+++ /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/lleq_drop.ma".
-include "basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma lleq_aaa_trans: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
- ∀L1. L1 ≡[T, 0] L2 → ⦃G, L1⦄ ⊢ T ⁝ A.
-#G #L2 #T #A #H elim H -G -L2 -T -A /2 width=1 by aaa_sort/
-[ #I #G #L2 #K2 #V2 #A #i #HLK2 #_ #IHV2 #L1 #H elim (lleq_fwd_lref_dx … H … HLK2) -L2
- [ #H elim (ylt_yle_false … H) //
- | * /3 width=5 by aaa_lref/
- ]
-| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
- /3 width=2 by aaa_abbr/
-| #a #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_bind_O … H) -H
- /3 width=1 by aaa_abst/
-| #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=3 by aaa_appl/
-| #G #L2 #V #T #A #_ #_ #IHV #IHT #L1 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by aaa_cast/
-]
-qed-.
-
-lemma aaa_lleq_conf: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
- ∀L1. L2 ≡[T, 0] L1 → ⦃G, L1⦄ ⊢ T ⁝ A.
-/3 width=3 by lleq_aaa_trans, lleq_sym/ qed-.
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-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 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 inversion lemmas ***************************************************)
lemma lleq_ind: ∀R:relation4 ynat term lenv lenv. (
#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_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-.
-
-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-.
-
(* Basic forward lemmas *****************************************************)
-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
∃K1. ⬇[i] L1 ≡ K1.
/2 width=7 by llpx_sn_fwd_drop_dx/ qed-.
-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_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,s. |L1| = |L2| → L1 ≡[⋆s, l] L2.
-/2 width=1 by llpx_sn_sort/ qed.
-
-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,k. ⬆[l, k] T ≡ U →
- lt ≤ l + k → 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-.
+++ /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/lleq_drop.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Main properties **********************************************************)
-
-theorem lleq_trans: ∀l,T. Transitive … (lleq l T).
-/2 width=3 by lleq_llpx_sn_trans/ qed-.
-
-theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-(* Advanced properies on negated lazy equivalence *****************************)
-
-(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *)
-lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L →
- ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥).
-/3 width=3 by lleq_canc_sn/ qed-.
-
-lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L →
- ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥).
-/3 width=3 by lleq_trans/ qed-.
(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
-| llpx_sn_sort: ∀L1,L2,l,s. |L1| = |L2| → llpx_sn R l (⋆s) L1 L2
-| llpx_sn_skip: ∀L1,L2,l,i. |L1| = |L2| → yinj i < l → llpx_sn R l (#i) L1 L2
| llpx_sn_lref: ∀I,L1,L2,K1,K2,V1,V2,l,i. l ≤ yinj i →
⬇[i] L1 ≡ K1.ⓑ{I}V1 → ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
llpx_sn R (yinj 0) V1 K1 K2 → R K1 V1 V2 → llpx_sn R l (#i) L1 L2
-| llpx_sn_free: ∀L1,L2,l,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → llpx_sn R l (#i) L1 L2
-| llpx_sn_gref: ∀L1,L2,l,p. |L1| = |L2| → llpx_sn R l (§p) L1 L2
-| llpx_sn_bind: ∀a,I,L1,L2,V,T,l.
- llpx_sn R l V L1 L2 → llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R l (ⓑ{a,I}V.T) L1 L2
-| llpx_sn_flat: ∀I,L1,L2,V,T,l.
- llpx_sn R l V L1 L2 → llpx_sn R l T L1 L2 → llpx_sn R l (ⓕ{I}V.T) L1 L2
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact llpx_sn_inv_bind_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
- ∀a,I,V,T. X = ⓑ{a,I}V.T →
- llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #l * -L1 -L2 -X -l
-[ #L1 #L2 #l #s #_ #b #J #W #U #H destruct
-| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #l #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #l #p #_ #b #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #l #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
-| #I #L1 #L2 #V #T #l #_ #_ #b #J #W #U #H destruct
-]
-qed-.
-
-lemma llpx_sn_inv_bind: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R l V L1 L2 ∧ llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
-
-fact llpx_sn_inv_flat_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 →
- ∀I,V,T. X = ⓕ{I}V.T →
- llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2.
-#R #L1 #L2 #X #l * -L1 -L2 -X -l
-[ #L1 #L2 #l #s #_ #J #W #U #H destruct
-| #L1 #L2 #l #i #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #l #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #l #p #_ #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #l #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #V #T #l #HV #HT #J #W #U #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma llpx_sn_inv_flat: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
- llpx_sn R l V L1 L2 ∧ llpx_sn R l T L1 L2.
-/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma llpx_sn_fwd_length: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l //
-#I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #HLK1 #HLK2 #_ #_ #HK12
-lapply (drop_fwd_length … HLK1) -HLK1
-lapply (drop_fwd_length … HLK2) -HLK2
-normalize //
-qed-.
-
lemma llpx_sn_fwd_drop_sn: ∀R,L1,L2,T,l. llpx_sn R l T L1 L2 →
∀K1,i. ⬇[i] L1 ≡ K1 → ∃K2. ⬇[i] L2 ≡ K2.
#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length … H) -H
#HL12 lapply (drop_fwd_length_le2 … HLK2) -HLK2 /2 width=1 by drop_O1_le/
qed-.
-fact llpx_sn_fwd_lref_aux: ∀R,L1,L2,X,l. llpx_sn R l X L1 L2 → ∀i. X = #i →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < l
- | ∃∃I,K1,K2,V1,V2. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
- ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & l ≤ yinj i.
-#R #L1 #L2 #X #l * -L1 -L2 -X -l
-[ #L1 #L2 #l #s #_ #j #H destruct
-| #L1 #L2 #l #i #_ #Hil #j #H destruct /2 width=1 by or3_intro1/
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
- /3 width=9 by or3_intro2, ex5_5_intro/
-| #L1 #L2 #l #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
-| #L1 #L2 #l #p #_ #j #H destruct
-| #a #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
-| #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
-]
-qed-.
