--- /dev/null
+lemma cpg_delift: ∀c,h,I,G,K,V,T1,L,i. ⬇*[i] L ≡ (K.ⓑ{I}V) →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2.
+#h #c #I #G #K #V #T1 elim T1 -T1
+[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_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/relocation/drops.ma".
+include "basic_2/rt_transition/cpg.ma".
+
+(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with length for local environments ****************************)
+
+lemma cpg_inv_lref1_ge: ∀h,c,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, c] T2 → |L| ≤ i → T2 = #i.
+#h #c #G #L #T2 #i #H elim (cpg_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-.
#g #Hg #H destruct /3 width=3 by drops_inv_drop1/
| #f1 #I #K1 #K #V1 #V #_ #HV1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
#g2 #g #Hf #H1 #H2 destruct
- [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_div/
+ [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, lifts_div3/
| /4 width=3 by drops_inv_drop1, drops_drop/
]
]
X = ⓕ{I}V1.T1.
/2 width=3 by lifts_inv_flat2_aux/ qed-.
+(* Advanced inversion lemmas ************************************************)
+
(* Basic_2A1: includes: lift_inv_pair_xy_x *)
lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≡ V → ⊥.
#f #J #V elim V -V
]
qed-.
+lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≡ Y → Y = #(l+i).
+#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
+qed-.
+
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: includes: lift_inv_O2 *)
]
qed-.
+(* Properties with uniform relocation ***************************************)
+
+lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≡ U → ⬆*[n1+n2] T ≡ U.
+/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
+
(* Basic_2A1: removed theorems 14:
lifts_inv_nil lifts_inv_cons
lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
(* Basic_1: includes: lift_gen_lift *)
(* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div: ∀f2,T,T2. ⬆*[f2] T2 ≡ T → ∀T1,f. ⬆*[f] T1 ≡ T →
- ∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
+theorem lift_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≡ T → ∀g2,Tg. ⬆*[g2] Tg ≡ T →
+ ∀f1,g1. H_at_div f2 g2 f1 g1 →
+ ∃∃T0. ⬆*[f1] T0 ≡ Tf & ⬆*[g1] T0 ≡ Tg.
+#f2 #Tf #T #H elim H -f2 -Tf -T
+[ #f2 #s #g2 #Tg #H #f1 #g1 #_
+ lapply (lifts_inv_sort2 … H) -H #H destruct
+ /2 width=3 by ex2_intro/
+| #f2 #jf #j #Hf2 #g2 #Tg #H #f1 #g1 #H0
+ elim (lifts_inv_lref2 … H) -H #jg #Hg2 #H destruct
+ elim (H0 … Hf2 Hg2) -H0 -j /3 width=3 by lifts_lref, ex2_intro/
+| #f2 #l #g2 #Tg #H #f1 #g1 #_
+ lapply (lifts_inv_gref2 … H) -H #H destruct
+ /2 width=3 by ex2_intro/
+| #f2 #p #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
+ elim (lifts_inv_bind2 … H) -H #Vg #Tg #HVg #HTg #H destruct
+ elim (IHV … HVg … H0) -IHV -HVg
+ elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by at_div_pp/ ]
+ /3 width=5 by lifts_bind, ex2_intro/
+| #f2 #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
+ elim (lifts_inv_flat2 … H) -H #Vg #Tg #HVg #HTg #H destruct
+ elim (IHV … HVg … H0) -IHV -HVg
+ elim (IHT … HTg … H0) -IHT -HTg -H0
+ /3 width=5 by lifts_flat, ex2_intro/
+]
+qed-.
+
+lemma lifts_div4_one: ∀f,Tf,T. ⬆*[↑f] Tf ≡ T →
+ ∀T1. ⬆*[1] T1 ≡ T →
+ ∃∃T0. ⬆*[1] T0 ≡ Tf & ⬆*[f] T0 ≡ T1.
+/4 width=6 by lift_div4, at_div_id_dx, at_div_pn/ qed-.
