(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lazyeq_3.ma".
-include "basic_2/relocation/ldrop.ma".
+include "basic_2/notation/relations/lazyeq_4.ma".
+include "basic_2/multiple/llpx_sn.ma".
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-inductive lleq: term → relation lenv ≝
-| lleq_sort: ∀L1,L2,k. |L1| = |L2| → lleq (⋆k) L1 L2
-| lleq_lref: ∀I,L1,L2,K1,K2,V,i.
- ⇩[0, i] L1 ≡ K1.ⓑ{I}V → ⇩[0, i] L2 ≡ K2.ⓑ{I}V →
- lleq V K1 K2 → lleq (#i) L1 L2
-| lleq_free: ∀L1,L2,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → lleq (#i) L1 L2
-| lleq_gref: ∀L1,L2,p. |L1| = |L2| → lleq (§p) L1 L2
-| lleq_bind: ∀a,I,L1,L2,V,T.
- lleq V L1 L2 → lleq T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- lleq (ⓑ{a,I}V.T) L1 L2
-| lleq_flat: ∀I,L1,L2,V,T.
- lleq V L1 L2 → lleq T L1 L2 → lleq (ⓕ{I}V.T) L1 L2
-.
+definition ceq: relation3 lenv term term ≝ λL,T1,T2. T1 = T2.
+
+definition lleq: relation4 ynat term lenv lenv ≝ llpx_sn ceq.
interpretation
"lazy equivalence (local environment)"
- 'LazyEq T L1 L2 = (lleq T L1 L2).
+ 'LazyEq T l L1 L2 = (lleq l T L1 L2).
+
+definition lleq_transitive: predicate (relation3 lenv term term) ≝
+ λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1, 0] L2 → R L1 T1 T2.
-(* Basic_properties *********************************************************)
+(* Basic inversion lemmas ***************************************************)
-lemma lleq_sym: ∀T. symmetric … (lleq T).
-#T #L1 #L2 #H elim H -T -L1 -L2
-/2 width=7 by lleq_sort, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
+lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
+ ∀L1,L2,l,k. |L1| = |L2| → R l (⋆k) L1 L2
+ ) → (
+ ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → R l (#i) L1 L2
+ ) → (
+ ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
+ ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, yinj O] K2 → R (yinj O) V K1 K2 → R l (#i) L1 L2
+ ) → (
+ ∀L1,L2,l,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R l (#i) L1 L2
+ ) → (
+ ∀L1,L2,l,p. |L1| = |L2| → R l (§p) L1 L2
+ ) → (
+ ∀a,I,L1,L2,V,T,l.
+ L1 ≡[V, l]L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V →
+ R l V L1 L2 → R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → R l (ⓑ{a,I}V.T) L1 L2
+ ) → (
+ ∀I,L1,L2,V,T,l.
+ L1 ≡[V, l]L2 → L1 ≡[T, l] L2 →
+ R l V L1 L2 → R l T L1 L2 → R l (ⓕ{I}V.T) L1 L2
+ ) →
+ ∀l,T,L1,L2. L1 ≡[T, l] L2 → R l T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/
qed-.
-lemma lleq_refl: ∀T. reflexive … (lleq T).
-#T #L @(f2_ind … rfw … L T)
-#n #IH #L * * /3 width=1 by lleq_sort, lleq_gref, lleq_bind, lleq_flat/
-#i #H elim (lt_or_ge i (|L|)) /2 width=1 by lleq_free/
-#HiL elim (ldrop_O1_lt … HiL) -HiL destruct /4 width=7 by lleq_lref, ldrop_fwd_rfw/
-qed.
+lemma lleq_inv_bind: ∀a,I,L1,L2,V,T,l. L1 ≡[ⓑ{a,I}V.T, l] L2 →
+ L1 ≡[V, l] L2 ∧ L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind/ qed-.
-(* Basic inversion lemmas ***************************************************)
+lemma lleq_inv_flat: ∀I,L1,L2,V,T,l. L1 ≡[ⓕ{I}V.T, l] L2 →
+ L1 ≡[V, l] L2 ∧ L1 ≡[T, l] L2.
+/2 width=2 by llpx_sn_inv_flat/ qed-.
-fact lleq_inv_lref_aux: ∀X,L1,L2. L1 ⋕[X] L2 → ∀i. X = #i →
- (|L1| ≤ i ∧ |L2| ≤ i) ∨
- ∃∃I,K1,K2,V. ⇩[0, i] L1 ≡ K1.ⓑ{I}V &
- ⇩[0, i] L2 ≡ K2.ⓑ{I}V &
- K1 ⋕[V] K2.