-
lemma llpx_sn_fwd_lref: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
∨∨ |L1| ≤ i ∧ |L2| ≤ i
| yinj i < l
llpx_sn R (yinj 0) V1 K1 K2 &
R K1 V1 V2 & l ≤ yinj i.
/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
-
-lemma llpx_sn_fwd_bind_sn: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R l V L1 L2.
-#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_bind_dx: ∀R,a,I,L1,L2,V,T,l. llpx_sn R l (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R (⫯l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
- llpx_sn R l V L1 L2.
-#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_flat_dx: ∀R,I,L1,L2,V,T,l. llpx_sn R l (ⓕ{I}V.T) L1 L2 →
- llpx_sn R l T L1 L2.
-#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat … H) -H //
-qed-.
-
-lemma llpx_sn_fwd_pair_sn: ∀R,I,L1,L2,V,T,l. llpx_sn R l (②{I}V.T) L1 L2 →
- llpx_sn R l V L1 L2.
-#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma llpx_sn_refl: ∀R. (∀L. reflexive … (R L)) → ∀T,L,l. llpx_sn R l T L L.
-#R #HR #T #L @(f2_ind … rfw … L T) -L -T
-#x #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
-#i #Hx elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
-#HiL #l elim (ylt_split i l) /2 width=1 by llpx_sn_skip/
-elim (drop_O1_lt … HiL) -HiL destruct /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
-qed-.
-
-lemma llpx_sn_Y: ∀R,T,L1,L2. |L1| = |L2| → llpx_sn R (∞) T L1 L2.
-#R #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#x #IH #L1 * * /3 width=1 by llpx_sn_sort, llpx_sn_skip, llpx_sn_gref, llpx_sn_flat/
-#a #I #V1 #T1 #Hx #L2 #HL12
-@llpx_sn_bind /2 width=1 by/ (**) (* explicit constructor *)
-@IH -IH // normalize /2 width=1 by eq_f2/
-qed-.
-
-lemma llpx_sn_ge_up: ∀R,L1,L2,U,lt. llpx_sn R lt U L1 L2 → ∀T,l,k. ⬆[l, k] T ≡ U →
- lt ≤ l + k → llpx_sn R l U L1 L2.
-#R #L1 #L2 #U #lt #H elim H -L1 -L2 -U -lt
-[ #L1 #L2 #lt #s #HL12 #X #l #k #H #_ >(lift_inv_sort2 … H) -H /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #lt #i #HL12 #Hilt #X #l #k #H #Hltlm
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
- elim (ylt_yle_false … Hilt) -Hilt
- @(yle_trans … Hltlm) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
-| #I #L1 #L2 #K1 #K2 #W1 #W2 #lt #i #Hlti #HLK1 #HLK2 #HW1 #HW12 #_ #X #l #k #H #_
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct
- [ lapply (llpx_sn_fwd_length … HW1) -HW1 #HK12
- lapply (drop_fwd_length … HLK1) lapply (drop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I -W1 -W2 -lt /3 width=1 by llpx_sn_skip, ylt_inj/
- | /3 width=9 by llpx_sn_lref, yle_fwd_plus_sn1/
- ]
-| /2 width=1 by llpx_sn_free/
-| #L1 #L2 #lt #p #HL12 #X #l #k #H #_ >(lift_inv_gref2 … H) -H /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #k #H #Hltlm destruct
- elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
- @(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
- @(IHT … HTU) /2 width=1 by yle_succ/
-| #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #k #H #Hltlm destruct
- elim (lift_inv_flat2 … H) -H #HVW #HTU #H destruct
- /3 width=4 by llpx_sn_flat/
-]
-qed-.
-
-(**) (* the minor premise comes first *)
-lemma llpx_sn_ge: ∀R,L1,L2,T,l1,l2. l1 ≤ l2 →
- llpx_sn R l1 T L1 L2 → llpx_sn R l2 T L1 L2.
-#R #L1 #L2 #T #l1 #l2 * -l1 -l2 (**) (* destructed yle *)
-/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
-qed-.
-
-lemma llpx_sn_bind_O: ∀R,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
- llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2.
-/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
-
-lemma llpx_sn_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T,l. llpx_sn R1 l T L1 L2 → llpx_sn R2 l T L1 L2.
-#R1 #R2 #HR12 #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
-/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
-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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⦻ * break [ term 46 R , break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RelationStar $R $T $L1 $L2 }.
(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E.A→B→C→D→E→Prop.
-
-definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
-≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
-
(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝
| lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
lexs RN RP (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
-interpretation "general entrywise extension (local environment)"
+interpretation "generic entrywise extension (local environment)"
'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2).
-definition lpx_sn_confluent: relation6 (relation3 lenv term term)
- (relation3 lenv term term) … ≝
- λR1,R2,RN1,RP1,RN2,RP2.
- ∀f,L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+definition lexs_confluent: relation6 (relation3 lenv term term)
+ (relation3 lenv term term) … ≝
+ λR1,R2,RN1,RP1,RN2,RP2.
+ ∀f,L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. L0 ⦻*[RN1, RP1, f] L1 → ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-definition lexs_transitive: relation4 (relation3 lenv term term)
+definition lexs_transitive: relation5 (relation3 lenv term term)
(relation3 lenv term term) … ≝
- λR1,R2,RN,RP.
+ λR1,R2,R3,RN,RP.
∀f,L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⦻*[RN, RP, f] L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+ ∀T2. R2 L2 T T2 → R3 L1 T1 T2.
(* Basic inversion lemmas ***************************************************)
/2 width=1 by lexs_next, lexs_push/
qed-.
+(* Basic forward lemmas *****************************************************)
+
+lemma lexs_fwd_pair: ∀RN,RP,I1,I2,L1,L2,V1,V2,f.