+
+theorem lifts_div3: ∀f2,T,T2. ⬆*[f2] T2 ≡ T → ∀f,T1. ⬆*[f] T1 ≡ T →
+ ∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
#f2 #T #T2 #H elim H -f2 -T -T2
-[ #f2 #s #T1 #f #H >(lifts_inv_sort2 … H) -T1 //
-| #f2 #i2 #i #Hi2 #T1 #f #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
+[ #f2 #s #f #T1 #H >(lifts_inv_sort2 … H) -T1 //
+| #f2 #i2 #i #Hi2 #f #T1 #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
#i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
-| #f2 #l #T1 #f #H >(lifts_inv_gref2 … H) -T1 //
-| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #T1 #f #H
+| #f2 #l #f #T1 #H >(lifts_inv_gref2 … H) -T1 //
+| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
/4 width=3 by lifts_bind, after_O2/
-| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #T1 #f #H
+| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
/3 width=3 by lifts_flat/
]
(* Basic_1: was: lift1_lift1 (left to right) *)
(* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
(* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
-theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≡ T → ∀T2,f2. ⬆*[f2] T ≡ T2 →
+theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≡ T → ∀f2,T2. ⬆*[f2] T ≡ T2 →
∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1 ≡ T2.
#f1 #T1 #T #H elim H -f1 -T1 -T
-[ #f1 #s #T2 #f2 #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i1 #i #Hi1 #T2 #f2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
+[ #f1 #s #f2 #T2 #H >(lifts_inv_sort1 … H) -T2 //
+| #f1 #i1 #i #Hi1 #f2 #T2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
#i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
-| #f1 #l #T2 #f2 #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #T2 #f2 #H
+| #f1 #l #f2 #T2 #H >(lifts_inv_gref1 … H) -T2 //
+| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/4 width=3 by lifts_bind, after_O2/
-| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #T2 #f2 #H
+| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/3 width=3 by lifts_flat/
]
qed-.
(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-theorem lifts_conf: ∀f1,T,T1. ⬆*[f1] T ≡ T1 → ∀T2,f. ⬆*[f] T ≡ T2 →
+theorem lifts_conf: ∀f1,T,T1. ⬆*[f1] T ≡ T1 → ∀f,T2. ⬆*[f] T ≡ T2 →
∀f2. f2 ⊚ f1 ≡ f → ⬆*[f2] T1 ≡ T2.
#f1 #T #T1 #H elim H -f1 -T -T1
-[ #f1 #s #T2 #f #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i #i1 #Hi1 #T2 #f #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
+[ #f1 #s #f #T2 #H >(lifts_inv_sort1 … H) -T2 //
+| #f1 #i #i1 #Hi1 #f #T2 #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
#i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
-| #f1 #l #T2 #f #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #T2 #f #H
+| #f1 #l #f #T2 #H >(lifts_inv_gref1 … H) -T2 //
+| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/4 width=3 by lifts_bind, after_O2/
-| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #T2 #f #H
+| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
/3 width=3 by lifts_flat/
]
(* Basic_2A1: includes: lift_inj *)
lemma lifts_inj: ∀f,T1,U. ⬆*[f] T1 ≡ U → ∀T2. ⬆*[f] T2 ≡ U → T1 = T2.
#f #T1 #U #H1 #T2 #H2 lapply (isid_after_dx 𝐈𝐝 … f)
-/3 width=6 by lifts_div, lifts_fwd_isid/
+/3 width=6 by lifts_div3, lifts_fwd_isid/
qed-.
(* Basic_2A1: includes: lift_mono *)
⬆*[1] V2 ≡ W2 → cpg h (↓c) G (L.ⓓV1) (#0) W2
| cpg_ell : ∀c,G,L,V1,V2,W2. cpg h c G L V1 V2 →
⬆*[1] V2 ≡ W2 → cpg h ((↓c)+𝟘𝟙) G (L.ⓛV1) (#0) W2
-| cpt_lref : ∀c,I,G,L,V,T,U,i. cpg h c G L (#i) T →
+| cpg_lref : ∀c,I,G,L,V,T,U,i. cpg h c G L (#i) T →
⬆*[1] T ≡ U → cpg h c G (L.ⓑ{I}V) (#⫯i) U
| cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2.
cpg h cV G L V1 V2 → cpg h cT G (L.ⓑ{I}V1) T1 T2 →
(* Basic inversion lemmas ***************************************************)
fact cpg_inv_atom1_aux: ∀c,h,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[c, h] T2 → ∀J. T1 = ⓪{J} →
- ∨∨ T2 = ⓪{J}
- | ∃∃s. J = Sort s & T2 = ⋆(next h s)
+ ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
+ | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & J = LRef 0 & c = ↓cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
| ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 &
L = K.ⓑ{I}V & J = LRef (⫯i).