-#X #L1 #L2 * -X -L1 -L2
-[ #L1 #L2 #k #_ #j #H destruct
-| #I #L1 #L2 #K1 #K2 #V #i #HLK1 #HLK2 #HK12 #j #H destruct /3 width=7 by ex3_4_intro, or_intror/
-| #L1 #L2 #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or_introl, conj/
-| #L1 #L2 #p #_ #j #H destruct
-| #a #I #L1 #L2 #V #T #_ #_ #j #H destruct
-| #I #L1 #L2 #V #T #_ #_ #j #H destruct
-]
-qed-.
+(* Basic forward lemmas *****************************************************)
-lemma lleq_inv_lref: ∀L1,L2,i. L1 ⋕[#i] L2 →
- (|L1| ≤ i ∧ |L2| ≤ i) ∨
- ∃∃I,K1,K2,V. ⇩[0, i] L1 ≡ K1.ⓑ{I}V &
- ⇩[0, i] L2 ≡ K2.ⓑ{I}V &
- K1 ⋕[V] K2.
-/2 width=3 by lleq_inv_lref_aux/ qed-.
-
-fact lleq_inv_bind_aux: ∀X,L1,L2. L1 ⋕[X] L2 → ∀a,I,V,T. X = ⓑ{a,I}V.T →
- L1 ⋕[V] L2 ∧ L1.ⓑ{I}V ⋕[T] L2.ⓑ{I}V.
-#X #L1 #L2 * -X -L1 -L2
-[ #L1 #L2 #k #_ #b #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #p #_ #b #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
-| #I #L1 #L2 #V #T #_ #_ #b #J #W #U #H destruct
-]
+lemma lleq_fwd_length: ∀L1,L2,T,l. L1 ≡[T, l] L2 → |L1| = |L2|.
+/2 width=4 by llpx_sn_fwd_length/ qed-.
+
+lemma lleq_fwd_lref: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
+ ∨∨ |L1| ≤ i ∧ |L2| ≤ i
+ | yinj i < l
+ | ∃∃I,K1,K2,V. ⬇[i] L1 ≡ K1.ⓑ{I}V &
+ ⬇[i] L2 ≡ K2.ⓑ{I}V &
+ K1 ≡[V, yinj 0] K2 & l ≤ yinj i.
+#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref … H) /2 width=1 by or3_intro0, or3_intro1/
+* /3 width=7 by or3_intro2, ex4_4_intro/
qed-.
-lemma lleq_inv_bind: ∀a,I,L1,L2,V,T. L1 ⋕[ ⓑ{a,I}V.T] L2 →
- L1 ⋕[V] L2 ∧ L1.ⓑ{I}V ⋕[T] L2.ⓑ{I}V.
-/2 width=4 by lleq_inv_bind_aux/ qed-.
-
-fact lleq_inv_flat_aux: ∀X,L1,L2. L1 ⋕[X] L2 → ∀I,V,T. X = ⓕ{I}V.T →
- L1 ⋕[V] L2 ∧ L1 ⋕[T] L2.
-#X #L1 #L2 * -X -L1 -L2
-[ #L1 #L2 #k #_ #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #p #_ #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #V #T #HV #HT #J #W #U #H destruct /2 width=1 by conj/
-]
-qed-.
+lemma lleq_fwd_drop_sn: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K1,i. ⬇[i] L1 ≡ K1 →
+ ∃K2. ⬇[i] L2 ≡ K2.
+/2 width=7 by llpx_sn_fwd_drop_sn/ qed-.
-lemma lleq_inv_flat: ∀I,L1,L2,V,T. L1 ⋕[ ⓕ{I}V.T] L2 →
- L1 ⋕[V] L2 ∧ L1 ⋕[T] L2.
-/2 width=4 by lleq_inv_flat_aux/ qed-.
+lemma lleq_fwd_drop_dx: ∀L1,L2,T,l. L1 ≡[l, T] L2 → ∀K2,i. ⬇[i] L2 ≡ K2 →
+ ∃K1. ⬇[i] L1 ≡ K1.
+/2 width=7 by llpx_sn_fwd_drop_dx/ qed-.
-(* Basic forward lemmas *****************************************************)
+lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,l.
+ L1 ≡[ⓑ{a,I}V.T, l] L2 → L1 ≡[V, l] L2.
+/2 width=4 by llpx_sn_fwd_bind_sn/ qed-.
-lemma lleq_fwd_length: ∀L1,L2,T. L1 ⋕[T] L2 → |L1| = |L2|.