+ L1.ⓑ{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 →
+ L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2.
+#RN #RP #I1 #I2 #L2 #L2 #V1 #V2 #f #Hf
+elim (pn_split f) * #g #H destruct
+[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf
+/2 width=1 by conj/
+qed-.
+
(* Basic properties *********************************************************)
lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⦻*[RN, RP, f] L2).
#RN1 #RP1 #RN2 #RP2 #HRN #HRP #L1 #L2 #f #H elim H -L1 -L2 -f
/3 width=1 by lexs_atom, lexs_next, lexs_push/
qed-.
-
-(* Basic_2A1: removed theorems 17:
- llpx_sn_inv_bind llpx_sn_inv_flat
- llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
- llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
- llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
-*)
(* Main properties **********************************************************)
-(* Basic_2A1: includes: lpx_sn_trans *)
-theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RP →
- lexs_transitive RP RP RN RP →
- Transitive … (lexs RN RP f).
-#RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
+theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
+ lexs_transitive RN1 RN2 RN RN1 RP1 →
+ lexs_transitive RP1 RP2 RP RN1 RP1 →
+ ∀L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
+ ∀L2. L0 ⦻*[RN2, RP2, f] L2 →
+ L1 ⦻*[RN, RP, f] L2.
+#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -L1 -L0 -f
[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
| #I #K1 #K #V1 #V #f #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
#K2 #V2 #HK2 #HV2 #H destruct /4 width=6 by lexs_next/
]
qed-.
+(* Basic_2A1: includes: lpx_sn_trans *)
+theorem lexs_trans (RN) (RP) (f): lexs_transitive RN RN RN RN RP →
+ lexs_transitive RP RP RP RN RP →
+ Transitive … (lexs RN RP f).
+/2 width=9 by lexs_trans_gen/ qed-.
+
(* Basic_2A1: includes: lpx_sn_conf *)
theorem lexs_conf: ∀RN1,RP1,RN2,RP2.
- lpx_sn_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
- lpx_sn_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
+ lexs_confluent RN1 RN2 RN1 RP1 RN2 RP2 →
+ lexs_confluent RP1 RP2 RN1 RP1 RN2 RP2 →
∀f. confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f).
#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L0 generalize in match f; -f
@(f_ind … lw … L0) -L0 #x #IH *
right_cancellable … (lexs RN RP f).
/3 width=3 by/ qed-.
-theorem lexs_meet: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
- ∀f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+lemma lexs_meet: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⦻*[RN, RP, f1] L2 →
+ ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f. f1 ⋒ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
-#g2 #H #H2 #f #Hf destruct
-[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
-#H2 #H2V #_
-[ elim (sand_inv_npx … Hf) | elim (sand_inv_ppx … Hf) | elim (sand_inv_nnx … Hf) | elim (sand_inv_pnx … Hf) ] -Hf
-/3 width=5 by lexs_next, lexs_push/
+#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
+[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
+| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
+] -Hf /3 width=5 by lexs_next, lexs_push/
qed-.
-theorem lexs_join: ∀RN,RP,L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
- ∀f2. L1 ⦻*[RN, RP, f2] L2 →
- ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
+lemma lexs_join: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⦻*[RN, RP, f1] L2 →
+ ∀f2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f. f1 ⋓ f2 ≡ f → L1 ⦻*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-#I #L1 #L2 #V1 #V2 #f1 #_ #H1V #IH #f2 elim (pn_split f2) *
-#g2 #H #H2 #f #Hf destruct
-[1,3: elim (lexs_inv_push … H2) |2,4: elim (lexs_inv_next … H2) ] -H2
-#H2 #H2V #_
-[ elim (sor_inv_npx … Hf) | elim (sor_inv_ppx … Hf) | elim (sor_inv_nnx … Hf) | elim (sor_inv_pnx … Hf) ] -Hf
-/3 width=5 by lexs_next, lexs_push/
+#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
+[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
+| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+] -Hf /3 width=5 by lexs_next, lexs_push/
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/static/lfeq_lreq.ma".
+include "basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties with equivalence on referred entries **************************)
+
+lemma lfeq_aaa_trans: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+ ∀L1. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ T ⁝ A.
+#G #L2 #T #A #H elim H -G -L2 -T -A /2 width=1 by aaa_sort/
+[ #I #G #L2 #V2 #A #_ #IH #L1 #H
+ elim (lfeq_inv_zero_pair_dx … H) -H /3 width=1 by aaa_zero/
+| #I #G #L2 #V2 #A #i #_ #IH #L1 #H
+ elim (lfeq_inv_lref_pair_dx … H) -H /3 width=1 by aaa_lref/
+| #p #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H
+ elim (lfeq_inv_bind … H) -H /3 width=2 by aaa_abbr/
+| #p #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H
+ elim (lfeq_inv_bind … H) -H /3 width=1 by aaa_abst/
+| #G #L2 #V #T #B #A #_ #_ #IHV #IHT #L1 #H
+ elim (lfeq_inv_flat … H) -H /3 width=3 by aaa_appl/
+| #G #L2 #V #T #A #_ #_ #IHV #IHT #L1 #H
+ elim (lfeq_inv_flat … H) -H /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma aaa_lfeq_conf: ∀G,L2,T,A. ⦃G, L2⦄ ⊢ T ⁝ A →
+ ∀L1. L2 ≡[T] L1 → ⦃G, L1⦄ ⊢ T ⁝ A.
+/3 width=3 by lfeq_aaa_trans, lfeq_sym/ 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/lazyeq_6.ma".
+include "basic_2/static/lfeq_lreq.ma".
+include "basic_2/static/lfeq_fqup.ma".
+
+(* EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *****************************)
+
+inductive ffeq (G) (L1) (T): relation3 genv lenv term ≝
+| fleq_intro: ∀L2. L1 ≡[T] L2 → ffeq G L1 T G L2 T
+.
