(* Advanced properties ******************************************************)
lemma frees_atom_drops:
- â\88\80b,L,i. â¬\87*[b,𝐔❴i❵] L ≘ ⋆ →
+ â\88\80b,L,i. â\87©*[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 *
lemma frees_pair_drops:
∀f,K,V. K ⊢ 𝐅+⦃V⦄ ≘ f →
- â\88\80i,I,L. â¬\87*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅+⦃#i⦄ ≘ ⫯*[i] ↑f.
+ â\88\80i,I,L. â\87©*[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:
- â\88\80f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â\88\80I,K,i,L. â¬\87*[i] L ≘ K.ⓤ{I} →
+ â\88\80f. ð\9d\90\88â¦\83fâ¦\84 â\86\92 â\88\80I,K,i,L. â\87©*[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/
#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 →
- â\88\80i,L. â¬\87*[i] L ≘ K → L ⊢ 𝐅+⦃#(i+j)⦄ ≘ ⫯*[i] f.
+ â\88\80i,L. â\87©*[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 →
- â\88¨â\88¨ â\88\83â\88\83g. â¬\87*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
- | â\88\83â\88\83g,I,K,V. K â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 â\89\98 g & â¬\87*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
- | â\88\83â\88\83g,I,K. â¬\87*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+ â\88¨â\88¨ â\88\83â\88\83g. â\87©*[Ⓕ,𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+ | â\88\83â\88\83g,I,K,V. K â\8a¢ ð\9d\90\85+â¦\83Vâ¦\84 â\89\98 g & â\87©*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
+ | â\88\83â\88\83g,I,K. â\87©*[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
lemma frees_lifts:
∀b,f1,K,T. K ⊢ 𝐅+⦃T⦄ ≘ f1 →
- â\88\80f,L. â¬\87*[b,f] L â\89\98 K â\86\92 â\88\80U. â¬\86*[f] T ≘ U →
+ â\88\80f,L. â\87©*[b,f] L â\89\98 K â\86\92 â\88\80U. â\87§*[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
qed-.
lemma frees_lifts_SO:
- â\88\80b,L,K. â¬\87*[b,ð\9d\90\94â\9d´1â\9dµ] L â\89\98 K â\86\92 â\88\80T,U. â¬\86*[1] T ≘ U →
+ â\88\80b,L,K. â\87©*[b,ð\9d\90\94â\9d´1â\9dµ] L â\89\98 K â\86\92 â\88\80T,U. â\87§*[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 *)
lemma frees_fwd_coafter:
∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
- â\88\80f,K. â¬\87*[b,f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T ≘ U →
+ â\88\80f,K. â\87©*[b,f] L â\89\98 K â\86\92 â\88\80T. â\87§*[f] T ≘ U →
∀f1. K ⊢ 𝐅+⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
lemma frees_inv_lifts_ex:
∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
- â\88\80f,K. â¬\87*[b,f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T ≘ U →
+ â\88\80f,K. â\87©*[b,f] L â\89\98 K â\86\92 â\88\80T. â\87§*[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/
lemma frees_inv_lifts_SO:
∀b,f,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f →
- â\88\80K. â¬\87*[b,ð\9d\90\94â\9d´1â\9dµ] L â\89\98 K â\86\92 â\88\80T. â¬\86*[1] T ≘ U →
+ â\88\80K. â\87©*[b,ð\9d\90\94â\9d´1â\9dµ] L â\89\98 K â\86\92 â\88\80T. â\87§*[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
lemma frees_inv_lifts:
∀b,f2,L,U. L ⊢ 𝐅+⦃U⦄ ≘ f2 →
- â\88\80f,K. â¬\87*[b,f] L â\89\98 K â\86\92 â\88\80T. â¬\86*[f] T ≘ U →
+ â\88\80f,K. â\87©*[b,f] L â\89\98 K â\86\92 â\88\80T. â\87§*[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/
(* Note: this is used by rex_conf and might be modified *)
lemma frees_inv_drops_next:
∀f1,L1,T1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 →
- â\88\80I2,L2,V2,n. â¬\87*[n] L1 ≘ L2.ⓑ{I2}V2 →
+ â\88\80I2,L2,V2,n. â\87©*[n] L1 ≘ L2.ⓑ{I2}V2 →
∀g1. ↑g1 = ⫱*[n] f1 →
∃∃g2. L2 ⊢ 𝐅+⦃V2⦄ ≘ g2 & g2 ⊆ g1.
#f1 #L1 #T1 #H elim H -f1 -L1 -T1