-#L1 #L2 #T #H elim H -L1 -L2 -T //
-#I #L1 #L2 #K1 #K2 #V #i #HLK1 #HLK2 #_ #IHK12
-lapply (ldrop_fwd_length … HLK1) -HLK1
-lapply (ldrop_fwd_length … HLK2) -HLK2
-normalize //
-qed-.
+lemma lleq_fwd_bind_dx: ∀a,I,L1,L2,V,T,l.
+ L1 ≡[ⓑ{a,I}V.T, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_dx/ qed-.
+
+lemma lleq_fwd_flat_sn: ∀I,L1,L2,V,T,l.
+ L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[V, l] L2.
+/2 width=3 by llpx_sn_fwd_flat_sn/ qed-.
+
+lemma lleq_fwd_flat_dx: ∀I,L1,L2,V,T,l.
+ L1 ≡[ⓕ{I}V.T, l] L2 → L1 ≡[T, l] L2.
+/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lleq_sort: ∀L1,L2,l,k. |L1| = |L2| → L1 ≡[⋆k, l] L2.
+/2 width=1 by llpx_sn_sort/ qed.
-lemma lleq_fwd_ldrop_sn: ∀L1,L2,T. L1 ⋕[T] L2 → ∀K1,i. ⇩[0, i] L1 ≡ K1 →
- ∃K2. ⇩[0, i] L2 ≡ K2.
-#L1 #L2 #T #H #K1 #i #HLK1 lapply (lleq_fwd_length … H) -H
-#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/ (**) (* full auto fails *)
+lemma lleq_skip: ∀L1,L2,l,i. yinj i < l → |L1| = |L2| → L1 ≡[#i, l] L2.
+/2 width=1 by llpx_sn_skip/ qed.
+
+lemma lleq_lref: ∀I,L1,L2,K1,K2,V,l,i. l ≤ yinj i →
+ ⬇[i] L1 ≡ K1.ⓑ{I}V → ⬇[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ≡[V, 0] K2 → L1 ≡[#i, l] L2.
+/2 width=9 by llpx_sn_lref/ qed.
+
+lemma lleq_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ≡[#i, l] L2.
+/2 width=1 by llpx_sn_free/ qed.
+
+lemma lleq_gref: ∀L1,L2,l,p. |L1| = |L2| → L1 ≡[§p, l] L2.
+/2 width=1 by llpx_sn_gref/ qed.
+
+lemma lleq_bind: ∀a,I,L1,L2,V,T,l.
+ L1 ≡[V, l] L2 → L1.ⓑ{I}V ≡[T, ⫯l] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, l] L2.
+/2 width=1 by llpx_sn_bind/ qed.
+
+lemma lleq_flat: ∀I,L1,L2,V,T,l.
+ L1 ≡[V, l] L2 → L1 ≡[T, l] L2 → L1 ≡[ⓕ{I}V.T, l] L2.
+/2 width=1 by llpx_sn_flat/ qed.
+
+lemma lleq_refl: ∀l,T. reflexive … (lleq l T).
+/2 width=1 by llpx_sn_refl/ qed.
+
+lemma lleq_Y: ∀L1,L2,T. |L1| = |L2| → L1 ≡[T, ∞] L2.
+/2 width=1 by llpx_sn_Y/ qed.
+
+lemma lleq_sym: ∀l,T. symmetric … (lleq l T).
+#l #T #L1 #L2 #H @(lleq_ind … H) -l -T -L1 -L2
+/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
qed-.
+
+lemma lleq_ge_up: ∀L1,L2,U,lt. L1 ≡[U, lt] L2 →
+ ∀T,l,m. ⬆[l, m] T ≡ U →
+ lt ≤ l + m → L1 ≡[U, l] L2.
+/2 width=6 by llpx_sn_ge_up/ qed-.
+
+lemma lleq_ge: ∀L1,L2,T,l1. L1 ≡[T, l1] L2 → ∀l2. l1 ≤ l2 → L1 ≡[T, l2] L2.
+/2 width=3 by llpx_sn_ge/ qed-.
+
+lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[V, 0] L2 → L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V →
+ L1 ≡[ⓑ{a,I}V.T, 0] L2.
+/2 width=1 by llpx_sn_bind_O/ qed-.
+
+(* Advanceded properties on lazy pointwise extensions ************************)
+
+lemma llpx_sn_lrefl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T,l. L1 ≡[T, l] L2 → llpx_sn R l T L1 L2.
+/2 width=3 by llpx_sn_co/ qed-.
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