+
+interpretation
+ "equivalence on referred entries (closure)"
+ 'LazyEq G1 L1 T1 G2 L2 T2 = (ffeq G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma ffeq_refl: tri_reflexive … ffeq.
+/2 width=1 by fleq_intro/ qed.
+
+lemma ffeq_sym: tri_symmetric … ffeq.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1 /3 width=1 by fleq_intro, lfeq_sym/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ffeq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L1 ≡[T1] L2 & T1 = T2.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 6:
+ fleq_refl fleq_sym fleq_inv_gen
+ fleq_trans fleq_canc_sn fleq_canc_dx
+*)
--- /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/static/lfeq_lfeq.ma".
+include "basic_2/static/ffeq.ma".
+
+(* EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *****************************)
+
+(* Main properties **********************************************************)
+
+theorem ffeq_trans: tri_transitive … ffeq.
+#G1 #G #L1 #L #T1 #T * -G -L -T
+#L #HL1 #G2 #L2 #T2 * -G2 -L2 -T2 /3 width=3 by fleq_intro, lfeq_trans/
+qed-.
+
+theorem ffeq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2.
+ ⦃G, L, T⦄ ≡ ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄.
+/3 width=5 by ffeq_trans, ffeq_sym/ qed-.
+
+theorem ffeq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T.
+ ⦃G1, L1, T1⦄ ≡ ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄.
+/3 width=5 by ffeq_trans, ffeq_sym/ qed-.
(* *)
(**************************************************************************)
-include "ground_2/relocation/rtmap_id.ma".
include "basic_2/s_computation/fqup_weight.ma".
include "basic_2/static/frees.ma".
+++ /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/lazyeq_6.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/static/frees_fqup.ma".
-include "basic_2/static/frees_lreq.ma".
-
-(* RANGED EQUIVALENCE FOR CLOSURES ******************************************)
-
-inductive freq (G) (L1) (T): relation3 genv lenv term ≝
-| fleq_intro: ∀L2,f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ≡[f] L2 → freq G L1 T G L2 T
-.
-
-interpretation
- "ranged equivalence (closure)"
- 'LazyEq G1 L1 T1 G2 L2 T2 = (freq G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma freq_refl: tri_reflexive … freq.
-#G #L #T elim (frees_total L T) /2 width=3 by fleq_intro/
-qed.
-
-lemma freq_sym: tri_symmetric … freq.
-#G1 #L1 #T1 #G2 #L2 #T2 * /4 width=3 by fleq_intro, frees_lreq_conf, lreq_sym/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma freq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄ →
- ∃∃f. G1 = G2 & L1 ⊢ 𝐅*⦃T1⦄ ≡ f & L1 ≡[f] L2 & T1 = T2.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=3 by ex4_intro/
-qed-.
-
-(* Basic_2A1: removed theorems 6:
- fleq_refl fleq_sym fleq_inv_gen
- fleq_trans fleq_canc_sn fleq_canc_dx
-*)
+++ /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/relocation/lreq_lreq.ma".
-include "basic_2/static/frees_frees.ma".
-include "basic_2/static/freq.ma".
-
-(* RANGED EQUIVALENCE FOR CLOSURES *****************************************)
-
-(* Main properties **********************************************************)
-
-theorem freq_trans: tri_transitive … freq.
-#G1 #G #L1 #L #T1 #T * -G -L -T
-#L #f1 #H1T11 #Hf1 #G2 #L2 #T2 * -G2 -L2 -T2 #L2 #f2 #HT12 #Hf2
-lapply (frees_lreq_conf … H1T11 … Hf1) #HT11
-lapply (frees_mono … HT12 … HT11) -HT11 -HT12
-/4 width=7 by fleq_intro, lreq_eq_repl_back, lreq_trans/
-qed-.
-
-theorem freq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2.
- ⦃G, L, T⦄ ≡ ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄.
-/3 width=5 by freq_trans, freq_sym/ qed-.
-
-theorem freq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T.
- ⦃G1, L1, T1⦄ ≡ ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡ ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡ ⦃G2, L2, T2⦄.
-/3 width=5 by freq_trans, freq_sym/ 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/lazyeq_3.ma".
+include "basic_2/static/lfxs.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+
+definition lfeq: relation3 term lenv lenv ≝ lfxs ceq.
+
+interpretation
+ "equivalence on referred entries (local environment)"
+ 'LazyEq T L1 L2 = (lfeq T L1 L2).
+
+definition lfeq_transitive: predicate (relation3 lenv term term) ≝
+ λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
+
+(* Basic properties ***********************************************************)
+
+lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆.
+/2 width=1 by lfxs_atom/ qed.
+
+lemma lfeq_sort: ∀I,L1,L2,V1,V2,s.
+ L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_sort/ qed.
+
+lemma lfeq_zero: ∀I,L1,L2,V.
+ L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V.
+/2 width=1 by lfxs_zero/ qed.
+
+lemma lfeq_lref: ∀I,L1,L2,V1,V2,i.
+ L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_lref/ qed.
+
+lemma lfeq_gref: ∀I,L1,L2,V1,V2,l.
+ L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2.
+/2 width=1 by lfxs_gref/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆.
+/2 width=3 by lfxs_inv_atom_sn/ qed-.
+
+lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆.
+/2 width=3 by lfxs_inv_atom_dx/ qed-.
+
+lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V. L1 ≡[V] L2 &
+ Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V.
+#Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
+/3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
+qed-.
+
+lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#Y1 #Y2 #i #H elim (lfxs_inv_lref … H) -H *
+/3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
+qed-.
+
+lemma lfeq_inv_bind: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 →
+ L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
+#I #L1 #L2 #V #T #p #H elim (lfxs_inv_bind … H) -H /2 width=3 by conj/
+qed-.
+
+lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
+ L1 ≡[V] L2 ∧ L1 ≡[T] L2.
+#I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H /2 width=3 by conj/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 →
+ ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V.
