--- /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/predeval_4.ma".
+include "basic_2/computation/cprs.ma".
+include "basic_2/computation/csx.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
+
+definition cpre: relation4 genv lenv term term ≝
+ λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive parallel reduction (term)"
+ 'PRedEval G L T1 T2 = (cpre G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: nf2_sn3 *)
+lemma csx_cpre: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
+* #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
+#T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
+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/computation/cprs_cprs.ma".
+include "basic_2/computation/cpre.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
+
+(* Main properties *********************************************************)
+
+(* Basic_1: was: nf2_pr3_confluence *)
+theorem cpre_mono: ∀G,L,T,T1. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
+#G #L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2
+elim (cprs_conf … H1T1 … H1T2) -T #T #HT1
+>(cprs_inv_cnr1 … HT1 H2T1) -T1 #HT2
+>(cprs_inv_cnr1 … HT2 H2T2) -T2 //
+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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedEval $G $L $T1 $T2 }.
--- /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/predeval_6.ma".
+include "basic_2/computation/cpxs.ma".
+include "basic_2/computation/csx.ma".
+
+(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION ON TERMS ****)
+
+definition cpxe: ∀h. sd h → relation4 genv lenv term term ≝
+ λh,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 ∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T2⦄.
+
+interpretation "evaluation for context-sensitive extended parallel reduction (term)"
+ 'PRedEval h o G L T1 T2 = (cpxe h o G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma csx_cpxe: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] 𝐍⦃T2⦄.
+#h #o #G #L #T1 #H @(csx_ind … H) -T1
+#T1 #_ #IHT1 elim (cnx_dec h o G L T1) /3 width=3 by ex_intro, conj/
+* #T #H1T1 #H2T1 elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1
+#T2 * /4 width=3 by cpxs_strap2, ex_intro, conj/
+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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * [ break term 46 h , break term 46 o ] 𝐍 ⦃ break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'PRedEval $h $o $G $L $T1 $T2 }.
--- /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/reduction/cpx_lreq.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) (lreq 0 (∞)).
+/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
+qed-.
lemma lpxs_pair2: ∀h,o,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, o] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V2.
/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
-
--- /dev/null
+fact lreq_lpxs_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
+ ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
+ (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
+[ #l #k #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+ /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
+]
+qed-.
+
+lemma lreq_lpxs_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
+ ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
+ (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
+/2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.
K1 ≡[V, 0] K2 → L1 ≡[T, l] L2.
/2 width=9 by llpx_sn_inv_S/ qed-.
-lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
- L1 ≡[V, 0] L2 ∧ L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_inv_bind_O/ qed-.
-
(* Advanced forward lemmas **************************************************)
lemma lleq_fwd_lref_dx: ∀L1,L2,l,i. L1 ≡[#i, l] L2 →
#L1 #L2 #l #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
qed-.
-
-lemma lleq_fwd_bind_O_dx: ∀a,I,L1,L2,V,T. L1 ≡[ⓑ{a,I}V.T, 0] L2 →
- L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V.
-/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_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 → ⊥).
-/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
-
-lemma nlleq_inv_flat: ∀I,L1,L2,V,T,l. (L1 ≡[ⓕ{I}V.T, l] L2 → ⊥) →
- (L1 ≡[V, l] L2 → ⊥) ∨ (L1 ≡[T, l] L2 → ⊥).
-/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
-
-lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ≡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
- (L1 ≡[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ⊥).
-/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.
llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2.
/2 width=9 by llpx_sn_inv_S_aux/ qed-.
-lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
-/3 width=9 by drop_pair, conj, llpx_sn_inv_S/
-qed-.
-
-(* More advanced forward lemmas *********************************************)
-
-lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
-qed-.
-
(* Advanced properties ******************************************************)
lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
-
-(* Inversion lemmas on negated lazy pointwise extension *********************)
-
-lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀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) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
-/4 width=1 by llpx_sn_bind, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀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 → ⊥).
-#R #HR #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
-/4 width=1 by llpx_sn_flat, or_intror, or_introl/
-qed-.
-
-lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
- (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
-#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
-/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
-qed-.
(* Properties on transitive_closure *****************************************)
-lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
-#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
-[ /2 width=1 by lpx_sn_refl/
-| /3 width=1 by TC_reflexive, lpx_sn_refl/
-| /3 width=5 by lpx_sn_pair, step/
-]
-qed-.
-
-lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
- TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
∀L1,L2. lpx_sn (LTC … R) L1 L2 →
TC … (lpx_sn R) L1 L2.
(* Inversion lemmas on transitive closure ***********************************)
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
∀S:relation lenv.
S (⋆) (⋆) → (
]
qed-.
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
∀L1,L2. TC … (lpx_sn R) L1 L2 →
lpx_sn (LTC … R) L1 L2.
/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-qed-.
--- /dev/null
+(* lfxs_pair_repl_dx is not generic enough **********************************)
+
+fact tc_lfxs_pair_repl_dx_aux: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,Y,V1,T. L1.ⓑ{I}V1 ⦻**[R, T] Y →
+ ∀L2,V2. Y = L2.ⓑ{I}V2 →
+ L1.ⓑ{I}V1 ⦻**[R, T] L2.ⓑ{I}V1.
+#R #HR #I #L1 #Y #V1 #T #H elim H -Y
+/3 width=6 by lfxs_pair_repl_dx, step, inj/
+#Y #Y2 #HY #HY2 #IH #L2 #V2 #H destruct
+elim (lfxs_fwd_dx … HY2) #L #V #H destruct
+lapply (IH ???) [4: |*: // ] -IH #HL1
+@(step … HL1) -HL1 //
+
+/3 width=6 by lfxs_pair_repl_dx, step, inj/
+
+lemma tc_lfxs_pair_repl_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2,V1,V2,T.
+ L1.ⓑ{I}V1 ⦻**[R, T] L2.ⓑ{I}V2 →
+ L1.ⓑ{I}V1 ⦻**[R, T] L2.ⓑ{I}V1.
+
+(* does this really hold? ***************************************************)
+
+lemma pippo1: ∀R. s_r_confluent1 … R (lfxs R) →
+ s_r_transitive … R (lfxs R) →
+ s_r_transitive … R (tc_lfxs R).
+#R #H1R #H2R @s_r_trans_LTC2 @s_r_trans_LTC1 //
+qed-.
+
+definition s_r_confluent2: ∀A,B. relation2 (A→relation B) (B→relation A) ≝ λA,B,R1,R2.
+ ∀L2,T1,T2. R1 L2 T1 T2 → ∀L1. R2 T1 L1 L2 → R2 T2 L1 L2.
+
+
+axiom s_r_conf2_LTC2: ∀R. s_r_transitive … R (lfxs R) →
+ s_r_confluent1 … R (lfxs R) →
+ s_r_confluent2 … R (tc_lfxs R).
+(*
+#R #H1R #H2R #L2 #T1 #T2 #HT12 #L1 #H
+@(TC_ind_dx ??????? H) -L1 /3 width=3 by inj/
+#L1 #L #HL1 #HL2 #IH @(tc_lfxs_step_sn … IH) -IH //
+*)
+
+lemma tc_lfxs_inv_zero: ∀R. s_r_confluent1 … R (lfxs R) →
+ s_r_transitive … R (lfxs R) →
+ ∀Y1,Y2. Y1 ⦻**[R, #0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, V1] L2 & LTC … R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #H1R #H2R #Y1 #Y2 #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_zero … H) -H *
+ /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H *
+ [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct
+ @or_intror @ex4_5_intro [6,7: |*: /width=7/ ]
+ [
+ | -HL2 @(trans_TC … HV10) @(pippo1 … HV2) // -V2 @pippo2 //
+
+ lapply (H2R … HV2 … HL2) -HL2 #HL02
+ /4 width=8 by ex3_5_intro, step, or_intror/
+ ]
+ ]
+]
+qed-.
+
+(* has no meaning without the former ****************************************)
+
+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_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-.
+++ /dev/null
-(*
-lemma lfxss_zero: ∀R,I,L1,L2,V1. L1 ⦻**[R, V1] L2 → ∀V2.
- R L1 V1 V2 → L1.ⓑ{I}V1 ⦻**[R, #0] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #H
-
-elim H -L2 /3 width=5 by lfxs_zero, inj/
-#L #L2 #H0 #HL2 #IH #V2 #HV12
-lapply (IH … HV12) -IH #HL1
-@(step … HL1) -HL1 @lfxs_zero
-
-axiom pippo_fwd: ∀R,I,Y1,L2,V2,T. Y1 ⦻*[R, T] L2.ⓑ{I}V2 →
- ∃∃L1,V1. Y1 = L1.ⓑ{I}V1.