#c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1 by or5_intro0/
-| #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex2_intro/
+[ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
+| #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex3_intro/
| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro2, ex5_4_intro/
| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro3, ex5_4_intro/
| #c #I #G #L #V #T #U #i #HT #HTU #J #H destruct /3 width=9 by or5_intro4, ex4_5_intro/
qed-.
lemma cpg_inv_atom1: ∀c,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[c, h] T2 →
- ∨∨ T2 = ⓪{J}
- | ∃∃s. J = Sort s & T2 = ⋆(next h s)
+ ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
+ | ∃∃s. J = Sort s & T2 = ⋆(next h s) & c = 𝟘𝟙
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & J = LRef 0 & c = ↓cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
/2 width=3 by cpg_inv_atom1_aux/ qed-.
lemma cpg_inv_sort1: ∀c,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[c, h] T2 →
- T2 = ⋆s ∨ T2 = ⋆(next h s).
+ (T2 = ⋆s ∧ c = 𝟘𝟘) ∨ (T2 = ⋆(next h s) ∧ c = 𝟘𝟙).
#c #h #G #L #T2 #s #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #s0 #H destruct /2 width=1 by or_intror/
+elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
+[ #s0 #H destruct /3 width=1 by or_intror, conj/
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
]
qed-.
lemma cpg_inv_zero1: ∀c,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[c, h] T2 →
- ∨∨ T2 = #0
+ ∨∨ (T2 = #0 ∧ c = 𝟘𝟘)
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & c = ↓cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓛV1 & c = (↓cV)+𝟘𝟙.
#c #h #G #L #T2 #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or3_intro0/ *
+elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #HV12 #HVT2 #H1 #_ #H2 destruct /3 width=8 by or3_intro1, or3_intro2, ex4_4_intro/
| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
qed-.
lemma cpg_inv_lref1: ∀c,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[c, h] T2 →
- (T2 = #⫯i) ∨
+ (T2 = #(⫯i) ∧ c = 𝟘𝟘) ∨
∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[c, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
#c #h #G #L #T2 #i #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
| #I #K #V1 #V2 #j #HV2 #HVT2 #H1 #H2 destruct /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
-lemma cpg_inv_gref1: ∀c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[c, h] T2 → T2 = §l.
+lemma cpg_inv_gref1: ∀c,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[c, h] T2 → T2 = §l ∧ c = 𝟘𝟘.
#c #h #G #L #T2 #l #H
-elim (cpg_inv_atom1 … H) -H // *
+elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
[ #s #H destruct
|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
/3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
qed-.
-lemma cpg_inv_abst1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[c, h] U2 →
+lemma cpg_inv_abst1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[c, h] U2 →
∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[cT, h] T2 &
- U2 = ⓛ{p} V2. T2 & c = (↓cV)+cT.
+ U2 = ⓛ{p}V2.T2 & c = (↓cV)+cT.
#c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
[ /3 width=8 by ex4_4_intro/
| #c #T #_ #_ #_ #H destruct
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops.ma".
+include "basic_2/relocation/drops_drops.ma".
+include "basic_2/s_computation/fqup_weight.ma".
+include "basic_2/s_computation/fqup_drops.ma".
include "basic_2/rt_transition/cpg.ma".
(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* Advanced properties ******************************************************)
+
+lemma cpg_delta_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓓV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
+ ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ➡[↓c, h] T2.
+#c #h #G #K #V #V2 #i elim i -i
+[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_delta/
+| #i #IH #L0 #T0 #H0 #HV2 #HVT2
+ elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct
+ elim (lifts_split_trans … HVT2 (𝐔❴⫯i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
+]
+qed.
