(* Advanced properties ******************************************************)
-lemma frees_atom_drops: ∀b,L,i. ⬇*[b,𝐔❴i❵] L ≘ ⋆ →
- ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
+lemma frees_atom_drops:
+ ∀b,L,i. ⇩*[b,𝐔❨i❩] L ≘ ⋆ →
+ ∀f. 𝐈❪f❫ → L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i]↑f.
#b #L elim L -L /2 width=1 by frees_atom/
#L #I #IH *
[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
]
qed.
-lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
- ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_pair_drops:
+ ∀f,K,V. K ⊢ 𝐅+❪V❫ ≘ f →
+ ∀i,I,L. ⇩[i] L ≘ K.ⓑ[I]V → L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i] ↑f.
#f #K #V #Hf #i elim i -i
[ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
| #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
]
qed.
-lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
- L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_unit_drops:
+ ∀f. 𝐈❪f❫ → ∀I,K,i,L. ⇩[i] L ≘ K.ⓤ[I] →
+ L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i] ↑f.
#f #Hf #I #K #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
| #i #IH #Y #H elim (drops_inv_succ … H) -H
#J #L #HLK #H destruct /3 width=1 by frees_lref/
]
qed.
-(*
-lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
-#f #K #s #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
-]
-qed.
-*)
-lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
+
+lemma frees_lref_pushs:
+ ∀f,K,j. K ⊢ 𝐅+❪#j❫ ≘ f →
+ ∀i,L. ⇩[i] L ≘ K → L ⊢ 𝐅+❪#(i+j)❫ ≘ ⫯*[i] f.
#f #K #j #Hf #i elim i -i
[ #L #H lapply (drops_fwd_isid … H ?) -H //
| #i #IH #L #H elim (drops_inv_succ … H) -H
#I #Y #HYK #H destruct /3 width=1 by frees_lref/
]
qed.
-(*
-lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
-#f #K #l #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
-]
-qed.
-*)
+
(* Advanced inversion lemmas ************************************************)
-lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
- ∨∨ ∃∃g. ⬇*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
- | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
- ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
- | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+lemma frees_inv_lref_drops:
+ ∀L,i,f. L ⊢ 𝐅+❪#i❫ ≘ f →
+ ∨∨ ∃∃g. ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆ & 𝐈❪g❫ & f = ⫯*[i] ↑g
+ | ∃∃g,I,K,V. K ⊢ 𝐅+❪V❫ ≘ g & ⇩[i] L ≘ K.ⓑ[I]V & f = ⫯*[i] ↑g
+ | ∃∃g,I,K. ⇩[i] L ≘ K.ⓤ[I] & 𝐈❪g❫ & f = ⫯*[i] ↑g.
#L elim L -L
[ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
[ elim (frees_inv_atom … H) -H #f #Hf #H destruct
(* Properties with generic slicing for local environments *******************)
-lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀f,L. ⬇*[b,f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
- ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+lemma frees_lifts:
+ ∀b,f1,K,T. K ⊢ 𝐅+❪T❫ ≘ f1 →
+ ∀f,L. ⇩*[b,f] L ≘ K → ∀U. ⇧*[f] T ≘ U →
+ ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅+❪U❫ ≘ f2.
#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
]
qed-.
-lemma frees_lifts_SO: ∀b,L,K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
- ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+lemma frees_lifts_SO:
+ ∀b,L,K. ⇩*[b,𝐔❨1❩] L ≘ K → ∀T,U. ⇧[1] T ≘ U →
+ ∀f. K ⊢ 𝐅+❪T❫ ≘ f → L ⊢ 𝐅+❪U❫ ≘ ⫯f.
#b #L #K #HLK #T #U #HTU #f #Hf
@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
qed.
(* Forward lemmas with generic slicing for local environments ***************)
-lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
+lemma frees_fwd_coafter:
+ ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+ ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+ ∀f1. K ⊢ 𝐅+❪T❫ ≘ f1 → f ~⊚ f1 ≘ f2.
/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
-lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts_ex:
+ ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+ ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+ ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅+❪T❫ ≘ f1.
#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
/3 width=9 by frees_fwd_coafter, ex2_intro/
qed-.
-lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
- ∀K. ⬇*[b,𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
- K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
+lemma frees_inv_lifts_SO:
+ ∀b,f,L,U. L ⊢ 𝐅+❪U❫ ≘ f →
+ ∀K. ⇩*[b,𝐔❨1❩] L ≘ K → ∀T. ⇧[1] T ≘ U →
+ K ⊢ 𝐅+❪T❫ ≘ ⫱f.
#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
qed-.
-lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b,f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts:
+ ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+ ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+ ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅+❪T❫ ≘ f1.
#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
/3 width=7 by frees_eq_repl_back, coafter_inj/
qed-.
(* Note: this is used by rex_conf and might be modified *)
-lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
- ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
- ∀g1. ↑g1 = ⫱*[n] f1 →
- ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
+lemma frees_inv_drops_next:
+ ∀f1,L1,T1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
+ ∀I2,L2,V2,i. ⇩[i] L1 ≘ L2.ⓑ[I2]V2 →
+ ∀g1. ↑g1 = ⫱*[i] f1 →
+ ∃∃g2. L2 ⊢ 𝐅+❪V2❫ ≘ g2 & g2 ⊆ g1.
#f1 #L1 #T1 #H elim H -f1 -L1 -T1
-[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
- lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -s
+ lapply (isid_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
elim (isid_inv_next … Hf1) -Hf1 //
-| #f1 #i #_ #I2 #L2 #V2 #n #H
+| #f1 #i #_ #I2 #L2 #V2 #j #H
elim (drops_inv_atom1 … H) -H #H destruct
| #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
[ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
#H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
/2 width=3 by ex2_intro/
- | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ | -Hf1 #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
#HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
/2 width=3 by ex2_intro/
]
| #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
[ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
- | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
+ | #j #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
]
| #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
[ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
- | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ | #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
#HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
/2 width=3 by ex2_intro/
]
-| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
- lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -l
+ lapply (isid_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
elim (isid_inv_next … Hf1) -Hf1 //
-| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
#gV1 #gT1 #Hg1
[ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
| -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
/3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
]
-| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
#gV1 #gT1 #Hg1
[ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1