(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
inductive frees: relation4 ynat lenv term nat ≝
-| frees_eq: ∀L,U,d,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees d L U i
-| frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
+| frees_eq: ∀L,U,l,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees l L U i
+| frees_be: ∀I,L,K,U,W,l,i,j. l ≤ yinj j → j < i →
(∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
- frees 0 K W (i-j-1) → frees d L U i.
+ frees 0 K W (i-j-1) → frees l L U i.
interpretation
"context-sensitive free variables (term)"
- 'FreeStar L i d U = (frees d L U i).
+ 'FreeStar L i l U = (frees l L U i).
definition frees_trans: predicate (relation3 lenv term term) ≝
λR. ∀L,U1,U2,i. R L U1 U2 → L ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L ⊢ i ϵ 𝐅*[0]⦃U1⦄.
(* Basic inversion lemmas ***************************************************)
-lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
+lemma frees_inv: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
(∀T. ⬆[i, 1] T ≡ U → ⊥) ∨
- ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
+ ∃∃I,K,W,j. l ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
⬇[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+#L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
qed-.
-lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
-#L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+lemma frees_inv_sort: ∀L,l,i,k. L ⊢ i ϵ 𝐅*[l]⦃⋆k⦄ → ⊥.
+#L #l #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
qed-.
-lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
-#L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
+lemma frees_inv_gref: ∀L,l,i,p. L ⊢ i ϵ 𝐅*[l]⦃§p⦄ → ⊥.
+#L #l #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
qed-.
-lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
+lemma frees_inv_lref: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
j = i ∨
- ∃∃I,K,W. d ≤ yinj j & j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #x #i #H elim (frees_inv … H) -H
+ ∃∃I,K,W. l ≤ yinj j & j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #x #i #H elim (frees_inv … H) -H
[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
-| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
+| * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
>(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
]
qed-.
-lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
-#L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
+lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → j = i.
+#L #l #j #i #H #Hj elim (frees_inv_lref … H) -H //
* #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 … HLK) -I
#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
qed-.
-lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
-#L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
-* #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
+lemma frees_inv_lref_skip: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → j = i.
+#L #l #j #i #H #Hjl elim (frees_inv_lref … H) -H //
+* #I #K #W #Hlj elim (ylt_yle_false … Hlj) -Hlj //
qed-.
-lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
-#L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
-* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
+lemma frees_inv_lref_ge: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → j = i.
+#L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -l /2 width=3 by lt_to_le_to_lt/
qed-.
-lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
- ∃∃I,K,W. d ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
+lemma frees_inv_lref_lt: ∀L,l,j,i.L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → j < i →
+ ∃∃I,K,W. l ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
+#L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
[ #H elim (lt_refl_false j) //
| * /2 width=5 by ex3_3_intro/
]
qed-.
-lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ .
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
[ #HnX elim (nlift_inv_bind … HnX) -HnX
/4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
[ /4 width=9 by frees_be, or_introl/
| #HnT @or_intror @(frees_be … HnT) -HnT
[4,5,6: /2 width=1 by drop_drop, yle_succ, lt_minus_to_plus/
]
qed-.
-lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
-#J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_inv_flat: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[l]⦃U⦄ .
+#J #L #V #U #l #i #H elim (frees_inv … H) -H
[ #HnX elim (nlift_inv_flat … HnX) -HnX
/4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
/4 width=9 by frees_be, or_intror, or_introl/
]
qed-.
(* Basic properties *********************************************************)
-lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
+lemma frees_lref_eq: ∀L,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
-lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
- K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
+lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
+ K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
-lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_bind_sn: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
qed.
-lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
-#a #J #L #V #U #d #i #H elim (frees_inv … H) -H
+lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯l]⦃U⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
+#a #J #L #V #U #l #i #H elim (frees_inv … H) -H
[ /4 width=9 by frees_eq, nlift_bind_dx/
-| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
- elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
+| * #I #K #W #j #Hlj #Hji #HnU #HLK #HW
+ elim (yle_inv_succ1 … Hlj) -Hlj <yminus_SO2 #Hyj #H
lapply (ylt_O … H) -H #Hj
>(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
/4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, lt_plus_to_minus/
]
qed.
-lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_flat_sn: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
qed.
-lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
- L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
-#I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
+lemma frees_flat_dx: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
+ L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
+#I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
qed.
-lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
- ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
-#L #U #d1 #i #H elim H -L -U -d1 -i
+lemma frees_weak: ∀L,U,l1,i. L ⊢ i ϵ 𝐅*[l1]⦃U⦄ →
+ ∀l2. l2 ≤ l1 → L ⊢ i ϵ 𝐅*[l2]⦃U⦄.
+#L #U #l1 #i #H elim H -L -U -l1 -i
/3 width=9 by frees_be, frees_eq, yle_trans/
qed-.
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