+
+lemma cpg_ell_drops: ∀c,h,G,K,V,V2,i,L,T2. ⬇*[i] L ≡ K.ⓛV → ⦃G, K⦄ ⊢ V ➡[c, h] V2 →
+ ⬆*[⫯i] V2 ≡ T2 → ⦃G, L⦄ ⊢ #i ➡[(↓c)+𝟘𝟙, h] T2.
+#c #h #G #K #V #V2 #i elim i -i
+[ #L #T2 #HLK lapply (drops_fwd_isid … HLK ?) // #H destruct /3 width=3 by cpg_ell/
+| #i #IH #L0 #T0 #H0 #HV2 #HVT2
+ elim (drops_inv_succ … H0) -H0 #I #L #V0 #HLK #H destruct
+ elim (lifts_split_trans … HVT2 (𝐔❴⫯i❵) (𝐔❴1❵) ?) -HVT2 /3 width=3 by cpg_lref/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpg_inv_lref1_drops: ∀c,h,G,i,L,T2. ⦃G, L⦄ ⊢ #i ➡[c, h] T2 →
+ ∨∨ T2 = #i ∧ c = 𝟘𝟘
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+ ⬆*[⫯i] V2 ≡ T2 & c = ↓cV
+ | ∃∃cV,K,V,V2. ⬇*[i] L ≡ K.ⓛV & ⦃G, K⦄ ⊢ V ➡[cV, h] V2 &
+ ⬆*[⫯i] V2 ≡ T2 & c = (↓cV) + 𝟘𝟙.
+#c #h #G #i elim i -i
+[ #L #T2 #H elim (cpg_inv_zero1 … H) -H * /3 width=1 by or3_intro0, conj/
+ /4 width=8 by drops_refl, ex4_4_intro, or3_intro2, or3_intro1/
+| #i #IH #L #T2 #H elim (cpg_inv_lref1 … H) -H * /3 width=1 by or3_intro0, conj/
+ #I #K #V #V2 #H #HVT2 #H0 destruct elim (IH … H) -IH -H
+ [ * #H1 #H2 destruct lapply (lifts_inv_lref1_uni … HVT2) -HVT2 #H destruct /3 width=1 by or3_intro0, conj/ ] *
+ #cV #L #W #W2 #HKL #HW2 #HWV2 #H destruct
+ lapply (lifts_trans … HWV2 … HVT2 ??) -V2
+ /4 width=8 by drops_drop, ex4_4_intro, or3_intro2, or3_intro1/
+]
+qed-.
+
(* Properties with generic slicing for local environments *******************)
-lemma cpg_delift: ∀c,h,I,G,K,V,T1,L,i. ⬇*[i] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, 𝟘𝟘] T2 & ⬆*[↑1] T ≡ T2.
-#h #c #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #l /2 width=4 by cpg_atom, lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpg_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i) /3 width=7 by cpg_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpg_bind, drop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpg_flat, lift_flat, ex2_2_intro/
+lemma cpg_lifts: ∀c,h,G. d_liftable2 (cpg h c G).
