ⬇[?] ? ≡ ? )" "gget_gget" * ]
+ [ "fquq ( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )" "fquq_length" + "fquq_weight" * ]
+ [ "fqu ( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )" "fqu_length" + "fqu_weight" * ]
}
]
- [ { "context-sensitive ordinary rt-substitution" * } {
- [ "cpy ( ⦃?,?⦄ ⊢ ? ▶[?,?] ? )" "cpy_lift" + "cpy_nlift" + "cpy_cpy" * ]
- }
- ]
- [ { "local env. ref. for rt-substitution" * } {
- [ "lsuby ( ? ⊆[?,?] ? )" "lsuby_lsuby" * ]
- }
- ]
- [ { "pointwise extension of a relation" * } {
- [ "lpx_sn" "lpx_sn_alt" + "lpx_sn_tc" + "lpx_sn_drop" + "lpx_sn_lpx_sn" * ]
- }
- ]
- [ "lleq ( ? ≡[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_lreq" + "lleq_drop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ]
-*)
- class "yellow"
+ }
+ ]
+ class "orange"
[ { "relocation" * } {
- [ { "ranged equivalence for closures" * } {
- [ "freq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )" "freq_freq" * ]
- }
- ]
- [ { "context-sensitive free variables" * } {
- [ "frees ( ? ⊢ 𝐅*⦃?⦄ ≡ ? )" "frees_weight" + "frees_lreq" + "frees_frees" * ]
- }
- ]
[ { "generic slicing for local environments" * } {
[ "drops_vector ( ⬇*[?,?] ? ≡ ? )" * ]
[ "drops ( ⬇*[?,?] ? ≡ ? )" "drops_lstar" + "drops_weight" + "drops_length" + "drops_ceq" + "drops_lexs" + "drops_lreq" + "drops_drops" * ]
}
]
[ { "generic relocation for terms" * } {
- [ "lifts_vector ( ⬆*[?] ? ≡ ? )" "lifts_lift_vector" * ]
+ [ "lifts_vector ( ⬆*[?] ? ≡ ? )" "lifts_lifts_vector" * ]
[ "lifts ( ⬆*[?] ? ≡ ? )" "lifts_simple" + "lifts_weight" + "lifts_lifts" * ]
}
]
]
}
]
- class "orange"
- [ { "" * } {
- [ { "" * } {
- [ * ]
- }
- ]
- }
- ]
class "red"
[ { "grammar" * } {
+ [ { "append for local environments" * } {
+ [ "append ( ? @@ ? )" "append_length" * ]
+ }
+ ]
[ { "context-sensitive equivalences for terms" * } {
[ "ceq" "ceq_ceq" * ]
}
]
[ { "internal syntax" * } {
[ "genv" * ]
- [ "lenv" "lenv_weight ( ♯{?} )" "lenv_length ( |?| )" "lenv_append ( ? @@ ? )" * ]
+ [ "lenv" "lenv_weight ( ♯{?} )" "lenv_length ( |?| )" * ]
[ "term" "term_weight ( ♯{?} )" "term_simple ( 𝐒⦃?⦄ )" "term_vector ( Ⓐ?.? )" * ]
[ "item" * ]
}
class "capitalize italic" { 0 }
class "italic" { 1 }
+(*
+ [ { "local env. ref. for degree assignment" * } {
+ [ "lsubd ( ? ⊢ ? ⫃▪[?,?] ? )" "lsubd_da" + "lsubd_lsubd" * ]
+ }
+ ]
+ [ { "degree assignment" * } {
+ [ "da ( ⦃?,?⦄ ⊢ ? ▪[?,?] ? )" "da_lift" + "da_aaa" + "da_da" * ]
+ }
+ ]
+ [ { "context-sensitive multiple rt-substitution" * } {
+ [ "cpys ( ⦃?,?⦄ ⊢ ? ▶*[?,?] ? )" "cpys_alt ( ⦃?,?⦄ ⊢ ? ▶▶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
+ }
+ ]
+ [ { "iterated structural successor for closures" * } {
+ [ "fqus ( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_alt" + "fqus_fqus" * ]
+ [ "fqup ( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_fqup" * ]
+ }
+ ]
+ [ { "pointwise union for local environments" * } {
+ [ "llor ( ? ⋓[?,?] ? ≡ ? )" "llor_alt" + "llor_drop" * ]
+ }
+ ]
+ [ { "lazy pointwise extension of a relation" * } {
+ [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt_rec" + "llpx_sn_tc" + "llpx_sn_lreq" + "llpx_sn_drop" + "llpx_sn_lpx_sn" + "llpx_sn_frees" + "llpx_sn_llor" * ]
+ }
+ ]
+ [ { "global env. slicing" * } {
+ [ "gget ( ⬇[?] ? ≡ ? )" "gget_gget" * ]
+ }
+ ]
+ [ { "context-sensitive ordinary rt-substitution" * } {
+ [ "cpy ( ⦃?,?⦄ ⊢ ? ▶[?,?] ? )" "cpy_lift" + "cpy_nlift" + "cpy_cpy" * ]
+ }
+ ]
+ [ { "local env. ref. for rt-substitution" * } {
+ [ "lsuby ( ? ⊆[?,?] ? )" "lsuby_lsuby" * ]
+ }
+ ]
+ [ { "pointwise extension of a relation" * } {
+ [ "lpx_sn" "lpx_sn_alt" + "lpx_sn_tc" + "lpx_sn_drop" + "lpx_sn_lpx_sn" * ]
+ }
+ ]
+ [ "lleq ( ? ≡[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_lreq" + "lleq_drop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ]
+*)
lemma lt_le_false: ∀x,y. x < y → y ≤ x → ⊥.
/3 width=4 by lt_refl_false, lt_to_le_to_lt/ qed-.
+lemma succ_inv_refl_sn: ∀x. ⫯x = x → ⊥.
+#x #H @(lt_le_false x (⫯x)) //
+qed-.
+
lemma lt_inv_O1: ∀n. 0 < n → ∃m. ⫯m = n.
* /2 width=2 by ex_intro/
#H cases (lt_le_false … H) -H //
include "ground_2/notation/relations/rafter_3.ma".
include "ground_2/relocation/rtmap_istot.ma".
-include "ground_2/relocation/rtmap_uni.ma".
(* RELOCATION MAP ***********************************************************)
(**************************************************************************)
include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/rtmap_id.ma".
+include "ground_2/relocation/rtmap_uni.ma".
(* RELOCATION MAP ***********************************************************)
lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≡ i.
/2 width=1 by isid_inv_at/ qed.
+
+(* Properties with uniform relocations **************************************)
+
+lemma at_uni: ∀n,i. @⦃i,𝐔❴n❵⦄ ≡ n+i.
+#n elim n -n /2 width=5 by at_next/
+qed.
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