-
-fact pippo: ∀R,I,L1,Y,T,V. L1.ⓑ{I}V ⦻**[R, T] Y →
- ∀L2,V1. Y = L2.ⓑ{I}V1 →
- ∀V2. R L1 V V2 →
- L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #Y #T #V #H elim H -Y
-[ /3 width=2 by lfxs_pair_repl_dx, inj/
-| #L #Y #_ #HL2 #IH #L2 #V1 #H #V2 #HV2 destruct
- elim (pippo_fwd … HL2) #L0 #V0 #H destruct
- @step [2: @(IH … HV2) // | skip ] -IH -HV2
-
-lemma lfxss_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V1 →
- ∀V2. R L1 V V2 →
- L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
-/3 width=5 by lexs_pair_repl, ex2_intro/
-qed-.
-*)
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-definition tc_lfxs (R) (T): relation lenv ≝ TC … (lfxs R T).
+definition tc_lfxs (R): term → relation lenv ≝ LTC … (lfxs R).
interpretation "iterated extension on referred entries (local environment)"
'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
(* Basic properties *********************************************************)
+lemma tc_lfxs_step_dx: ∀R,L1,L,T. L1 ⦻**[R, T] L →
+ ∀L2. L ⦻*[R, T] L2 → L1 ⦻**[R, T] L2.
+#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
+qed-.
+
+lemma tc_lfxs_step_sn: ∀R,L1,L,T. L1 ⦻*[R, T] L →
+ ∀L2. L ⦻**[R, T] L2 → L1 ⦻**[R, T] L2.
+#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
+qed-.
+
lemma tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
/2 width=1 by inj/ qed.
lemma tc_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 #H elim H -L2
-/3 width=4 by lfxs_sort, step, inj/
+/3 width=4 by lfxs_sort, tc_lfxs_step_dx, inj/
+qed.
+
+lemma tc_lfxs_zero: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2,V. L1 ⦻**[R, V] L2 →
+ L1.ⓑ{I}V ⦻**[R, #0] L2.ⓑ{I}V.
+#R #HR #I #L1 #L2 #V #H elim H -L2
+/3 width=5 by lfxs_zero, tc_lfxs_step_dx, inj/
qed.
lemma tc_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 #H elim H -L2
-/3 width=4 by lfxs_lref, step, inj/
+/3 width=4 by lfxs_lref, tc_lfxs_step_dx, inj/
qed.
lemma tc_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 #H elim H -L2
-/3 width=4 by lfxs_gref, step, inj/
+/3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/
qed.
lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
(∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
∀T. symmetric … (tc_lfxs R T).
#R #H1R #H2R #T #L1 #L2 #H elim H -L2
-/4 width=3 by lfxs_sym, TC_strap, inj/
+/4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
qed-.
lemma tc_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 #H elim H -L2
-/4 width=5 by lfxs_co, step, inj/
+/4 width=5 by lfxs_co, tc_lfxs_step_dx, inj/
qed-.
(* Basic inversion lemmas ***************************************************)
+(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆.
#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
qed-.
+(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆.
#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
/3 width=3 by inj, lfxs_inv_atom_dx/
| #I #L #L2 #V #V2 #HL2 #H #H2 * *
[ #H1 #H0 destruct
| #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
- /4 width=8 by ex3_5_intro, step, or_intror/
+ /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
]
]
]
qed-.
-(*
-lemma tc_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 &
+
+lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻**[R, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, §l] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 #H elim H -Y2
-[ #Y2 #H elim (lfxs_inv_zero … H) -H *
- /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/
-| #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H *
- [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
- | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * *
+#R #Y1 #Y2 #l #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_gref … H) -H *
+ /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_gref … H) -H *
+ [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #H #H2 * *
[ #H1 #H0 destruct
- | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct
- @or_intror @ex4_5_intro [6,7: |*: /width=7/ ]
-
- /4 width=8 by ex3_5_intro, step, or_intror/
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
+ /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
]
]
]
qed-.
-
-
-
-
-#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/
+lemma tc_lfxs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
+ L1 ⦻**[R, V] L2 ∧ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V.
+#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
+[ #L2 #H elim (lfxs_inv_bind … V ? H) -H /3 width=1 by inj, conj/
+| #L #L2 #_ #H * elim (lfxs_inv_bind … V ? H) -H /3 width=3 by tc_lfxs_step_dx, conj/
]
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/
+lemma tc_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 #H elim H -L2
+[ #L2 #H elim (lfxs_inv_flat … H) -H /3 width=1 by inj, conj/
+| #L #L2 #_ #H * elim (lfxs_inv_flat … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
]
qed-.
-lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| lapply (frees_inv_gref … H1) -H1 #Hf
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T * #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_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
+lemma tc_lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻**[R, ⋆s] Y2 →
+ ∃∃L2,V2. L1 ⦻**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #s #H elim (tc_lfxs_inv_sort … H) -H *
[ #H destruct
| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
+lemma tc_lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻**[R, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #s #H elim (tc_lfxs_inv_sort … H) -H *
[ #_ #H destruct
| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-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-.
-
-lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
+lemma tc_lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻**[R, §l] Y2 →
+ ∃∃L2,V2. L1 ⦻**[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #l #H elim (tc_lfxs_inv_gref … H) -H *
[ #H destruct
| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
+lemma tc_lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻**[R, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻**[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #l #H elim (tc_lfxs_inv_gref … H) -H *
[ #_ #H destruct
| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
(* Basic forward lemmas *****************************************************)
-lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
-/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
+lemma tc_lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻**[R, ②{I}V.T] L2 → L1 ⦻**[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim H -L2
+/3 width=5 by lfxs_fwd_pair_sn, tc_lfxs_step_dx, inj/
qed-.
-lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
+lemma tc_lfxs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
+ L1.ⓑ{I}V ⦻**[R, T] L2.ⓑ{I}V.
+#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind … H) -H //
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 //
+lemma tc_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 (tc_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
-*)
+(* Basic_2A1: removed theorems 2:
+ TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2
*)
--- /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/i_static/tc_lfxs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Advanced properties ******************************************************)
+
+lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (tc_lfxs R T).
+/3 width=1 by lfxs_refl, inj/ qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
+lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⦻**[R, T] L.ⓑ{I}V2.
+#R #HR #L #V1 #V2 #H elim H -V2
+/3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,T. ∀R0:predicate …. R0 L1 →
+ (∀L,L2. L1 ⦻**[R, T] L → L ⦻*[R, T] L2 → R0 L → R0 L2) →
+ ∀L2. L1 ⦻**[R, T] L2 → R0 L2.
+#R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
+qed-.
+
+lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀L2,T. ∀R0:predicate …. R0 L2 →
+ (∀L1,L. L1 ⦻*[R, T] L → L ⦻**[R, T] L2 → R0 L → R0 L1) →
+ ∀L1. L1 ⦻**[R, T] L2 → R0 L1.
+#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 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/i_static/tc_lfxs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
+lemma tc_lfxs_fwd_length: ∀R,L1,L2,T. L1 ⦻**[R, T] L2 → |L1| = |L2|.
+#R #L1 #L2 #T #H elim H -L2
+[ #L2 #HL12 >(lfxs_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lfxs_fwd_length … HL2) -HL2 //
+]
+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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * 𝐍 ⦃ break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedEval $G $L $T1 $T2 }.
+++ /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( ⦃ term 46 G, break term 46 L ⦄ ⊢ break term 46 T1 ➡ * [ break term 46 h , break term 46 o ] 𝐍 ⦃ break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'PRedEval $h $o $G $L $T1 $T2 }.
+++ /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/predeval_4.ma".
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/csx.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
-
-definition cpre: relation4 genv lenv term term ≝
- λG,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 ∧ ⦃G, L⦄ ⊢ ➡ 𝐍⦃T2⦄.
-
-interpretation "evaluation for context-sensitive parallel reduction (term)"
- 'PRedEval G L T1 T2 = (cpre G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: nf2_sn3 *)
-lemma csx_cpre: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ➡* 𝐍⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1
-#T1 #_ #IHT1 elim (cnr_dec G L T1) /3 width=3 by ex_intro, conj/
-* #T #H1T1 #H2T1 elim (IHT1 … H2T1) -IHT1 -H2T1 /2 width=2 by cpr_cpx/
-#T2 * /4 width=3 by cprs_strap2, ex_intro, conj/
-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/computation/cprs_cprs.ma".
-include "basic_2/computation/cpre.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS *************)
-
-(* Main properties *********************************************************)
-
-(* Basic_1: was: nf2_pr3_confluence *)
-theorem cpre_mono: ∀G,L,T,T1. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T1⦄ → ∀T2. ⦃G, L⦄ ⊢ T ➡* 𝐍⦃T2⦄ → T1 = T2.