+#c #h #G #K #T generalize in match c; -c
+@(fqup_wf_ind_eq … G K T) -G -K -T #G0 #K0 #T0 #IH #G #K * *
+[ #s #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort1 … H1) -H1 #H destruct
+ elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
+ /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
+| #i1 #HG #HK #HT #c #T2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
+ #cV #K0 #V #V2 #HK0 #HV2 #HVT2 #H destruct
+ elim (lifts_inv_lref1 … H1) -H1 #i2 #Hf #H destruct
+ lapply (drops_trans … HLK … HK0 ??) -HLK [3,6: |*: // ] #H
+ elim (drops_split_trans … H) -H [1,6: |*: /2 width=6 by after_uni_dx/ ] #Y #HL0 #HY
+ lapply (drops_inv_tls_at … Hf … HY) -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #L0 #W #HLK0 #HVW #H destruct
+ elim (IH … HV2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -K -K0 -V #W2 #HVW2 #HW2
+ elim (lifts_total W2 (𝐔❴⫯i2❵)) #U2 #HWU2
+ lapply (lifts_trans … HVW2 … HWU2 ??) -HVW2 [3,6: |*: // ] #HVU2
+ lapply (lifts_conf … HVT2 … HVU2 f ?) -V2 [1,3: /2 width=3 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_gen, ex2_intro/
+| #l #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref1 … H1) -H1 #H destruct
+ elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
+ /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
+| #p #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cT #T2 #HT12 #HXT2 #H1 #H2 #H3 destruct
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #U2 #HTU2 #HU12
+ lapply (lifts_trans … HXT2 … HTU2 ??) -T2 [3: |*: // ] #HXU2
+ elim (lifts_split_trans … HXU2 f (𝐔❴⫯O❵)) [2: /2 width=1 by after_uni_one_dx/ ]
+ /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| #I #V1 #T1 #HG #HK #HT #c #X2 #H2 #b #f #L #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat1 … H1) -H1 #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_flat1 … H2) -H2 *
+ [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
+ | #cT #HT12 #H1 #H2 destruct
+ elim (IH … HT12 … HLK … HTU1) -IH -HT12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cV #HV12 #H1 #H2 destruct
+ elim (IH … HV12 … HLK … HVW1) -IH -HV12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ | #cV #cY #cT #a #V2 #Y1 #Y2 #T0 #T2 #HV12 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 //
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cV #cY #cT #a #V2 #V20 #Y1 #Y2 #T0 #T2 #HV12 #HV20 #HY12 #HT12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind1 … HTU1) -HTU1 #Z1 #U0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HV12 … HLK … HVW1) -HV12 -HVW1 // #W2 #HVW2 #HW12
+ elim (IH … HY12 … HLK … HYZ1) -HY12 //
+ elim (IH … HT12 … HTU1) -IH -HT12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ elim (lifts_total W2 (𝐔❴1❵)) #W20 #HW20
+ lapply (lifts_trans … HVW2 … HW20 ??) -HVW2 [3: |*: // ] #H
+ lapply (lifts_conf … HV20 … H (↑f) ?) -V2 /2 width=3 by after_uni_one_sn/
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, ex2_intro/
]
]
qed-.
-lemma cpg_inv_lref1: ∀h,c,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, c] T2 →
- T2 = #i ∨
- ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, c] V2 &
- ⬆[O, i+1] V2 ≡ T2.
-#h #c #G #L #T2 #i #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #s #d #_ #_ #H destruct
-| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma cpg_inv_lift1: ∀c,h,G. d_deliftable2_sn (cpg h c G).
+#c #h #G #L #U generalize in match c; -c
+@(fqup_wf_ind_eq … G L U) -G -L -U #G0 #L0 #U0 #IH #G #L * *
+[ #s #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_sort2 … H1) -H1 #H destruct
+ elim (cpg_inv_sort1 … H2) -H2 * #H1 #H2 destruct
+ /2 width=3 by cpg_atom, cpg_ess, lifts_sort, ex2_intro/