+#I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/
+qed-.
+
+lemma lfeq_inv_zero_pair_dx: ∀I,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V →
+ ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V.
+#I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H
+#L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma lfeq_inv_lref_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 →
+ ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2.
+/2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
+
+lemma lfeq_inv_lref_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1.
+/2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfeq_fwd_bind_sn: ∀I,L1,L2,V,T,p. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2.
+/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+
+lemma lfeq_fwd_bind_dx: ∀I,L1,L2,V,T,p.
+ L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
+/2 width=2 by lfxs_fwd_bind_dx/ qed-.
+
+lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2.
+/2 width=3 by lfxs_fwd_flat_sn/ qed-.
+
+lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2.
+/2 width=3 by lfxs_fwd_flat_dx/ qed-.
+
+lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
+
+(* Advanceded forward lemmas with generic extension on referred entries *****)
+
+lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2.
+/2 width=3 by lfxs_co/ qed-.
+
+(* Basic_2A1: removed theorems 30:
+ lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
+ lleq_fwd_drop_sn lleq_fwd_drop_dx
+ lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx
+ lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat
+ lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl
+ lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div
+*)
--- /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/static/lfxs_fqup.ma".
+include "basic_2/static/lfeq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+
+(* Advanced properties ******************************************************)
+
+lemma lfeq_refl: ∀T. reflexive … (lfeq T).
+/2 width=1 by lfxs_refl/ 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/static/lfxs_length.ma".
+include "basic_2/static/lfeq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lfeq_fwd_length: ∀L1,L2,T. L1 ≡[T] L2 → |L1| = |L2|.
+/2 width=3 by lfxs_fwd_length/ 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/relocation/lreq_lreq.ma".
+include "basic_2/static/frees_frees.ma".
+include "basic_2/static/lfxs_lfxs.ma".
+include "basic_2/static/lfeq_lreq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+
+(* Main properties **********************************************************)
+
+theorem lfeq_bind: ∀I,L1,L2,V1,V2,T,p.
+ L1 ≡[V1] L2 → L1.ⓑ{I}V1 ≡[T] L2.ⓑ{I}V2 →
+ L1 ≡[ⓑ{p,I}V1.T] L2.
+/2 width=2 by lfxs_bind/ qed.
+
+theorem lfeq_flat: ∀I,L1,L2,V,T. L1 ≡[V] L2 → L1 ≡[T] L2 →
+ L1 ≡[ⓕ{I}V.T] L2.
+/2 width=1 by lfxs_flat/ qed.
+
+(* Note: /2 width=3 by lfeq_lfxs_trans/ *)
+theorem lfeq_trans: ∀T. Transitive … (lfeq T).
+#T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (frees_lreq_conf … Hf1 … HL1) #H0
+lapply (frees_mono … Hf2 … H0) -Hf2 -H0
+/4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/
+qed-.
+
+theorem lfeq_canc_sn: ∀T. left_cancellable … (lfeq T).
+/3 width=3 by lfeq_trans, lfeq_sym/ qed-.
+
+theorem lfeq_canc_dx: ∀T. right_cancellable … (lfeq T).
+/3 width=3 by lfeq_trans, lfeq_sym/ qed-.
+
+(* Advanced properies on negated lazy equivalence *****************************)
+
+(* Note: for use in auto, works with /4 width=8/ so lfeq_canc_sn is preferred *)
+lemma lfeq_nlfeq_trans: ∀T,L1,L. L1 ≡[T] L →
+ ∀L2. (L ≡[T] L2 → ⊥) → (L1 ≡[T] L2 → ⊥).
+/3 width=3 by lfeq_canc_sn/ qed-.
+
+lemma nlfeq_lfeq_div: ∀T,L2,L. L2 ≡[T] L →
+ ∀L1. (L1 ≡[T] L → ⊥) → (L1 ≡[T] L2 → ⊥).
+/3 width=3 by lfeq_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/static/frees_lreq.ma".
+include "basic_2/static/lfeq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
+
+(* Inversion lemmas with ranged equivalence for local environments **********)
+
+lemma lfeq_inv_lreq: ∀L1,L2,T. L1 ≡[T] L2 → ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ≡[f] L2.
+#L1 #L2 #T * /2 width=3 by ex2_intro/
+qed-.
+
+(* Properties with ranged equivalence for local environments ****************)
+
+lemma lreq_lfeq: ∀L1,L2,T,f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ≡[f] L2 → L1 ≡[T] L2.
+/2 width=3 by ex2_intro/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lfeq_sym: ∀T. symmetric … (lfeq T).
+#T #L1 #L2 #H elim (lfeq_inv_lreq … H) -H
+/3 width=3 by lreq_lfeq, frees_lreq_conf, lreq_sym/
+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 "ground_2/relocation/rtmap_id.ma".
+include "basic_2/notation/relations/relationstar_4.ma".
+include "basic_2/grammar/ceq.ma".
+include "basic_2/relocation/lexs.ma".
+include "basic_2/static/frees.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition lfxs (R) (T): relation lenv ≝
+ λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⦻*[R, cfull, f] L2.
+
+interpretation "generic extension on referred entries (local environment)"
+ 'RelationStar R T L1 L2 = (lfxs R T L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
+/3 width=3 by lexs_atom, frees_atom, ex2_intro/
+qed.
+
+lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
+ L1 ⦻*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻*[R, ⋆s] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
+qed.
+
+lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
+qed.
+
+lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
+ L1 ⦻*[R, #i] L2 → L1.ⓑ{I}V1 ⦻*[R, #⫯i] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
+qed.
+
+lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
+ L1 ⦻*[R, §l] L2 → L1.ⓑ{I}V1 ⦻*[R, §l] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
+qed.
+
+lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⦻*[R1, T] L2 → L1 ⦻*[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
+#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
+qed-.
+
+lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
+qed-.