-#G #L #T #T1 * #H1T1 #H2T1 #T2 * #H1T2 #H2T2
-elim (cprs_conf … H1T1 … H1T2) -T #T #HT1
->(cprs_inv_cnr1 … HT1 H2T1) -T1 #HT2
->(cprs_inv_cnr1 … HT2 H2T2) -T2 //
-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/predeval_6.ma".
-include "basic_2/computation/cpxs.ma".
-include "basic_2/computation/csx.ma".
-
-(* EVALUATION FOR CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION ON TERMS ****)
-
-definition cpxe: ∀h. sd h → relation4 genv lenv term term ≝
- λh,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2 ∧ ⦃G, L⦄ ⊢ ⬈[h, o] 𝐍⦃T2⦄.
-
-interpretation "evaluation for context-sensitive extended parallel reduction (term)"
- 'PRedEval h o G L T1 T2 = (cpxe h o G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma csx_cpxe: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, o] T1 → ∃T2. ⦃G, L⦄ ⊢ T1 ⬈*[h, o] 𝐍⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1
-#T1 #_ #IHT1 elim (cnx_dec h o G L T1) /3 width=3 by ex_intro, conj/
-* #T #H1T1 #H2T1 elim (IHT1 … H1T1 H2T1) -IHT1 -H2T1
-#T2 * /4 width=3 by cpxs_strap2, ex_intro, conj/
-qed-.
qed.
lemma lfpx_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpx h G).
-#h #G #L2 #T1 #T2 #H #L1 #HL12 @(cpxs_ind … H) -T2
-/4 width=7 by lfpx_cpx_trans, cpxs_trans, lfpx_cpxs_conf/ (**) (* lfpx_fqup slows this down *)
+/3 width=6 by lfpx_cpx_conf, lfpx_cpx_trans, s_r_trans_LTC1/
qed-.
(* Advanced properties ******************************************************)
+++ /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/reduction/cpx_lreq.ma".
-include "basic_2/computation/cpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma lreq_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) (lreq 0 (∞)).
-/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
-qed-.
@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
-lapply (tndeq_tdeq_canc_dx … H … HV02) -H #HnTV0
+lapply (tdneq_tdeq_canc_dx … H … HV02) -H #HnTV0
elim (tdeq_dec h o T1 T0) #H
[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
(**************************************************************************)
include "basic_2/notation/relations/predtysnstar_5.ma".
+include "basic_2/i_static/tc_lfxs.ma".
include "basic_2/rt_transition/lfpx.ma".
(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
definition lfpxs: ∀h. relation4 genv term lenv lenv ≝
- λh,G,T. TC … (lfpx h G T).
+ λh,G. tc_lfxs (cpx h G).
interpretation
"uncounted parallel rt-computation on referred entries (local environment)"
/2 width=1 by inj/ qed.
(* Basic_2A1: was just: lpxs_strap1 *)
-lemma lfpxs_strap1: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
-/2 width=3 by step/ qed.
+lemma lfpxs_step_dx: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by tc_lfxs_step_dx/ qed.
(* Basic_2A1: was just: lpxs_strap2 *)
-lemma lfpxs_strap2: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
-/2 width=3 by TC_strap/ qed.
-(*
-(* Basic_2A1: was just: lpxs_pair_refl *)
-lemma lfpxs_pair_refl: ∀h,G,T,L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈*[h, T] L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+lemma lfpxs_step_sn: ∀h,G,T,L1,L,L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by tc_lfxs_step_sn/ qed.
(* Basic inversion lemmas ***************************************************)
-(* Basic_2A1: was just: lpxs_inv_atom1 *)
-lemma lfpxs_inv_atom1: ∀h,G,L2.T. ⦃G, ⋆⦄ ⊢ ⬈*[h, T] L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
+(* Basic_2A1: uses: lpxs_inv_atom1 *)
+lemma lfpxs_inv_atom1: ∀h,I,G,L2. ⦃G, ⋆⦄ ⊢ ⬈*[h, ⓪{I}] L2 → L2 = ⋆.
+/2 width=3 by tc_lfxs_inv_atom_sn/ qed-.
-(* Basic_2A1: was just: lpxs_inv_atom2 *)
-lemma lfpxs_inv_atom2: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-*)
+(* Basic_2A1: uses: lpxs_inv_atom2 *)
+lemma lfpxs_inv_atom2: ∀h,I,G,L1. ⦃G, L1⦄ ⊢ ⬈*[h, ⓪{I}] ⋆ → L1 = ⋆.
+/2 width=3 by tc_lfxs_inv_atom_dx/ qed-.
-(* Basic_2A1: removed theorems 1:
- lpxs_pair
-*)
+(* Basic forward lemmas *****************************************************)
+
+lemma lfpxs_fwd_pair_sn: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈*[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, V] L2.
+/2 width=3 by tc_lfxs_fwd_pair_sn/ qed-.
+
+lemma lfpxs_flat_dx: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈*[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, T] L2.
+/2 width=3 by tc_lfxs_fwd_flat_dx/ qed-.
(* Properties with uncounted context-sensitive rt-computation for terms *****)
+(* Basic_2A1: uses: lpxs_pair lpxs_pair_refl *)
lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
-#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
-/3 width=3 by lfpxs_strap1, lfpx_pair/
-qed.
+/2 width=1 by tc_lfxs_pair/ qed.
-(* Basic_2A1: was just: lpxs_cpx_trans *)
+(* Basic_2A1: uses: lpxs_cpx_trans *)
lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
-/3 width=5 by s_r_trans_LTC2, lfpx_cpxs_trans/ qed-.
+#h #G @s_r_trans_LTC2 @lfpx_cpxs_trans (**) (* auto fails *)
+qed-.
(* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_LTC1 *)
-(* Basic_2A1: was just: lpxs_cpxs_trans *)
+(* Basic_2A1: uses: lpxs_cpxs_trans *)
lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
#h #G @s_r_to_s_rs_trans @s_r_trans_LTC2
@s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_fqup.ma".
+include "basic_2/i_static/tc_lfxs_fqup.ma".
include "basic_2/rt_computation/lfpxs.ma".
(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lpxs_refl *)
+lemma lfpxs_refl: ∀h,G,T. reflexive … (lfpxs h G T).
+/2 width=1 by tc_lfxs_refl/ qed.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lfpxs_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈*[h, ⓑ{p,I}V.T] L2 →
+ ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈*[h, T] L2.ⓑ{I}V.
+/2 width=2 by tc_lfxs_fwd_bind_dx/ qed-.
+
(* Advanced eliminators *****************************************************)
-(* Basic_2A1: was just: lpxs_ind *)
-lemma lfpxs_ind: ∀h,G,L1,T. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → R L2.
-#h #G #L1 #T #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
+(* Basic_2A1: uses: lpxs_ind *)
+lemma lfpxs_ind_sn: ∀h,G,L1,T. ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L → ⦃G, L⦄ ⊢ ⬈[h, T] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → R L2.
+#h #G @tc_lfxs_ind_sn // (**) (* auto fails *)
+qed-.
-(* Basic_2A1: was just: lpxs_ind_dx *)
+(* Basic_2A1: uses: lpxs_ind_dx *)
lemma lfpxs_ind_dx: ∀h,G,L2,T. ∀R:predicate lenv. R L2 →
(∀L1,L. ⦃G, L1⦄ ⊢ ⬈[h, T] L → ⦃G, L⦄ ⊢ ⬈*[h, T] L2 → R L → R L1) →
∀L1. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → R L1.
-#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: was just: lpxs_refl *)
-lemma lfpxs_refl: ∀h,G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, T] L.
-/2 width=1 by lfpx_lfpxs/ qed.
+#h #G @tc_lfxs_ind_dx // (**) (* auto fails *)
+qed-.
#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
/3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
qed-.
+
+lemma lpxs_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
+ #K0 #V0 #H1KL1 #_ #H destruct
+ elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
+ #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
+ /2 width=4 by fqu_lref_O, ex3_intro/
+| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
+ [ elim (lleq_inv_bind … H)
+ | elim (lleq_inv_flat … H)
+ ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
+| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
+ /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
+| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
+ /2 width=4 by fqu_flat_dx, ex3_intro/
+| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+ elim (drop_O1_le (Ⓕ) (k+1) K1)
+ [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
+ #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
+ #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
+ /3 width=4 by fqu_drop, ex3_intro/
+ | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
+ lapply (lleq_fwd_length … H2KL1) //
+ ]
+]
+qed-.
+
+lemma lpxs_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fquq_inv_gen … H) -H
+[ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+ /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
+
+lemma lpxs_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
+ /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
+ #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
+ /3 width=5 by fqup_strap1, ex3_intro/
+]
+qed-.
+
+lemma lpxs_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
+ ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
+elim (fqus_inv_gen … H) -H
+[ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
+ /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
+]
+qed-.
