-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/freestar_4.ma".
-include "basic_2/substitution/lift_neg.ma".
-include "basic_2/substitution/drop.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-inductive frees: relation4 ynat lenv term ynat ≝
-| 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 → yinj j < i →
- (∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
- frees 0 K W (⫰(i-j)) → frees l L U i.
-
-interpretation
- "context-sensitive free variables (term)"
- '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,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
- (∀T. ⬆[i, 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) ϵ 𝐅*[yinj 0]⦃W⦄.
-#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,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,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,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
- yinj j = i ∨
- ∃∃I,K,W. l ≤ yinj j & yinj j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) 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 #Hlj #Hji #Hnx #HLK #HW
- lapply (nlift_inv_lref_be_SO … Hnx) -Hnx #H
- lapply (yinj_inj … H) -H #H destruct
- /3 width=5 by ex4_3_intro, or_intror/
-]
-qed-.
-
-lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → yinj 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,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → yinj 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,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → yinj j = i.
-#L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
-* #I #K #W #_ #Hji elim (ylt_yle_false … Hji Hij)
-qed-.
-
-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) ϵ 𝐅*[yinj 0]⦃W⦄.
-#L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
-[ #H elim (ylt_yle_false … Hji) //
-| * /2 width=5 by ex3_3_intro/
-]
-qed-.
-
-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 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 #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: lapply (yle_succ … Hlj) // (**)
- |5: lapply (ylt_succ … Hji) // (**)
- |6: /2 width=4 by drop_drop/
- |7: <yminus_succ in HW; // (**)
- |*: skip
- ]
- ]
-]
-qed-.
-
-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 #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,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
-/4 width=7 by frees_eq, lift_inv_lref2_be, ylt_inj/ qed.
-
-lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
- K ⊢ ⫰(i-j) ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
-/4 width=9 by frees_be, lift_inv_lref2_be, ylt_inj/ qed.
-
-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,l,i. L.ⓑ{I}W ⊢ ⫯i 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 #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj
- #Hj <Hj >yminus_succ
- lapply (ylt_O … Hj) -Hj #Hj #H
- lapply (ylt_inv_succ … H) -H #Hji #HnU #HLK #HW
- @(frees_be … Hlj Hji … HW) -HW -Hlj -Hji (**) (* explicit constructor *)
- [2: #X #H lapply (nlift_bind_dx … H) /2 width=2 by/ (**)
- |3: lapply (drop_inv_drop1_lt … HLK ?) -HLK //
- |*: skip
-]
-qed.
-
-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,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,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-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
- L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[0]⦃U⦄ .
-#a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
-/3 width=3 by frees_weak, or_intror, or_introl/
-qed-.
+(* A Basic_A2 lemma we do not need so far *)
+axiom frees_pair_flat: ∀L,T,f1,I1,V1. L.ⓑ{I1}V1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
+ ∀f2,I2,V2. L.ⓑ{I2}V2 ⊢ 𝐅*⦃T⦄ ≡ f2 →
+ ∀f0. f1 ⋓ f2 ≡ f0 →
+ ∀I0,I. L.ⓑ{I0}ⓕ{I}V1.V2 ⊢ 𝐅*⦃T⦄ ≡ f0.
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