(* Advanced properties ******************************************************)
(* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
-lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
+lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅+⦃T⦄ ≘ f.
#L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
#G0 #L0 #T0 #IH #G #L * *
[ /3 width=2 by frees_sort, ex_intro/
(* Advanced main properties *************************************************)
-theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
+theorem frees_bind_void:
+ ∀f1,L,V. L ⊢ 𝐅+⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅+⦃T⦄ ≘ f2 →
+ ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f.
#f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
(* Advanced inversion lemmas ************************************************)
-lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+lemma frees_inv_bind_void:
+ ∀f,p,I,L,V,T. L ⊢ 𝐅+⦃ⓑ{p,I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅+⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅+⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
#f #p #I #L #V #T #H
elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
elim (frees_total (L.ⓧ) T) #f0 #Hf0
]
qed-.
-lemma frees_ind_void: ∀Q:relation3 ….
- (
- ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
- ) → (
- ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
- ) → (
- ∀f,I,L,V.
- L ⊢ 𝐅*⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
- ) → (
- ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
- ) → (
- ∀f,I,L,i.
- L ⊢ 𝐅*⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
- ) → (
- ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
- ) → (
- ∀f1,f2,f,p,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
- Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
- ) → (
- ∀f1,f2,f,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
- Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
- ) →
- ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f → Q L T f.
+lemma frees_ind_void (Q:relation3 …):
+ (
+ ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
+ ) → (
+ ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
+ ) → (
+ ∀f,I,L,V.
+ L ⊢ 𝐅+⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
+ ) → (
+ ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
+ ) → (
+ ∀f,I,L,i.
+ L ⊢ 𝐅+⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
+ ) → (
+ ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
+ ) → (
+ ∀f1,f2,f,p,I,L,V,T.
+ L ⊢ 𝐅+⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅+⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
+ Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
+ ) → (
+ ∀f1,f2,f,I,L,V,T.
+ L ⊢ 𝐅+⦃V⦄ ≘ f1 → L ⊢𝐅+⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
+ Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
+ ) →
+ ∀L,T,f. L ⊢ 𝐅+⦃T⦄ ≘ f → Q L T f.
#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
@(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
[ #s #HG #HL #HT #f #H destruct -IH