-(* Basic forward lemmas *****************************************************)
+(**************************************************************************)
+(* ___ *)
+(* ||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/i_static/tc_lfxs_length.ma".
+include "basic_2/rt_computation/lfpxs.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
(* Forward lemmas with length for local environments ************************)
-(* Basic_2A1: was just: lpxs_fwd_length *)
-lemma lfpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h, o] L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ qed-.
+(* Basic_2A1: uses: lpxs_fwd_length *)
+lemma lfpxs_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → |L1| = |L2|.
+/2 width=3 by tc_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/static/lfdeq_lfdeq.ma".
+include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_computation/lfpxs_fqup.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+(* Properties with degree-based equivalence on referred entries *************)
+
+(* Basic_2A1: was: lleq_lpxs_trans *)
+lemma lfdeq_lfpxs_trans: ∀h,o,G,L2,K2,T. ⦃G, L2⦄ ⊢ ⬈*[h, T] K2 →
+ ∀L1. L1 ≡[h, o, T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h, T] K1 & K1 ≡[h, o, T] K2.
+#h #o #G #L2 #K2 #T #H @(lfpxs_ind_sn … H) -K2 /2 width=3 by ex2_intro/
+#K #K2 #_ #HK2 #IH #L1 #HT elim (IH … HT) -L2
+#L #HL1 #HT elim (lfdeq_lfpx_trans … HK2 … HT) -K
+/3 width=3 by lfpxs_step_dx, ex2_intro/
+qed-.
+
+(* Basic_2A1: was: lpxs_nlleq_inv_step_sn *)
+lemma lfpxs_lfdneq_inv_step_sn: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) →
+ ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h, T] L & L1 ≡[h, o, T] L → ⊥ & ⦃G, L⦄ ⊢ ⬈*[h, T] L0 & L0 ≡[h, o, T] L2.
+#h #o #G #L1 #L2 #T #H @(lfpxs_ind_dx … H) -L1
+[ #H elim H -H //
+| #L1 #L #H1 #H2 #IH2 #H12 elim (lfdeq_dec h o L1 L T) #H
+ [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lfdeq_trans/ -H12
+ #L0 #L3 #H1 #H2 #H3 #H4 lapply (lfdeq_lfdneq_trans … H … H2) -H2
+ #H2 elim (lfdeq_lfpx_trans … H1 … H) -L
+ #L #H1 #H lapply (lfdneq_lfdeq_div … H … H2) -H2
+ #H2 elim (lfdeq_lfpxs_trans … H3 … H) -L0
+ /3 width=8 by lfdeq_trans, ex4_2_intro/
+ | -H12 -IH2 /3 width=6 by ex4_2_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/rt_computation/lfpxs.ma".
+
+(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENV.S ON REFERRED ENTRIES ****)
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: uses: lpxs_trans *)
+theorem lfpxs_trans: ∀h,G,T. Transitive … (lfpxs h G T).
+#h #G #T #L1 #L #HL1 #L2 #HL2 @(trans_TC … HL1 HL2) (**) (* auto fails *)
+qed-.
(* Basic eliminators ********************************************************)
-(* Basic_2A1: was: lsx_ind *)
+(* Basic_2A1: uses: lsx_ind *)
lemma lfsx_ind: ∀h,o,G,T. ∀R:predicate lenv.
(∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
(∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → R L2) →
(* Basic properties *********************************************************)
-(* Basic_2A1: was: lsx_intro *)
+(* Basic_2A1: uses: lsx_intro *)
lemma lfsx_intro: ∀h,o,G,L1,T.
(∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
/5 width=1 by lfdeq_sym, SN_intro/ qed.
-(* Basic_2A1: was: lsx_sort *)
+(* Basic_2A1: uses: lsx_sort *)
lemma lfsx_sort: ∀h,o,G,L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
#h #o #G #L1 #s @lfsx_intro
#L2 #H #Hs elim Hs -Hs elim (lfpx_inv_sort … H) -H *
]
qed.
-(* Basic_2A1: was: lsx_gref *)
+(* Basic_2A1: uses: lsx_gref *)
lemma lfsx_gref: ∀h,o,G,L,p. G ⊢ ⬈*[h, o, §p] 𝐒⦃L⦄.
#h #o #G #L1 #s @lfsx_intro
#L2 #H #Hs elim Hs -Hs elim (lfpx_inv_gref … H) -H *
]
qed.
-(* Basic_2A1: removed theorems 2:
+(* Basic_2A1: removed theorems 9:
lsx_ge_up lsx_ge
+ lsxa_ind lsxa_intro lsxa_lleq_trans lsxa_lpxs_trans lsxa_intro_lpx lsx_lsxa lsxa_inv_lsx
*)
(* Advanced properties ******************************************************)
-(* Basic_2A1: was: lsx_atom *)
+(* Basic_2A1: uses: lsx_atom *)
lemma lfsx_atom: ∀h,o,G,T. G ⊢ ⬈*[h, o, T] 𝐒⦃⋆⦄.
#h #o #G #T @lfsx_intro
#Y #H #HI lapply (lfpx_inv_atom_sn … H) -H
--- /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/rt_computation/lfpxs_lfdeq.ma".
+include "basic_2/rt_computation/lfpxs_cpxs.ma".
+include "basic_2/rt_computation/lfpxs_lfpxs.ma".
+include "basic_2/rt_computation/lfsx_lfsx.ma".
+
+(* STRONGLY NORMALIZING LOCAL ENV.S FOR UNCOUNTED PARALLEL RT-TRANSITION ****)
+
+(* Properties with uncounted rt-computation on referred entries *************)
+
+(* Basic_2A1: uses: lsx_intro_alt *)
+lemma lfsx_intro_lfpxs: ∀h,o,G,L1,T.
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄) →
+ G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄.
+/4 width=1 by lfpx_lfpxs, lfsx_intro/ qed-.
+
+(* Basic_2A1: uses: lsx_lpxs_trans *)
+lemma lfsx_lfpxs_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
+#h #o #G #L1 #T #HL1 #L2 #H @(lfpxs_ind … H) -L2
+/2 width=3 by lfsx_lfpx_trans/
+qed-.
+
+(* Eliminators with uncounted rt-computation on referred entries ************)
+
+lemma lfsx_ind_lfpxs_lfdeq: ∀h,o,G,T. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h, T] L0 → ∀L2. L0 ≡[h, o, T] L2 → R L2.
+#h #o #G #T #R #IH #L1 #H @(lfsx_ind … H) -L1
+#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
+@IH -IH /3 width=3 by lfsx_lfpxs_trans, lfsx_lfdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
+lapply (lfdeq_lfdneq_trans … HL02 … HnLK2) -HnLK2 #H
+elim (lfdeq_lfpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
+lapply (lfdneq_lfdeq_canc_dx … H … HK02) -H #HnLK0
+elim (lfdeq_dec h o L1 L0 T) #H
+[ lapply (lfdeq_lfdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
+ lapply (lfpxs_trans … HL10 … HLK0) -L0 #H10
+ elim (lfpxs_lfdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by lfdeq_trans/
+| elim (lfpxs_lfdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
+ elim (lfdeq_lfpxs_trans … HLK0 … HKL0) -L0
+ /3 width=8 by lfpxs_trans, lfdeq_trans/
+]
+qed-.
+
+(* Basic_2A1: uses: lsx_ind_alt *)
+lemma lfsx_ind_lfpxs: ∀h,o,G,T. ∀R:predicate lenv.
+ (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h, T] L2 → (L1 ≡[h, o, T] L2 → ⊥) → R L2) →
+ R L1
+ ) →
+ ∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → R L.
+#h #o #G #T #R #IH #L #HL
+@(lfsx_ind_lfpxs_lfdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
+qed-.
+
+(* Advanced properties ******************************************************)
+
+fact lsx_bind_lpxs_aux: ∀h,o,p,I,G,L1,V. G ⊢ ⬈*[h, o, V] 𝐒⦃L1⦄ →
+ ∀Y,T. G ⊢ ⬈*[h, o, T] 𝐒⦃Y⦄ →
+ ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ⬈*[h, ⓑ{p,I}V.T] L2 →
+ G ⊢ ⬈*[h, o, ⓑ{p,I}V.T] 𝐒⦃L2⦄.
