(* Advanced properties ******************************************************)
-lemma frees_dec: ∀L,U,d,i. Decidable (frees d L U i).
+lemma frees_dec: ∀L,U,l,i. Decidable (frees l L U i).
#L #U @(f2_ind … rfw … L U) -L -U
#n #IH #L * *
[ -IH /3 width=5 by frees_inv_sort, or_intror/
-| #j #Hn #d #i elim (lt_or_eq_or_gt i j) #Hji
+| #j #Hn #l #i elim (lt_or_eq_or_gt i j) #Hji
[ -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_ge … H ?) -L -d /2 width=1 by lt_to_le/
+ lapply (frees_inv_lref_ge … H ?) -L -l /2 width=1 by lt_to_le/
| -n /2 width=1 by or_introl/
- | elim (ylt_split j d) #Hdi
+ | elim (ylt_split j l) #Hli
[ -n @or_intror #H elim (lt_refl_false i)
lapply (frees_inv_lref_skip … H ?) -L //
| elim (lt_or_ge j (|L|)) #Hj
[ elim (drop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
- @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -d
+ @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -l
lapply (drop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
| -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_free … H ?) -d //
+ lapply (frees_inv_lref_free … H ?) -l //
]
]
]
| -IH /3 width=5 by frees_inv_gref, or_intror/
-| #a #I #W #U #Hn #d #i destruct
- elim (IH L W … d i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
- elim (IH (L.ⓑ{I}W) U … (⫯d) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+| #a #I #W #U #Hn #l #i destruct
+ elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
+ elim (IH (L.ⓑ{I}W) U … (⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
@or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
-| #I #W #U #Hn #d #i destruct
- elim (IH L W … d i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
- elim (IH L U … d i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
+| #I #W #U #Hn #l #i destruct
+ elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
+ elim (IH L U … l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
@or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
]
qed-.
-lemma frees_S: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[yinj d]⦃U⦄ → ∀I,K,W. ⬇[d] L ≡ K.ⓑ{I}W →
- (K ⊢ i-d-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯d]⦃U⦄.
-#L #U #d #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
-* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
-lapply (yle_inv_inj … Hdj) -Hdj #Hdj
-elim (le_to_or_lt_eq … Hdj) -Hdj
+lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
+ (K ⊢ i-l-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[⫯l]⦃U⦄.
+#L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
+* #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
+lapply (yle_inv_inj … Hlj) -Hlj #Hlj
+elim (le_to_or_lt_eq … Hlj) -Hlj
[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
| -Hji -HnU #H destruct
lapply (drop_mono … HLK0 … HLK) #H destruct -I
(* Properties on relocation *************************************************)
-lemma frees_lift_ge: ∀K,T,d,i. K ⊢ i ϵ𝐅*[d]⦃T⦄ →
- ∀L,s,d0,e0. ⬇[s, d0, e0] L ≡ K →
- ∀U. ⬆[d0, e0] T ≡ U → d0 ≤ i →
- L ⊢ i+e0 ϵ 𝐅*[d]⦃U⦄.
-#K #T #d #i #H elim H -K -T -d -i
-[ #K #T #d #i #HnT #L #s #d0 #e0 #_ #U #HTU #Hd0i -K -s
+lemma frees_lift_ge: ∀K,T,l,i. K ⊢ i ϵ𝐅*[l]⦃T⦄ →
+ ∀L,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
+ ∀U. ⬆[l0, m0] T ≡ U → l0 ≤ i →
+ L ⊢ i+m0 ϵ 𝐅*[l]⦃U⦄.