+
+lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_bind: ∀R,I,L1,L2,V1,V2,T,p. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #T #p * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
+/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
+ L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
+/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfxs_fwd_bind_sn: ∀R,I,L1,L2,V,T,p. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #I #L1 #L2 #V #T #p * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
+qed-.
+
+lemma lfxs_fwd_bind_dx: ∀R,I,L1,L2,V1,V2,T,p. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #T #p #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
+qed-.
+
+lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
+qed-.
+
+(* Basic_2A1: removed theorems 24:
+ llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
+ llpx_sn_bind llpx_sn_flat
+ llpx_sn_inv_bind llpx_sn_inv_flat
+ llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
+ llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
+ llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
+*)
--- /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/static/frees_fqup.ma".
+include "basic_2/static/lfxs.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Advanced properties ******************************************************)
+
+lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⦻*[R, T] L.
+#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_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/relocation/lexs_length.ma".
+include "basic_2/static/lfxs.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|.
+#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/
+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/relocation/lexs_lexs.ma".
+include "basic_2/static/lfxs.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Main properties ******************************************************)
+
+theorem lfxs_bind: ∀R,I,L1,L2,V1,V2,T,p.
+ L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
+ L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
+#R #I #L1 #L2 #V1 #V2 #T #p * #f1 #HV #Hf1 * #f2 #HT #Hf2
+elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
+qed.
+
+theorem lfxs_flat: ∀R,I,L1,L2,V,T.
+ L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 →
+ L1 ⦻*[R, ⓕ{I}V.T] L2.
+#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
+/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
+qed.
}
]
[ { "atomic arity assignment" * } {
- [ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + (* "aaa_lleq" + *) "aaa_aaa" * ]
+ [ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_lfeq" + "aaa_aaa" * ]
}
]
[ { "restricted ref. for local env." * } {
[ "lsubr ( ? ⫃ ? )" "lsubr_length" + "lsubr_drops" + "lsubr_lsubr" * ]
}
]
- [ { "ranged equivalence for closures" * } {
- [ "freq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )" "freq_freq" * ]
+ [ { "equivalence for closures on referred entries" * } {
+ [ "ffeq ( ⦃?,?,?⦄ ≡ ⦃?,?,?⦄ )" "ffeq_freq" * ]
+ }
+ ]
+ [ { "equivalence for local environments on referred entries" * } {
+ [ "lfeq ( ? ≡[?] ? )" "lfeq_length" + "lfeq_lreq" + "lfeq_fqup" + "lfeq_lfeq" * ]
+ }
+ ]
+ [ { "generic extension on referred entries" * } {
+ [ "lfxs ( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_fqup" + "lfxs_lfxs" * ]
}
]
[ { "context-sensitive free variables" * } {
[ "lreq ( ? ≡[?] ? )" "lreq_length" + "lreq_lreq" * ]
}
]
- [ { "generic entrywise extension of context-sensitive relations for terma" * } {
+ [ { "generic entrywise extension" * } {
[ "lexs ( ? ⦻*[?,?,?] ? )" "lexs_length" + "lexs_lexs" * ]
}
]
(* PROPERTIES OF RELATIONS **************************************************)
+definition relation5 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
+≝ λA,B,C,D,E.A→B→C→D→E→Prop.
+
+definition relation6 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
+≝ λA,B,C,D,E,F.A→B→C→D→E→F→Prop.
+
definition Decidable: Prop → Prop ≝ λR. R ∨ (R → ⊥).
definition Transitive: ∀A. ∀R: relation A. Prop ≝ λA,R.
(* This file was generated by xoa.native: do not edit *********************)
-(* multiple existental quantifier (1, 2) *)
+(* multiple inductive existental quantifier (1, 2) *)
notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) }.
-(* multiple existental quantifier (1, 3) *)
+(* multiple inductive existental quantifier (1, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) }.
-(* multiple existental quantifier (2, 2) *)
+(* multiple inductive existental quantifier (2, 2) *)
notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }.
-(* multiple existental quantifier (2, 3) *)
+(* multiple inductive existental quantifier (2, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) }.
-(* multiple existental quantifier (3, 1) *)
+(* multiple inductive existental quantifier (3, 1) *)
notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
-(* multiple existental quantifier (3, 2) *)
+(* multiple inductive existental quantifier (3, 2) *)
notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }.
-(* multiple existental quantifier (3, 3) *)
+(* multiple inductive existental quantifier (3, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }.
-(* multiple existental quantifier (3, 4) *)
+(* multiple inductive existental quantifier (3, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) }.
-(* multiple existental quantifier (4, 1) *)
+(* multiple inductive existental quantifier (3, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) }.
+
+(* multiple inductive existental quantifier (4, 1) *)
notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) (λ${ident x0}:$T0.$P3) }.
-(* multiple existental quantifier (4, 2) *)
+(* multiple inductive existental quantifier (4, 2) *)
notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) }.
-(* multiple existental quantifier (4, 3) *)
+(* multiple inductive existental quantifier (4, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }.
-(* multiple existental quantifier (4, 4) *)
+(* multiple inductive existental quantifier (4, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }.
-(* multiple existental quantifier (4, 5) *)
+(* multiple inductive existental quantifier (4, 5) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) }.
-(* multiple existental quantifier (5, 2) *)
+(* multiple inductive existental quantifier (5, 2) *)
notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P4) }.
-(* multiple existental quantifier (5, 3) *)
+(* multiple inductive existental quantifier (5, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }.
-(* multiple existental quantifier (5, 4) *)
+(* multiple inductive existental quantifier (5, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
-(* multiple existental quantifier (5, 5) *)
+(* multiple inductive existental quantifier (5, 5) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }.
-(* multiple existental quantifier (5, 6) *)
+(* multiple inductive existental quantifier (5, 6) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) }.
-(* multiple existental quantifier (6, 3) *)
+(* multiple inductive existental quantifier (6, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) }.