+| #i2 #HG #HL #HU #c #U2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (cpg_inv_lref1_drops … H2) -H2 *
+ [ #H1 #H2 destruct /2 width=3 by ex2_intro/ ]
+ #cW #L0 #W #W2 #HL0 #HW2 #HWU2 #H destruct
+ elim (lifts_inv_lref2 … H1) -H1 #i1 #Hf #H destruct
+ lapply (drops_split_div … HLK (𝐔❴i1❵) ???) -HLK [4,8: * |*: // ] #Y0 #HK0 #HLY0
+ lapply (drops_conf … HL0 … HLY0 ??) -HLY0 [3,6: |*: /2 width=6 by after_uni_dx/ ] #HLY0
+ lapply (drops_inv_tls_at … Hf … HLY0) -HLY0 #HLY0
+ elim (drops_inv_skip1 … HLY0) -HLY0 #K0 #V #HLK0 #HVW #H destruct
+ elim (IH … HW2 … HLK0 … HVW) -IH /2 width=2 by fqup_lref/ -L -L0 -W #V2 #HVW2 #HV2
+ lapply (lifts_trans … HVW2 … HWU2 ??) -W2 [3,6: |*: // ] #HVU2
+ elim (lifts_split_trans … HVU2 ? f) -HVU2 [1,4: |*: /2 width=4 by after_uni_succ_sn/ ]
+ /4 width=8 by cpg_ell_drops, cpg_delta_drops, drops_inv_F, ex2_intro/
+| #l #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct -IH
+ lapply (lifts_inv_gref2 … H1) -H1 #H destruct
+ elim (cpg_inv_gref1 … H2) -H2 #H1 #H2 destruct
+ /2 width=3 by cpg_atom, lifts_gref, ex2_intro/
+| #p #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_bind2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_bind1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /3 width=5 by cpg_bind, lifts_bind, ex2_intro/
+ | #cU #U2 #HU12 #HXU2 #H1 #H2 #H3 destruct
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ] #T2 #HTU2 #HT12
+ elim (lifts_div4_one … HTU2 … HXU2) -U2 /3 width=5 by cpg_zeta, ex2_intro/
+ ]
+| #I #W1 #U1 #HG #HL #HU #c #X2 #H2 #b #f #K #HLK #X1 #H1 destruct
+ elim (lifts_inv_flat2 … H1) -H1 #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (cpg_inv_flat1 … H2) -H2 *
+ [ #cW #cU #W2 #U2 #HW12 #HU12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=5 by cpg_flat, lifts_flat, ex2_intro/
+ | #cU #HU12 #H1 #H2 destruct
+ elim (IH … HU12 … HLK … HTU1) -IH -HU12 -HLK -HTU1 //
+ /3 width=3 by cpg_eps, ex2_intro/
+ | #cW #HW12 #H1 #H2 destruct
+ elim (IH … HW12 … HLK … HVW1) -IH -HW12 -HLK -HVW1 //
+ /3 width=3 by cpg_ee, ex2_intro/
+ | #cW #cZ #cU #a #W2 #Z1 #Z2 #U0 #U2 #HW12 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 //
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ /4 width=7 by cpg_beta, lifts_bind, lifts_flat, ex2_intro/
+ | #cW #cZ #cU #a #W2 #W20 #Z1 #Z2 #U0 #U2 #HW12 #HW20 #HZ12 #HU12 #H1 #H2 #H3 #H4 destruct
+ elim (lifts_inv_bind2 … HTU1) -HTU1 #Y1 #T0 #HYZ1 #HTU1 #H destruct
+ elim (IH … HW12 … HLK … HVW1) -HW12 -HVW1 // #V2 #HVW2 #HV12
+ elim (IH … HZ12 … HLK … HYZ1) -HZ12 //
+ elim (IH … HU12 … HTU1) -IH -HU12 -HTU1 [ |*: /3 width=3 by drops_skip/ ]
+ lapply (lifts_trans … HVW2 … HW20 ??) -W2 [3: |*: // ] #H
+ elim (lifts_split_trans … H ? (↑f)) -H [ |*: /2 width=3 by after_uni_one_sn/ ]
+ /4 width=9 by cpg_theta, lifts_bind, lifts_flat, 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/relocation/drops.ma".
-include "basic_2/rt_transition/cpg.ma".
-
-(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
-
-(* Properties with length for local environments ****************************)
-
-lemma cpg_inv_lref1_ge: ∀h,c,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, c] T2 → |L| ≤ i → T2 = #i.
-#h #c #G #L #T2 #i #H elim (cpg_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-.
(* Properties with restricted refinement for local environments *************)
-lemma lsubr_cpg_trans: ∀h,c,G. lsub_trans … (cpg h c G) lsubr.
-#h #c #G #L1 #T1 #T2 #H elim H -c -G -L1 -T1 -T2
+lemma lsubr_cpg_trans: ∀c,h,G. lsub_trans … (cpg h c G) lsubr.
+#c #h #G #L1 #T1 #T2 #H elim H -c -G -L1 -T1 -T2
[ //
| /2 width=2 by cpg_st/
| #c #G #L1 #V1 #V2 #W2 #_ #HVW2 #IH #X #H
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops.ma".