+#h #o #p #I #G #L1 #V #H @(lfsx_ind_lfpxs … H) -L1
+#L1 #HL1 #IHL1 #Y #T #H @(lfsx_ind_lfpxs … H) -Y
+#Y #HY #IHY #L2 #H #HL12 destruct @lfsx_intro_lfpxs
+#L0 #HL20 lapply (lfpxs_trans … HL12 … HL20)
+#HL10 #H elim (lfdneq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
+[ #HnV elim (lfdeq_dec h o L1 L2 V)
+ [ #HV @(IHL1 … L0) -IHL1
+ [2: /3 width=5 by lfdeq_canc_sn/
+ |5: // |3: skip
+ |6: //
+
+ lfdeq_lfdneq_trans/
+
+ /3 width=5 by lfsx_lfpxs_trans, lfpxs_pair, lfdeq_canc_sn/ (**) (* full auto too slow *)
+ | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
+ ]
+| /3 width=4 by lpxs_pair/
+]
+qed-.
+
+lemma lsx_bind: ∀h,o,a,I,G,L,V,l. G ⊢ ⬈*[h, o, V, l] L →
+ ∀T. G ⊢ ⬈*[h, o, T, ⫯l] L.ⓑ{I}V →
+ G ⊢ ⬈*[h, o, ⓑ{a,I}V.T, l] L.
+/2 width=3 by lsx_bind_lpxs_aux/ qed.
+
+lemma lsx_flat_lpxs: ∀h,o,I,G,L1,V,l. G ⊢ ⬈*[h, o, V, l] L1 →
+ ∀L2,T. G ⊢ ⬈*[h, o, T, l] L2 → ⦃G, L1⦄ ⊢ ⬈*[h, o] L2 →
+ G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L2.
+#h #o #I #G #L1 #V #l #H @(lsx_ind_lfpxs … H) -L1
+#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_lfpxs … H) -L2
+#L2 #HL2 #IHL2 #HL12 @lsx_intro_lfpxs
+#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
+#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (lleq_dec V L1 L2 l)
+ [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
+ | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
+ ]
+| /3 width=1 by/
+]
+qed-.
+
+lemma lsx_flat: ∀h,o,I,G,L,V,l. G ⊢ ⬈*[h, o, V, l] L →
+ ∀T. G ⊢ ⬈*[h, o, T, l] L → G ⊢ ⬈*[h, o, ⓕ{I}V.T, l] L.
+/2 width=3 by lsx_flat_lpxs/ qed.
(* Advanced properties ******************************************************)
-(* Basic_2A1: was just: lsx_lleq_trans *)
+(* Basic_2A1: uses: lsx_lleq_trans *)
lemma lfsx_lfdeq_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
∀L2. L1 ≡[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
#h #o #G #L1 #T #H @(lfsx_ind … H) -L1
/4 width=5 by lfdeq_repl/
qed-.
-(* Basic_2A1: was: lsx_lpx_trans *)
+(* Basic_2A1: uses: lsx_lpx_trans *)
lemma lfsx_lfpx_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
#h #o #G #L1 #T #H @(lfsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
(* Advanced forward lemmas **************************************************)
-(* Basic_2A1: includes: lsx_fwd_bind_sn lsx_fwd_flat_sn *)
-(* Basic_2A1: was: lsx_fwd_pair_sn *)
+(* Basic_2A1: uses: lsx_fwd_pair_sn lsx_fwd_bind_sn lsx_fwd_flat_sn *)
lemma lfsx_fwd_pair_sn: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L⦄ →
G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄.
#h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
/6 width=4 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_pair_sn/
qed-.
-(* Basic_2A1: was: lsx_fwd_flat_dx *)
+(* Basic_2A1: uses: lsx_fwd_flat_dx *)
lemma lfsx_fwd_flat_dx: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
#h #o #I #G #L #V #T #H @(lfsx_ind … H) -L
(* Advanced inversion lemmas ************************************************)
-(* Basic_2A1: was: lsx_inv_flat *)
+(* Basic_2A1: uses: lsx_inv_flat *)
lemma lfsx_inv_flat: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ →
G ⊢ ⬈*[h, o, V] 𝐒⦃L⦄ ∧ G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄.
/3 width=3 by lfsx_fwd_pair_sn, lfsx_fwd_flat_dx, conj/ 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_lleq.ma".
-include "basic_2/reduction/lpx_lleq.ma".
-include "basic_2/computation/cpxs_lreq.ma".
-include "basic_2/computation/lpxs_drop.ma".
-include "basic_2/computation/lpxs_cpxs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma lleq_lpxs_trans: ∀h,o,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, o] K2 →
- ∀L1,T,l. L1 ≡[T, l] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, o] K1 & K1 ≡[T, l] K2.
-#h #o #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
-#K #K2 #_ #HK2 #IH #L1 #T #l #HT elim (IH … HT) -L2
-#L #HL1 #HT elim (lleq_lpx_trans … HK2 … HT) -K
-/3 width=3 by lpxs_strap1, ex2_intro/
-qed-.
-
-lemma lpxs_nlleq_inv_step_sn: ∀h,o,G,L1,L2,T,l. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) →
- ∃∃L,L0. ⦃G, L1⦄ ⊢ ➡[h, o] L & L1 ≡[T, l] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, o] L0 & L0 ≡[T, l] L2.
-#h #o #G #L1 #L2 #T #l #H @(lpxs_ind_dx … H) -L1
-[ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
- [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
- #L0 #L3 #H1 #H2 #H3 #H4 lapply (lleq_nlleq_trans … H … H2) -H2
- #H2 elim (lleq_lpx_trans … H1 … H) -L
- #L #H1 #H lapply (nlleq_lleq_div … H … H2) -H2
- #H2 elim (lleq_lpxs_trans … H3 … H) -L0
- /3 width=8 by lleq_trans, ex4_2_intro/
- | -H12 -IH2 /3 width=6 by ex4_2_intro/
- ]
-]
-qed-.
-
-lemma lpxs_lleq_fqu_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 … H1) -H1
- #K0 #V0 #H1KL1 #_ #H destruct
- elim (lleq_inv_lref_ge_dx … H2 ? I L1 V1) -H2 //
- #K1 #H #H2KL1 lapply (drop_inv_O2 … H) -H #H destruct
- /2 width=4 by fqu_lref_O, ex3_intro/
-| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
- [ elim (lleq_inv_bind … H)
- | elim (lleq_inv_flat … H)
- ] -H /2 width=4 by fqu_pair_sn, ex3_intro/
-| #a #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_bind_O … H) -H
- /3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
-| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat … H) -H
- /2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #k #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (drop_O1_le (Ⓕ) (k+1) K1)
- [ #K #HK1 lapply (lleq_inv_lift_le … H2KL1 … HK1 HL1 … HTU1 ?) -H2KL1 //
- #H2KL elim (lpxs_drop_trans_O1 … H1KL1 … HL1) -L1
- #K0 #HK10 #H1KL lapply (drop_mono … HK10 … HK1) -HK10 #H destruct
- /3 width=4 by fqu_drop, ex3_intro/
- | lapply (drop_fwd_length_le2 … HL1) -L -T1 -o
- lapply (lleq_fwd_length … H2KL1) //
- ]
-]
-qed-.
-
-lemma lpxs_lleq_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
-elim (fquq_inv_gen … H) -H
-[ #H elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqu_fquq, ex3_intro/
-| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
-]
-qed-.
-
-lemma lpxs_lleq_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[ #G2 #L2 #T2 #H #K1 #H1KL1 #H2KL1 elim (lpxs_lleq_fqu_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqu_fqup, ex3_intro/
-| #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #K1 #H1KL1 #H2KL1 elim (IHT1 … H2KL1) // -L1
- #K #HT1 #H1KL #H2KL elim (lpxs_lleq_fqu_trans … HT2 … H1KL H2KL) -L
- /3 width=5 by fqup_strap1, ex3_intro/
-]
-qed-.
-
-lemma lpxs_lleq_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 → K1 ≡[T1, 0] L1 →
- ∃∃K2. ⦃G1, K1, T1⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2 & K2 ≡[T2, 0] L2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #K1 #H1KL1 #H2KL1
-elim (fqus_inv_gen … H) -H
-[ #H elim (lpxs_lleq_fqup_trans … H … H1KL1 H2KL1) -L1
- /3 width=4 by fqup_fqus, ex3_intro/
-| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
-]
-qed-.
-
-fact lreq_lpxs_trans_lleq_aux: ∀h,o,G,L1,L0,l,k. L1 ⩬[l, k] L0 → k = ∞ →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
- ∃∃L. L ⩬[l, k] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
- (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
-#h #o #G #L1 #L0 #l #k #H elim H -L1 -L0 -l -k
-[ #l #k #_ #L2 #H >(lpxs_inv_atom1 … H) -H
- /3 width=5 by ex3_intro, conj/
-| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
-| #I #L1 #L0 #V1 #k #HL10 #IHL10 #Hm #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- lapply (ysucc_inv_Y_dx … Hm) -Hm #Hm
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_pair/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
-| #I1 #I0 #L1 #L0 #V1 #V0 #l #k #HL10 #IHL10 #Hm #Y #H
- elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
- elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro … (L.ⓑ{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
- #T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_lreq_repl … H) -H /3 width=1 by lreq_sym, lreq_succ/
-]
-qed-.