+#K #T #l #i #H elim H -K -T -l -i
+[ #K #T #l #i #HnT #L #s #l0 #m0 #_ #U #HTU #Hl0i -K -s
@frees_eq #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
-| #I #K #K0 #T #V #d #i #j #Hdj #Hji #HnT #HK0 #HV #IHV #L #s #d0 #e0 #HLK #U #HTU #Hd0i
- elim (lt_or_ge j d0) #H1
+| #I #K #K0 #T #V #l #i #j #Hlj #Hji #HnT #HK0 #HV #IHV #L #s #l0 #m0 #HLK #U #HTU #Hl0i
+ elim (lt_or_ge j l0) #H1
[ elim (drop_trans_lt … HLK … HK0) // -K #L0 #W #HL0 #HLK0 #HVW
@(frees_be … HL0) -HL0 -HV
/3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
- [ #X #HXU >(plus_minus_m_m d0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
+ [ #X #HXU >(plus_minus_m_m l0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
elim (lift_div_le … HXU … HTU ?) -U /2 width=2 by monotonic_pred/
| >minus_plus <plus_minus // <minus_plus
/3 width=5 by monotonic_le_minus_l2/
(* Inversion lemmas on relocation *******************************************)
-lemma frees_inv_lift_be: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- ∀K,s,d0,e0. ⬇[s, d0, e0+1] L ≡ K →
- ∀T. ⬆[d0, e0+1] T ≡ U → d0 ≤ i → i ≤ d0 + e0 → ⊥.
-#L #U #d #i #H elim H -L -U -d -i
-[ #L #U #d #i #HnU #K #s #d0 #e0 #_ #T #HTU #Hd0i #Hide0
- elim (lift_split … HTU i e0) -HTU /2 width=2 by/
-| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hd0i #Hide0
- elim (lt_or_ge j d0) #H1
+lemma frees_inv_lift_be: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+ ∀K,s,l0,m0. ⬇[s, l0, m0+1] L ≡ K →
+ ∀T. ⬆[l0, m0+1] T ≡ U → l0 ≤ i → i ≤ l0 + m0 → ⊥.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #_ #T #HTU #Hl0i #Hilm0
+ elim (lift_split … HTU i m0) -HTU /2 width=2 by/
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hl0i #Hilm0
+ elim (lt_or_ge j l0) #H1
[ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
@(IHW … HKL0 … HVW)
[ /2 width=1 by monotonic_le_minus_l2/
| >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
]
- | elim (lift_split … HTU j e0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
+ | elim (lift_split … HTU j m0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
]
]
qed-.
-lemma frees_inv_lift_ge: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- ∀K,s,d0,e0. ⬇[s, d0, e0] L ≡ K →
- ∀T. ⬆[d0, e0] T ≡ U → d0 + e0 ≤ i →
- K ⊢ i-e0 ϵ𝐅*[d-yinj e0]⦃T⦄.
-#L #U #d #i #H elim H -L -U -d -i
-[ #L #U #d #i #HnU #K #s #d0 #e0 #HLK #T #HTU #Hde0i -L -s
- elim (le_inv_plus_l … Hde0i) -Hde0i #Hd0ie0 #He0i @frees_eq #X #HXT -K
+lemma frees_inv_lift_ge: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+ ∀K,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
+ ∀T. ⬆[l0, m0] T ≡ U → l0 + m0 ≤ i →
+ K ⊢ i-m0 ϵ𝐅*[l-yinj m0]⦃T⦄.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
+ elim (le_inv_plus_l … Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
-| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hde0i
- elim (lt_or_ge j d0) #H1
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hlm0i
+ elim (lt_or_ge j l0) #H1
[ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
- elim (le_inv_plus_l … Hde0i) #H0 #He0i
+ elim (le_inv_plus_l … Hlm0i) #H0 #Hm0i
@(frees_be … H) -H
[ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
| /2 width=3 by lt_to_le_to_lt/
| lapply (IHW … HKL0 … HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
>minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
]
- | elim (lt_or_ge j (d0+e0)) [ >commutative_plus |] #H2
+ | elim (lt_or_ge j (l0+m0)) [ >commutative_plus |] #H2
[ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
- elim (lift_split … HTU j (e0-1)) -HTU //
+ elim (lift_split … HTU j (m0-1)) -HTU //
[ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
#X #_ #H elim (HnU … H)
| >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
]
| lapply (drop_conf_ge … HLK … HLK0 ?) // -L #HK0
- elim (le_inv_plus_l … H2) -H2 #H2 #He0j
+ elim (le_inv_plus_l … H2) -H2 #H2 #Hm0j
@(frees_be … HK0)
[ /2 width=1 by monotonic_yle_minus_dx/
| /2 width=1 by monotonic_lt_minus_l/
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