+include "basic_2/grammar/term_simple.ma".
include "basic_2/rt_transition/cpg.ma".
(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
-(* Properties with generic slicing for local environments *******************)
+(* Properties with simple terms *********************************************)
(* Note: the main property of simple terms *)
-lemma cpg_inv_appl1_simple: ∀h,c,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, c] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, c] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, c] T2 &
- U = ⓐV2.T2.
-#h #c #G #L #V1 #T1 #U #H #HT1
-elim (cpg_inv_appl1 … H) -H *
-[ /2 width=5 by ex3_2_intro/
-| #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
+lemma cpg_inv_appl1_simple: ∀c,h,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[c, h] U → 𝐒⦃T1⦄ →
+ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[cV, h] V2 & ⦃G, L⦄ ⊢ T1 ➡[cT, h] T2 &
+ U = ⓐV2.T2 & c = (↓cV)+cT.
+#c #h #G #L #V1 #T1 #U #H #HT1 elim (cpg_inv_appl1 … H) -H *
+[ /2 width=8 by ex4_4_intro/
+| #cV #cW #cT #p #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H destruct
elim (simple_inv_bind … HT1)
-| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+| #cV #cW #cT #p #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H destruct
elim (simple_inv_bind … HT1)
]
qed-.
(* Advanced properties ******************************************************)
(* 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.
+lemma aaa_lref_drops: ∀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/
(* Advanced inversion lemmas ************************************************)
(* 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.
+lemma aaa_inv_lref_drops: ∀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
lapply (aaa_inv_sort … H) -H #H destruct
>(lifts_inv_sort1 … HX) -HX //
| #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
- elim (aaa_inv_lref_gen … H) -H #J #K1 #V1 #HLK1 #HA
+ elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
elim (drops_inv_skip2 … HY) -HY #K2 #V2 #HK21 #HV12 #H destruct
- /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_gen/
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
| #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
elim (aaa_inv_gref … H)
| #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
lapply (aaa_inv_sort … H) -H #H destruct
>(lifts_inv_sort2 … HX) -HX //
| #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
- elim (aaa_inv_lref_gen … H) -H #J #K2 #V2 #HLK2 #HA
+ elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
lapply (drops_inv_tls_at … Hf … HY) -HY #HY -Hf
elim (drops_inv_skip1 … HY) -HY #K1 #V1 #HK21 #HV12 #H destruct
- /4 width=12 by aaa_lref_gen, fqup_lref, drops_inv_F/
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
| #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
elim (aaa_inv_gref … H)
| #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
]
qed-.
+lemma after_inv_total: ∀f1,f2,f. f1 ⊚ f2 ≡ f → f1 ∘ f2 ≗ f.
+/2 width=4 by after_mono/ qed-.
+
(* Specific forward lemmas **************************************************)
lemma after_fwd_hd: ∀f1,f2,f,n2,n. f1 ⊚ n2@f2 ≡ n@f → f1@❴n2❵ = n.
]
qed-.
+lemma after_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. f2 ⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H
+ lapply (after_isid_inv_dx … Hg ?) -Hg
+ /4 width=5 by isid_after_sn, after_eq_repl_back_0, after_next/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct
+ elim (after_inv_pnx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ /3 width=5 by after_next/
+ | #g2 #Hg2 #H2 destruct
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ /3 width=5 by after_next/
+ ]
+]
+qed.
+
+lemma after_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
+ ∀f. 𝐔❴⫯i2❵ ⊚ ⫱*[⫯i2] f2 ≡ f → f2 ⊚ 𝐔❴⫯i1❵ ≡ f.
+#i2 elim i2 -i2
+[ #i1 #f2 #Hf2 #f #Hf
+ elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
+ elim (after_inv_nxx … Hf) -Hf [ |*: // ] #g #Hg #H destruct
+ lapply (after_isid_inv_sn … Hg ?) -Hg
+ /4 width=7 by isid_after_dx, after_eq_repl_back_0, after_push/
+| #i2 #IH #i1 #f2 #Hf2 #f #Hf
+ elim (after_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H destruct
+ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
+ [ #g2 #j1 #Hg2 #H1 #H2 destruct <tls_xn in Hg; /3 width=7 by after_push/
+ | #g2 #Hg2 #H2 destruct <tls_xn in Hg; /3 width=5 by after_next/
+ ]
+]
+qed-.