-
-lemma lreq_lpxs_trans_lleq: ∀h,o,G,L1,L0,l. L1 ⩬[l, ∞] L0 →
- ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, o] L2 →
- ∃∃L. L ⩬[l, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, o] L &
- (∀T. L0 ≡[T, l] L2 ↔ L1 ≡[T, l] L).
-/2 width=1 by lreq_lpxs_trans_lleq_aux/ 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/computation/lpxs.ma".
-
-(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
-
-(* Main properties **********************************************************)
-
-theorem lpxs_trans: ∀h,o,G. Transitive … (lpxs h o G).
-/2 width=3 by trans_TC/ 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/snalt_6.ma".
-include "basic_2/computation/lpxs_lleq.ma".
-include "basic_2/computation/lsx.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* alternative definition of lsx *)
-definition lsxa: ∀h. sd h → relation4 ynat term genv lenv ≝
- λh,o,l,T,G. SN … (lpxs h o G) (lleq l T).
-
-interpretation
- "extended strong normalization (local environment) alternative"
- 'SNAlt h o l T G L = (lsxa h o T l G L).
-
-(* Basic eliminators ********************************************************)
-
-lemma lsxa_ind: ∀h,o,G,T,l. ∀R:predicate lenv.
- (∀L1. G ⊢ ⬊⬊*[h, o, T, l] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⬊⬊*[h, o, T, l] L → R L.
-#h #o #G #T #l #R #H0 #L1 #H elim H -L1
-/5 width=1 by lleq_sym, SN_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsxa_intro: ∀h,o,G,L1,T,l.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
- G ⊢ ⬊⬊*[h, o, T, l] L1.
-/5 width=1 by lleq_sym, SN_intro/ qed.
-
-fact lsxa_intro_aux: ∀h,o,G,L1,T,l.
- (∀L,L2. ⦃G, L⦄ ⊢ ➡*[h, o] L2 → L1 ≡[T, l] L → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
- G ⊢ ⬊⬊*[h, o, T, l] L1.
-/4 width=3 by lsxa_intro/ qed-.
-
-lemma lsxa_lleq_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
- ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
-#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1
-#L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
-#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans … HLK2 … HL12) -HLK2
-/5 width=4 by lleq_canc_sn, lleq_trans/
-qed-.
-
-lemma lsxa_lpxs_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊⬊*[h, o, T, l] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊⬊*[h, o, T, l] L2.
-#h #o #T #G #L1 #l #H @(lsxa_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
-qed-.
-
-lemma lsxa_intro_lpx: ∀h,o,G,L1,T,l.
- (∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊⬊*[h, o, T, l] L2) →
- G ⊢ ⬊⬊*[h, o, T, l] L1.
-#h #o #G #L1 #T #l #IH @lsxa_intro_aux
-#L #L2 #H @(lpxs_ind_dx … H) -L
-[ #H destruct #H elim H //
-| #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
- #HnT #HL0 #HL2 #_ #HT #_ elim (lleq_lpx_trans … HL0 … HT) -L0
- #L0 #HL10 #HL0 @(lsxa_lpxs_trans … HL2) -HL2
- /5 width=3 by lsxa_lleq_trans, lleq_trans/
-]
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lsx_lsxa: ∀h,o,G,L,T,l. G ⊢ ⬊*[h, o, T, l] L → G ⊢ ⬊⬊*[h, o, T, l] L.
-#h #o #G #L #T #l #H @(lsx_ind … H) -L
-/4 width=1 by lsxa_intro_lpx/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lsxa_inv_lsx: ∀h,o,G,L,T,l. G ⊢ ⬊⬊*[h, o, T, l] L → G ⊢ ⬊*[h, o, T, l] L.
-#h #o #G #L #T #l #H @(lsxa_ind … H) -L
-/4 width=1 by lsx_intro, lpx_lpxs/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lsx_intro_alt: ∀h,o,G,L1,T,l.
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → G ⊢ ⬊*[h, o, T, l] L2) →
- G ⊢ ⬊*[h, o, T, l] L1.
-/6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
-
-lemma lsx_lpxs_trans: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → G ⊢ ⬊*[h, o, T, l] L2.
-/4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma lsx_ind_alt: ∀h,o,G,T,l. ∀R:predicate lenv.
- (∀L1. G ⊢ ⬊*[h, o, T, l] L1 →
- (∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → (L1 ≡[T, l] L2 → ⊥) → R L2) →
- R L1
- ) →
- ∀L. G ⊢ ⬊*[h, o, T, l] L → R L.
-#h #o #G #T #l #R #IH #L #H @(lsxa_ind h o G T l … L)
-/4 width=1 by lsxa_inv_lsx, lsx_lsxa/
-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/computation/lpxs_lpxs.ma".
-include "basic_2/computation/lsx_alt.ma".
-
-(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
-
-(* Advanced properties ******************************************************)
-
-fact lsx_bind_lpxs_aux: ∀h,o,a,I,G,L1,V,l. G ⊢ ⬊*[h, o, V, l] L1 →
- ∀Y,T. G ⊢ ⬊*[h, o, T, ⫯l] Y →
- ∀L2. Y = L2.ⓑ{I}V → ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
- G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L2.
-#h #o #a #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
-#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind_alt … H) -Y
-#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro_alt
-#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (nlleq_inv_bind … H) -H [ -HL1 -IHY | -HY -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 l)
- [ #HV @(IHL1 … L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
- | -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
- ]
-| /3 width=4 by lpxs_pair/
-]
-qed-.
-
-lemma lsx_bind: ∀h,o,a,I,G,L,V,l. G ⊢ ⬊*[h, o, V, l] L →
- ∀T. G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V →
- G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L.
-/2 width=3 by lsx_bind_lpxs_aux/ qed.
-
-lemma lsx_flat_lpxs: ∀h,o,I,G,L1,V,l. G ⊢ ⬊*[h, o, V, l] L1 →
- ∀L2,T. G ⊢ ⬊*[h, o, T, l] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
- G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L2.
-#h #o #I #G #L1 #V #l #H @(lsx_ind_alt … H) -L1
-#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_alt … H) -L2
-#L2 #HL2 #IHL2 #HL12 @lsx_intro_alt
-#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (nlleq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 l)
- [ #HV @(IHL1 … L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
- | -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
- ]
-| /3 width=1 by/
-]
-qed-.
-
-lemma lsx_flat: ∀h,o,I,G,L,V,l. G ⊢ ⬊*[h, o, V, l] L →
- ∀T. G ⊢ ⬊*[h, o, T, l] L → G ⊢ ⬊*[h, o, ⓕ{I}V.T, l] L.
-/2 width=3 by lsx_flat_lpxs/ qed.
cpxs.ma cpxs_tdeq.ma cpxs_theq.ma cpxs_theq_vector.ma cpxs_drops.ma cpxs_lsubr.ma cpxs_aaa.ma cpxs_lfpx.ma cpxs_cnx.ma cpxs_cpxs.ma
-lfpxs.ma lfpxs_fqup.ma lfpxs_cpxs.ma
+lfpxs.ma lfpxs_length.ma lfpxs_fqup.ma lfpxs_lfdeq.ma lfpxs_cpxs.ma lfpxs_lfpxs.ma
csx.ma csx_simple.ma csx_simple_theq.ma csx_drops.ma csx_lsubr.ma csx_gcp.ma csx_gcr.ma csx_lfpx.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
csx_vector.ma csx_cnx_vector.ma csx_csx_vector.ma
lfsx.ma lfsx_fqup.ma lfsx_lfsx.ma
(* Basic inversion lemmas ***************************************************)
+(* Basic_2A1: uses: lpr_inv_atom1 *)
lemma lfpr_inv_atom_sn: ∀h,I,G,Y2. ⦃G, ⋆⦄ ⊢ ➡[h, ⓪{I}] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
+(* Basic_2A1: uses: lpr_inv_atom2 *)
lemma lfpr_inv_atom_dx: ∀h,I,G,Y1. ⦃G, Y1⦄ ⊢ ➡[h, ⓪{I}] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lfpr_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
- â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, â\93\91{p,I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
-/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+lemma lfpr_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
+ â¦\83G, L1â¦\84 â\8a¢ â\9e¡[h, â\91¡{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
lemma lfpr_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ➡[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-lemma lfpr_fwd_flat_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
-/2 width=3 by lfxs_fwd_flat_sn/ qed-.
