+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_2/substitution/drop_lreq.ma".
-include "basic_2/multiple/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma lreq_frees_trans: ∀L2,U,l,i. L2 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
- ∀L1. L1 ⩬[l, ∞] L2 → L1 ⊢ i ϵ 𝐅*[l]⦃U⦄.
-#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
-#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
-elim (lreq_drop_trans_be … HL12 … HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
-lapply (lreq_inv_O_Y … HK12) -HK12 #H destruct /3 width=9 by frees_be/
-qed-.
-
-lemma frees_lreq_conf: ∀L1,U,l,i. L1 ⊢ i ϵ 𝐅*[l]⦃U⦄ →
- ∀L2. L1 ⩬[l, ∞] L2 → L2 ⊢ i ϵ 𝐅*[l]⦃U⦄.
-/3 width=3 by lreq_sym, lreq_frees_trans/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_2/notation/relations/lazyor_5.ma".
-include "basic_2/multiple/frees.ma".
-
-(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-
-definition llor: ynat → relation4 term lenv lenv lenv ≝ λl,T,L2,L1,L.
- ∧∧ |L1| = |L2| & |L1| = |L|
- & (∀I1,I2,I,K1,K2,K,V1,V2,V,i.
- ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 → ⬇[i] L ≡ K.ⓑ{I}V → ∨∨
- (∧∧ yinj i < l & I1 = I & V1 = V) |
- (∧∧ (L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ → ⊥) & I1 = I & V1 = V) |
- (∧∧ l ≤ yinj i & L1 ⊢ i ϵ 𝐅*[l]⦃T⦄ & I2 = I & V2 = V)
- ).
-
-interpretation
- "lazy union (local environment)"
- 'LazyOr L1 T l L2 L = (llor l T L2 L1 L).
-
-(* Basic properties *********************************************************)
-
-(* Note: this can be proved by llor_skip *)
-lemma llor_atom: ∀T,l. ⋆ ⋓[T, l] ⋆ ≡ ⋆.
-#T #l @and3_intro //
-#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
-elim (drop_inv_atom1 … HLK1) -HLK1 #H destruct
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_2/multiple/frees_append.ma".
-include "basic_2/multiple/llor.ma".
-
-(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Alternative definition ***************************************************)
-
-lemma llor_tail_frees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → l ≤ yinj (|L1|) →
- ∀I1,W1. ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ →
- ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I2}W2.L.
-#L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2
-@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
-#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
-lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
-lapply (lt_plus_SO_to_le … H) -H #H
-elim (le_to_or_lt_eq … H) -H #H
-[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
- elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
- elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
- lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
- elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
- [ /3 width=1 by and3_intro, or3_intro0/
- | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/
- | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/
- ]
-| -IH -HLK1 destruct
- lapply (drop_O1_inv_append1_le … HLK2 … (⋆) ?) // -HLK2 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
- /3 width=1 by or3_intro2, and4_intro/
-]
-qed.
-
-lemma llor_tail_cofrees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L →
- ∀I1,W1. (ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → ⊥) →
- ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I1}W1.L.
-#L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2
-@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
-#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
-lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
-lapply (lt_plus_SO_to_le … H) -H #H
-elim (le_to_or_lt_eq … H) -H #H
-[ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
- elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
- elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
- lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
- elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
- [ /3 width=1 by and3_intro, or3_intro0/
- | /7 width=2 by frees_inv_append, yle_inj, lt_to_le, or3_intro1, and3_intro/
- | /6 width=1 by frees_append, yle_inj, lt_to_le, or3_intro2, and4_intro/
- ]
-| -IH -HLK2 destruct
- lapply (drop_O1_inv_append1_le … HLK1 … (⋆) ?) // -HLK1 normalize #H destruct
- lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
- /4 width=1 by or3_intro1, and3_intro/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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 "basic_2/multiple/frees_lift.ma".
-include "basic_2/multiple/llor_alt.ma".
-
-(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma llor_skip: ∀L1,L2,U,l. |L1| = |L2| → yinj (|L1|) ≤ l → L1 ⋓[U, l] L2 ≡ L1.
-#L1 #L2 #U #l #HL12 #Hl @and3_intro // -HL12
-#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2
-lapply (drop_mono … HLK … HLK1) -HLK #H destruct
-lapply (drop_fwd_length_lt2 … HLK1) -K1
-/5 width=3 by ylt_yle_trans, ylt_inj, or3_intro0, and3_intro/
-qed.
-
-(* Note: lemma 1400 concludes the "big tree" theorem *)
-lemma llor_total: ∀L1,L2,T,l. |L1| = |L2| → ∃L. L1 ⋓[T, l] L2 ≡ L.
-#L1 @(lenv_ind_alt … L1) -L1
-[ #L2 #T #l #H >(length_inv_zero_sn … H) -L2 /2 width=2 by ex_intro/
-| #I1 #L1 #V1 #IHL1 #Y #T #l >ltail_length #H
- elim (length_inv_pos_sn_ltail … H) -H #I2 #L2 #V2 #HL12 #H destruct
- elim (ylt_split l (|ⓑ{I1}V1.L1|))
- [ elim (frees_dec (ⓑ{I1}V1.L1) T l (|L1|)) #HnU
- elim (IHL1 L2 T l) // -IHL1 -HL12
- [ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/
- | /4 width=2 by llor_tail_cofrees, ex_intro/
- ]
- | -IHL1 /4 width=3 by llor_skip, plus_to_minus, ex_intro/
- ]
-]
-qed-.
#L elim L -L
/4 width=3 by frees_eq_repl_back, frees_gref, frees_atom, eq_push_inv_isid/
qed.
+
+(* Basic_2A1: removed theorems 27:
+ frees_eq frees_be frees_inv
+ frees_inv_sort frees_inv_gref frees_inv_lref frees_inv_lref_free
+ frees_inv_lref_skip frees_inv_lref_ge frees_inv_lref_lt
+ frees_inv_bind frees_inv_flat frees_inv_bind_O
+ frees_lref_eq frees_lref_be frees_weak
+ frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
+ lreq_frees_trans frees_lreq_conf
+ llor_atom llor_skip llor_total
+ llor_tail_frees llor_tail_cofrees
+*)
(* Advanced properties ******************************************************)
+(* Notes: this replaces lemma 1400 concluding the "big tree" theorem *)
lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≡ f.
@(f2_ind … rfw) #n #IH #L *
[ cases L -L /3 width=2 by frees_atom, ex_intro/
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