+
+lemma after_uni_one_dx: ∀f2,f. ↑f2 ⊚ 𝐔❴⫯O❵ ≡ f → 𝐔❴⫯O❵ ⊚ f2 ≡ f.
+#f2 #f #H @(after_uni_succ_dx … (↑f2)) /2 width=3 by at_refl/
+qed.
+
+lemma after_uni_one_sn: ∀f1,f. 𝐔❴⫯O❵ ⊚ f1 ≡ f → ↑f1 ⊚ 𝐔❴⫯O❵ ≡ f.
+/3 width=3 by after_uni_succ_sn, at_refl/ qed-.
+
(* Forward lemmas on istot **************************************************)
lemma after_istot_fwd: ∀f2,f1,f. f2 ⊚ f1 ≡ f → 𝐓⦃f2⦄ → 𝐓⦃f1⦄ → 𝐓⦃f⦄.
interpretation "relational application (rtmap)"
'RAt i1 f i2 = (at f i1 i2).
+definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1.
+ ∀jf,jg,j. @⦃jf, f2⦄ ≡ j → @⦃jg, g2⦄ ≡ j →
+ ∃∃j0. @⦃j0, f1⦄ ≡ jf & @⦃j0, g1⦄ ≡ jg.
+
(* Basic inversion lemmas ***************************************************)
lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. 0 = i1 → ↑g = f → 0 = i2.
#Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/
qed-.
+theorem at_div_comm: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div g2 f2 g1 f1.
+#f2 #g2 #f1 #g1 #IH #jg #jf #j #Hg #Hf
+elim (IH … Hf Hg) -IH -j /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_pp: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (↑g2) (↑f1) (↑g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xpx … Hf) -Hf [1,2: * |*: // ]
+[ #H1 #H2 destruct -IH
+ lapply (at_inv_xpp … Hg ???) -Hg [4: |*: // ] #H destruct
+ /3 width=3 by at_refl, ex2_intro/
+| #xf #i #Hf2 #H1 #H2 destruct
+ lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+ elim (IH … Hf2 Hg2) -IH -i /3 width=9 by at_push, ex2_intro/
+]
+qed-.
+
+theorem at_div_nn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (⫯g2) (f1) (g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xnn … Hg ????) -Hg [5: |*: // ] #Hg2
+elim (IH … Hf2 Hg2) -IH -i /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_np: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (↑g2) (f1) (⫯g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+elim (IH … Hf2 Hg2) -IH -i /3 width=7 by at_next, ex2_intro/
+qed-.
+
+theorem at_div_pn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (⫯g2) (⫯f1) (g1).
+/4 width=6 by at_div_np, at_div_comm/ qed-.
+
(* Properties on tls ********************************************************)
lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≡ n → @⦃0, ⫱*[n]f⦄ ≡ 0.
lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≡ i.
/2 width=1 by isid_inv_at/ qed.
+(* Advanced forward lemmas on id ********************************************)
+
+lemma at_id_fwd: ∀i1,i2. @⦃i1, 𝐈𝐝⦄ ≡ i2 → i1 = i2.
+/2 width=4 by at_mono/ qed.
+
+(* Main properties on id ****************************************************)
+
+theorem at_div_id_dx: ∀f. H_at_div f 𝐈𝐝 𝐈𝐝 f.
+#f #jf #j0 #j #Hf #H0
+lapply (at_id_fwd … H0) -H0 #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_id_sn: ∀f. H_at_div 𝐈𝐝 f f 𝐈𝐝.
+/3 width=6 by at_div_id_dx, at_div_comm/ qed-.
+
(* Properties with uniform relocations **************************************)
lemma at_uni: ∀n,i. @⦃i,𝐔❴n❵⦄ ≡ n+i.
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