-
lemma lfpr_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ➡[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-lemma lfpr_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ➡[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ➡[h, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-(* Basic_2A1: removed theorems 16:
- lpr_inv_atom1 lpr_inv_pair1 lpr_inv_atom2 lpr_inv_pair2
- lpr_refl lpr_pair
- lpr_fwd_length lpr_lpx
- lpr_drop_conf drop_lpr_trans lpr_drop_trans_O1
+(* Basic_2A1: removed theorems 5:
+ lpr_inv_pair1 lpr_inv_pair2
cpr_conf_lpr lpr_cpr_conf_dx lpr_cpr_conf_sn
- fqu_lpr_trans fquq_lpr_trans
*)
(* Basic_1: removed theorems 7:
wcpr0_gen_sort wcpr0_gen_head
(* Properties with generic slicing for local environments *******************)
+(* Basic_2A1: uses: drop_lpr_trans *)
lemma drops_lfpr_trans: ∀h,G. dedropable_sn (cpm 0 h G).
/3 width=6 by lfxs_liftable_dedropable_sn, cpm_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
+(* Basic_2A1: uses: lpr_drop_conf *)
lemma lfpr_drops_conf: ∀h,G. dropable_sn (cpm 0 h G).
/2 width=5 by lfxs_dropable_sn/ qed-.
+(* Basic_2A1: uses: lpr_drop_trans_O1 *)
lemma lfpr_drops_trans: ∀h,G. dropable_dx (cpm 0 h G).
/2 width=5 by lfxs_dropable_dx/ qed-.
(* Advanced properties ******************************************************)
(* Note: lemma 250 *)
+(* Basic_2A1: uses: lpr_refl *)
lemma lfpr_refl: ∀h,G,T. reflexive … (lfpr h G T).
/2 width=1 by lfxs_refl/ qed.
+(* Basic_2A1: uses: lpr_pair *)
lemma lfpr_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ➡[h, T] L.ⓑ{I}V2.
/2 width=1 by lfxs_pair/ qed.
/3 width=9 by fqu_drop, drops_refl, drops_drop, ex3_2_intro/
qed-.
+(* Basic_2A1: uses: fqu_lpr_trans *)
lemma fqu_cpr_trans_sn: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐ ⦃G2, L2, U2⦄.
]
qed-.
+(* Basic_2A1: uses: fquq_lpr_trans *)
lemma fquq_cpr_trans_sn: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
∀U2. ⦃G2, L2⦄ ⊢ T2 ➡[h] U2 →
∃∃L,U1. ⦃G1, L1⦄ ⊢ ➡[h, T1] L & ⦃G1, L1⦄ ⊢ T1 ➡[h] U1 & ⦃G1, L, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
(* Forward lemmas with length for local environments ************************)
+(* Basic_2A1: uses: lpr_fwd_length *)
lemma lfpr_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → |L1| = |L2|.
/2 width=3 by lfxs_fwd_length/ qed-.
(* Fwd. lemmas with unc. rt-transition for local env.s on referred entries **)
+(* Basic_2A1: uses: lpr_lpx *)
lemma lfpr_fwd_lfpx: ∀h,T,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
/3 width=3 by cpm_fwd_cpx, lfxs_co/ qed-.
(* Basic inversion lemmas ***************************************************)
+(* Basic_2A1: uses: lpx_inv_atom1 *)
lemma lfpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
/2 width=3 by lfxs_inv_atom_sn/ qed-.
+(* Basic_2A1: uses: lpx_inv_atom2 *)
lemma lfpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
/2 width=3 by lfxs_inv_atom_dx/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lfpx_fwd_bind_sn: ∀h,p,I,G,L1,L2,V,T.
- â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\93\91{p,I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
+ â¦\83G, L1â¦\84 â\8a¢ â¬\88[h, â\91¡{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
lemma lfpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-lemma lfpx_fwd_flat_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=3 by lfxs_fwd_flat_sn/ qed-.
-
lemma lfpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-(* Basic_2A1: removed theorems 14:
- lpx_refl lpx_pair lpx_fwd_length
- lpx_inv_atom1 lpx_inv_pair1 lpx_inv_atom2 lpx_inv_pair2 lpx_inv_pair
- lpx_drop_conf drop_lpx_trans lpx_drop_trans_O1
- lpx_cpx_frees_trans cpx_frees_trans lpx_frees_trans
+(* Basic_2A1: removed theorems 3:
+ lpx_inv_pair1 lpx_inv_pair2 lpx_inv_pair
*)
(* Properties with atomic arity assignment for terms ************************)
(* Note: lemma 500 *)
-(* Basic_2A1: was: cpx_lpx_aaa_conf *)
+(* Basic_2A1: uses: cpx_lpx_aaa_conf *)
lemma cpx_aaa_conf_lfpx: ∀h,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
(* Properties with generic slicing for local environments *******************)
+(* Basic_2A1: uses: drop_lpx_trans *)
lemma drops_lfpx_trans: ∀h,G. dedropable_sn (cpx h G).
/3 width=6 by lfxs_liftable_dedropable_sn, cpx_lifts_sn/ qed-.
(* Inversion lemmas with generic slicing for local environments *************)
+(* Basic_2A1: uses: lpx_drop_conf *)
lemma lfpx_drops_conf: ∀h,G. dropable_sn (cpx h G).
/2 width=5 by lfxs_dropable_sn/ qed-.
+(* Basic_2A1: uses: lpx_drop_trans_O1 *)
lemma lfpx_drops_trans: ∀h,G. dropable_dx (cpx h G).
/2 width=5 by lfxs_dropable_dx/ qed-.
(* Advanced properties ******************************************************)
+(* Basic_2A1: uses: lpx_refl lpx_pair *)
lemma lfpx_refl: ∀h,G,T. reflexive … (lfpx h G T).
/2 width=1 by lfxs_refl/ qed.
+(* Basic_2A1: uses: lpx_refl lpx_pair *)
lemma lfpx_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
/2 width=1 by lfxs_pair/ qed.
(* Properties with context-sensitive free variables *************************)
-(* Basic_2A1: was: lpx_cpx_frees_trans *)
+(* Basic_2A1: uses: lpx_cpx_frees_trans *)
lemma cpx_frees_conf_lfpx: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
∀L2. L1 ⦻*[cpx h G, cfull, f1] L2 →
∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
]
qed-.
-(* Basic_2A1: was: cpx_frees_trans *)
+(* Basic_2A1: uses: cpx_frees_trans *)
lemma cpx_frees_conf: ∀h,G. R_frees_confluent (cpx h G).
/3 width=7 by cpx_frees_conf_lfpx, lexs_refl/ qed-.
-(* Basic_2A1: was: lpx_frees_trans *)
+(* Basic_2A1: uses: lpx_frees_trans *)
lemma lfpx_frees_conf: ∀h,G. lexs_frees_confluent (cpx h G) cfull.
/2 width=7 by cpx_frees_conf_lfpx/ qed-.
(* Forward lemmas with length for local environments ************************)
+(* Basic_2A1: uses: lpx_fwd_length *)
lemma lfpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
/2 width=3 by lfxs_fwd_length/ qed-.
elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
qed-.
-(* Basic_2A1: was just: cpx_lleq_conf *)
+(* Basic_2A1: uses: cpx_lleq_conf *)
lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
∀L2. L0 ≡[h, o, T0] L2 →
∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≡[h, o] T.
/2 width=3 by lfxs_refl, ex2_intro/
qed-.
-(* Basic_2A1: was just: lleq_cpx_trans *)
+(* Basic_2A1: uses: lleq_cpx_trans *)
lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≡[h, o, T0] L0 →
∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≡[h, o] T1.
lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
/3 width=6 by lfpx_frees_conf, cpx_tdeq_conf_lexs, frees_lfdeq_conf_lexs, lfxs_conf/ qed-.
-(* Basic_2A1: was just: lleq_lpx_trans *)
+(* Basic_2A1: uses: lleq_lpx_trans *)
lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
∀L1. L1 ≡[h, o, T] L2 →
∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≡[h, o, T] K2.
/3 width=5 by frees_tdeq_conf, ex2_intro/
qed-.
+(* Basic_2A1: uses: lleq_sym *)
lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
#h #o #T #L1 #L2 *
/4 width=7 by frees_tdeq_conf_lexs, lfxs_sym, tdeq_sym, ex2_intro/
lemma lfdeq_atom: ∀h,o,I. ⋆ ≡[h, o, ⓪{I}] ⋆.
/2 width=1 by lfxs_atom/ qed.
+(* Basic_2A1: uses: lleq_sort *)
lemma lfdeq_sort: ∀h,o,I,L1,L2,V1,V2,s.
L1 ≡[h, o, ⋆s] L2 → L1.ⓑ{I}V1 ≡[h, o, ⋆s] L2.ⓑ{I}V2.
/2 width=1 by lfxs_sort/ qed.
L1 ≡[h, o, #i] L2 → L1.ⓑ{I}V1 ≡[h, o, #⫯i] L2.ⓑ{I}V2.