-(* multiple existental quantifier (6, 4) *)
+(* multiple inductive existental quantifier (6, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) }.
-(* multiple existental quantifier (6, 5) *)
+(* multiple inductive existental quantifier (6, 5) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P5) }.
-(* multiple existental quantifier (6, 6) *)
+(* multiple inductive existental quantifier (6, 6) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }.
-(* multiple existental quantifier (6, 7) *)
+(* multiple inductive existental quantifier (6, 7) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) }.
-(* multiple existental quantifier (7, 3) *)
+(* multiple inductive existental quantifier (7, 3) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P6) }.
-(* multiple existental quantifier (7, 4) *)
+(* multiple inductive existental quantifier (7, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) }.
-(* multiple existental quantifier (7, 7) *)
+(* multiple inductive existental quantifier (7, 7) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 , ident x6 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.λ${ident x6}:$T6.$P6) }.
-(* multiple existental quantifier (8, 4) *)
+(* multiple inductive existental quantifier (8, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
non associative with precedence 20
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P6) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P7) }.
-(* multiple existental quantifier (8, 5) *)
+(* multiple inductive existental quantifier (8, 5) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6 break & term 19 P7)"
non associative with precedence 20
<key name="ex">3 2</key>
<key name="ex">3 3</key>
<key name="ex">3 4</key>
+ <key name="ex">3 5</key>
<key name="ex">4 1</key>
<key name="ex">4 2</key>
<key name="ex">4 3</key>
include "ground_2/notation/xoa_notation.ma".
-(* multiple existental quantifier (1, 2) *)
+(* multiple inductive existental quantifier (1, 2) *)
inductive ex1_2 (A0,A1:Type[0]) (P0:A0→A1→Prop) : Prop ≝
| ex1_2_intro: ∀x0,x1. P0 x0 x1 → ex1_2 ? ? ?
.
-interpretation "multiple existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
+interpretation "multiple inductive existental quantifier (1, 2)" 'Ex P0 = (ex1_2 ? ? P0).
-(* multiple existental quantifier (1, 3) *)
+(* multiple inductive existental quantifier (1, 3) *)
inductive ex1_3 (A0,A1,A2:Type[0]) (P0:A0→A1→A2→Prop) : Prop ≝
| ex1_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → ex1_3 ? ? ? ?
.
-interpretation "multiple existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0).
+interpretation "multiple inductive existental quantifier (1, 3)" 'Ex P0 = (ex1_3 ? ? ? P0).
-(* multiple existental quantifier (2, 2) *)
+(* multiple inductive existental quantifier (2, 2) *)
inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝
| ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ?
.
-interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
+interpretation "multiple inductive existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
-(* multiple existental quantifier (2, 3) *)
+(* multiple inductive existental quantifier (2, 3) *)
inductive ex2_3 (A0,A1,A2:Type[0]) (P0,P1:A0→A1→A2→Prop) : Prop ≝
| ex2_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → ex2_3 ? ? ? ? ?
.
-interpretation "multiple existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1).
+interpretation "multiple inductive existental quantifier (2, 3)" 'Ex P0 P1 = (ex2_3 ? ? ? P0 P1).
-(* multiple existental quantifier (3, 1) *)
+(* multiple inductive existental quantifier (3, 1) *)
inductive ex3 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
| ex3_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3 ? ? ? ?
.
-interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3 ? P0 P1 P2).
+interpretation "multiple inductive existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3 ? P0 P1 P2).
-(* multiple existental quantifier (3, 2) *)
+(* multiple inductive existental quantifier (3, 2) *)
inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝
| ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ?
.
-interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
+interpretation "multiple inductive existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
-(* multiple existental quantifier (3, 3) *)
+(* multiple inductive existental quantifier (3, 3) *)
inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝
| ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
+interpretation "multiple inductive existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
-(* multiple existental quantifier (3, 4) *)
+(* multiple inductive existental quantifier (3, 4) *)
inductive ex3_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2:A0→A1→A2→A3→Prop) : Prop ≝
| ex3_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → ex3_4 ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2).
+interpretation "multiple inductive existental quantifier (3, 4)" 'Ex P0 P1 P2 = (ex3_4 ? ? ? ? P0 P1 P2).
-(* multiple existental quantifier (4, 1) *)
+(* multiple inductive existental quantifier (3, 5) *)
+
+inductive ex3_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2:A0→A1→A2→A3→A4→Prop) : Prop ≝
+ | ex3_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → ex3_5 ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple inductive existental quantifier (3, 5)" 'Ex P0 P1 P2 = (ex3_5 ? ? ? ? ? P0 P1 P2).
+
+(* multiple inductive existental quantifier (4, 1) *)
inductive ex4 (A0:Type[0]) (P0,P1,P2,P3:A0→Prop) : Prop ≝
| ex4_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → P3 x0 → ex4 ? ? ? ? ?
.
-interpretation "multiple existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4 ? P0 P1 P2 P3).
+interpretation "multiple inductive existental quantifier (4, 1)" 'Ex P0 P1 P2 P3 = (ex4 ? P0 P1 P2 P3).
-(* multiple existental quantifier (4, 2) *)
+(* multiple inductive existental quantifier (4, 2) *)
inductive ex4_2 (A0,A1:Type[0]) (P0,P1,P2,P3:A0→A1→Prop) : Prop ≝
| ex4_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → ex4_2 ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3).
+interpretation "multiple inductive existental quantifier (4, 2)" 'Ex P0 P1 P2 P3 = (ex4_2 ? ? P0 P1 P2 P3).
-(* multiple existental quantifier (4, 3) *)
+(* multiple inductive existental quantifier (4, 3) *)
inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝
| ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
+interpretation "multiple inductive existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
-(* multiple existental quantifier (4, 4) *)
+(* multiple inductive existental quantifier (4, 4) *)
inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝
| ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
+interpretation "multiple inductive existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
-(* multiple existental quantifier (4, 5) *)
+(* multiple inductive existental quantifier (4, 5) *)
inductive ex4_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→A4→Prop) : Prop ≝
| ex4_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → ex4_5 ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3).
+interpretation "multiple inductive existental quantifier (4, 5)" 'Ex P0 P1 P2 P3 = (ex4_5 ? ? ? ? ? P0 P1 P2 P3).
-(* multiple existental quantifier (5, 2) *)
+(* multiple inductive existental quantifier (5, 2) *)
inductive ex5_2 (A0,A1:Type[0]) (P0,P1,P2,P3,P4:A0→A1→Prop) : Prop ≝
| ex5_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → P3 x0 x1 → P4 x0 x1 → ex5_2 ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
+interpretation "multiple inductive existental quantifier (5, 2)" 'Ex P0 P1 P2 P3 P4 = (ex5_2 ? ? P0 P1 P2 P3 P4).
-(* multiple existental quantifier (5, 3) *)
+(* multiple inductive existental quantifier (5, 3) *)
inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝
| ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
+interpretation "multiple inductive existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
-(* multiple existental quantifier (5, 4) *)
+(* multiple inductive existental quantifier (5, 4) *)
inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝
| ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
+interpretation "multiple inductive existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
-(* multiple existental quantifier (5, 5) *)
+(* multiple inductive existental quantifier (5, 5) *)
inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝
| ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4).
+interpretation "multiple inductive existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4).
-(* multiple existental quantifier (5, 6) *)
+(* multiple inductive existental quantifier (5, 6) *)
inductive ex5_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
| ex5_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → ex5_6 ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (5, 6)" 'Ex P0 P1 P2 P3 P4 = (ex5_6 ? ? ? ? ? ? P0 P1 P2 P3 P4).
+interpretation "multiple inductive existental quantifier (5, 6)" 'Ex P0 P1 P2 P3 P4 = (ex5_6 ? ? ? ? ? ? P0 P1 P2 P3 P4).
-(* multiple existental quantifier (6, 3) *)
+(* multiple inductive existental quantifier (6, 3) *)
inductive ex6_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→Prop) : Prop ≝
| ex6_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → ex6_3 ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (6, 3)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5).
+interpretation "multiple inductive existental quantifier (6, 3)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_3 ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (6, 4) *)
+(* multiple inductive existental quantifier (6, 4) *)
inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝
| ex6_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → ex6_4 ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5).
+interpretation "multiple inductive existental quantifier (6, 4)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_4 ? ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (6, 5) *)
+(* multiple inductive existental quantifier (6, 5) *)
inductive ex6_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→Prop) : Prop ≝
| ex6_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → ex6_5 ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+interpretation "multiple inductive existental quantifier (6, 5)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (6, 6) *)
+(* multiple inductive existental quantifier (6, 6) *)
inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
| ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+interpretation "multiple inductive existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (6, 7) *)
+(* multiple inductive existental quantifier (6, 7) *)
inductive ex6_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
| ex6_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → ex6_7 ? ? ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
+interpretation "multiple inductive existental quantifier (6, 7)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-(* multiple existental quantifier (7, 3) *)
+(* multiple inductive existental quantifier (7, 3) *)
inductive ex7_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→Prop) : Prop ≝
| ex7_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → P5 x0 x1 x2 → P6 x0 x1 x2 → ex7_3 ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (7, 3)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_3 ? ? ? P0 P1 P2 P3 P4 P5 P6).
+interpretation "multiple inductive existental quantifier (7, 3)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_3 ? ? ? P0 P1 P2 P3 P4 P5 P6).
-(* multiple existental quantifier (7, 4) *)
+(* multiple inductive existental quantifier (7, 4) *)
inductive ex7_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→Prop) : Prop ≝
| ex7_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → ex7_4 ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (7, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+interpretation "multiple inductive existental quantifier (7, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-(* multiple existental quantifier (7, 7) *)
+(* multiple inductive existental quantifier (7, 7) *)
inductive ex7_7 (A0,A1,A2,A3,A4,A5,A6:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→A6→Prop) : Prop ≝
| ex7_7_intro: ∀x0,x1,x2,x3,x4,x5,x6. P0 x0 x1 x2 x3 x4 x5 x6 → P1 x0 x1 x2 x3 x4 x5 x6 → P2 x0 x1 x2 x3 x4 x5 x6 → P3 x0 x1 x2 x3 x4 x5 x6 → P4 x0 x1 x2 x3 x4 x5 x6 → P5 x0 x1 x2 x3 x4 x5 x6 → P6 x0 x1 x2 x3 x4 x5 x6 → ex7_7 ? ? ? ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
+interpretation "multiple inductive existental quantifier (7, 7)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_7 ? ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-(* multiple existental quantifier (8, 4) *)
+(* multiple inductive existental quantifier (8, 4) *)
inductive ex8_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→Prop) : Prop ≝
| ex8_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → P5 x0 x1 x2 x3 → P6 x0 x1 x2 x3 → P7 x0 x1 x2 x3 → ex8_4 ? ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (8, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
+interpretation "multiple inductive existental quantifier (8, 4)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_4 ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
-(* multiple existental quantifier (8, 5) *)
+(* multiple inductive existental quantifier (8, 5) *)
inductive ex8_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4,P5,P6,P7:A0→A1→A2→A3→A4→Prop) : Prop ≝
| ex8_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → P5 x0 x1 x2 x3 x4 → P6 x0 x1 x2 x3 x4 → P7 x0 x1 x2 x3 x4 → ex8_5 ? ? ? ? ? ? ? ? ? ? ? ? ?
.
-interpretation "multiple existental quantifier (8, 5)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
+interpretation "multiple inductive existental quantifier (8, 5)" 'Ex P0 P1 P2 P3 P4 P5 P6 P7 = (ex8_5 ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6 P7).
(* multiple disjunction connective (3) *)
projects / helm.git / commitdiff