/2 width=1 by lfxs_lref/ qed.
+(* Basic_2A1: uses: lleq_gref *)
lemma lfdeq_gref: ∀h,o,I,L1,L2,V1,V2,l.
L1 ≡[h, o, §l] L2 → L1.ⓑ{I}V1 ≡[h, o, §l] L2.ⓑ{I}V2.
/2 width=1 by lfxs_gref/ qed.
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
/2 width=1 by lfxs_inv_lref/ qed-.
+(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
lemma lfdeq_inv_bind: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 →
L1 ≡[h, o, V] L2 ∧ L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_inv_bind/ qed-.
+(* Basic_2A1: uses: lleq_inv_flat *)
lemma lfdeq_inv_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 →
L1 ≡[h, o, V] L2 ∧ L1 ≡[h, o, T] L2.
/2 width=2 by lfxs_inv_flat/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lfdeq_fwd_bind_sn: ∀h,o,p,I,L1,L2,V,T. L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1 ≡[h, o, V] L2.
-/2 width=4 by lfxs_fwd_bind_sn/ qed-.
+(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
+lemma lfdeq_fwd_pair_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ②{I}V.T] L2 → L1 ≡[h, o, V] L2.
+/2 width=3 by lfxs_fwd_pair_sn/ qed-.
+(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
lemma lfdeq_fwd_bind_dx: ∀h,o,p,I,L1,L2,V,T.
L1 ≡[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V.
/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-lemma lfdeq_fwd_flat_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, V] L2.
-/2 width=3 by lfxs_fwd_flat_sn/ qed-.
-
+(* Basic_2A1: uses: lleq_fwd_flat_dx *)
lemma lfdeq_fwd_flat_dx: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ⓕ{I}V.T] L2 → L1 ≡[h, o, T] L2.
/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-lemma lfdeq_fwd_pair_sn: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, ②{I}V.T] L2 → L1 ≡[h, o, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
lemma lfdeq_fwd_dx: ∀h,o,I,L1,K2,V2. ∀T:term. L1 ≡[h, o, T] K2.ⓑ{I}V2 →
∃∃K1,V1. L1 = K1.ⓑ{I}V1.
/2 width=5 by lfxs_fwd_dx/ qed-.
-(* Basic_2A1: removed theorems 31:
- lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
+(* Basic_2A1: removed theorems 10:
+ lleq_ind 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
- lleq_dec
+ lleq_skip lleq_lref lleq_free
+ lleq_Y lleq_ge_up lleq_ge
+
*)
(* Advanced properties ******************************************************)
+(* Basic_2A1: uses: lleq_refl *)
lemma lfdeq_refl: ∀h,o,T. reflexive … (lfdeq h o T).
/2 width=1 by lfxs_refl/ qed.
(* Forward lemmas with length for local environments ************************)
+(* Basic_2A1: lleq_fwd_length *)
lemma lfdeq_fwd_length: ∀h,o,L1,L2. ∀T:term. L1 ≡[h, o, T] L2 → |L1| = |L2|.
/2 width=3 by lfxs_fwd_length/ qed-.
(* Advanced properties ******************************************************)
+(* Basic_2A1: uses: lleq_dec *)
lemma lfdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≡[h, o, T] L2).
/3 width=1 by lfxs_dec, tdeq_dec/ qed-.
(* Main properties **********************************************************)
+(* Basic_2A1: uses: lleq_bind lleq_bind_O *)
theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
L1 ≡[h, o, V1] L2 → L1.ⓑ{I}V1 ≡[h, o, T] L2.ⓑ{I}V2 →
L1 ≡[h, o, ⓑ{p,I}V1.T] L2.
/2 width=2 by lfxs_bind/ qed.
+(* Basic_2A1: uses: lleq_flat *)
theorem lfdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, V] L2 → L1 ≡[h, o, T] L2 →
L1 ≡[h, o, ⓕ{I}V.T] L2.
/2 width=1 by lfxs_flat/ qed.
+(* Basic_2A1: uses: lleq_trans *)
theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T).
#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
lapply (frees_tdeq_conf_lexs … Hf1 T … HL1) // #H0
/4 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ex2_intro/
qed-.
+(* Basic_2A1: uses: lleq_canc_sn *)
theorem lfdeq_canc_sn: ∀h,o,T. left_cancellable … (lfdeq h o T).
/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
+(* Basic_2A1: uses: lleq_canc_dx *)
theorem lfdeq_canc_dx: ∀h,o,T. right_cancellable … (lfdeq h o T).
/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
∀K1. L1 ≡[h, o, T] K1 → ∀K2. L2 ≡[h, o, T] K2 → K1 ≡[h, o, T] K2.
/3 width=3 by lfdeq_canc_sn, lfdeq_trans/ qed-.
-(* Advanced properies on negated lazy equivalence *****************************)
+(* Negated properties *******************************************************)
-(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred ************)
-lemma lfdeq_nlfdeq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L →
+(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred **********)
+(* Basic_2A1: uses: lleq_nlleq_trans *)
+lemma lfdeq_lfdneq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L →
∀L2. (L ≡[h, o, T] L2 → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
/3 width=3 by lfdeq_canc_sn/ qed-.
-lemma nlfdeq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L →
+(* Basic_2A1: uses: nlleq_lleq_div *)
+lemma lfdneq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L →
∀L1. (L1 ≡[h, o, T] L → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
/3 width=3 by lfdeq_trans/ qed-.
+
+theorem lfdneq_lfdeq_canc_dx: ∀h,o,L1,L. ∀T:term. (L1 ≡[h, o, T] L → ⊥) →
+ ∀L2. L2 ≡[h, o, T] L → L1 ≡[h, o, T] L2 → ⊥.
+/3 width=3 by lfdeq_trans/ qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
+lemma lfdneq_inv_bind: ∀h,o,p,I,L1,L2,V,T. (L1 ≡[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ≡[h, o, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V → ⊥).
+/3 width=2 by lfnxs_inv_bind, tdeq_dec/ qed-.
+
+(* Basic_2A1: uses: nlleq_inv_flat *)
+lemma lfdneq_inv_flat: ∀h,o,I,L1,L2,V,T. (L1 ≡[h, o, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ≡[h, o, V] L2 → ⊥) ∨ (L1 ≡[h, o, T] L2 → ⊥).
+/3 width=2 by lfnxs_inv_flat, tdeq_dec/ qed-.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
qed.
+(* Basic_2A1: uses: llpx_sn_sort *)
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/
#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
qed.
+(* Basic_2A1: uses: llpx_sn_gref *)
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/
/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/
qed-.
+(* Basic_2A1: uses: llpx_sn_co *)
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_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T * #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-.
+(* Basic_2A1: uses: llpx_sn_inv_flat *)
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
(* Basic forward lemmas *****************************************************)
-lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
+lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
+[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
qed-.
+(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T #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-.
-
+(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
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-.
-
lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 →
∃∃K1,V1. L1 = K1.ⓑ{I}V1.
#R #I #L1 #K2 #T #V2 * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
/2 width=3 by ex1_2_intro/
qed-.
-(* Basic_2A1: removed theorems 25:
- 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
- llpx_sn_dec
+(* Basic_2A1: removed theorems 9:
+ llpx_sn_skip llpx_sn_lref llpx_sn_free
+ llpx_sn_fwd_lref
+ llpx_sn_Y llpx_sn_ge_up llpx_sn_ge
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
*)
(* Advanced properties ******************************************************)
+(* Basic_2A1: uses: llpx_sn_refl *)
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.
(* Forward lemmas with length for local environments ************************)
+(* Basic_2A1: uses: llpx_sn_fwd_length *)
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-.
#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
qed-.
+(* Basic_2A1: uses: llpx_sn_dec *)
lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀L1,L2,T. Decidable (L1 ⦻*[R, T] L2).
#R #HR #L1 #L2 #T
(* Main properties **********************************************************)
+(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
qed.
+(* Basic_2A1: llpx_sn_flat *)
theorem lfxs_flat: ∀R,I,L1,L2,V,T.
L1 ⦻*[R, V] L2 → L1 ⦻*[R, T] L2 →
L1 ⦻*[R, ⓕ{I}V.T] L2.
elim (HR12 … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/
]
qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
+lemma lfnxs_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⦻*[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⦻*[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
+/4 width=2 by lfxs_bind, or_intror, or_introl/
+qed-.
+
+(* Basic_2A1: uses: nllpx_sn_inv_flat *)
+lemma lfnxs_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀I,L1,L2,V,T. (L1 ⦻*[R, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ⦻*[R, V] L2 → ⊥) ∨ (L1 ⦻*[R, T] L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
+/4 width=1 by lfxs_flat, or_intror, or_introl/
+qed-.
T1 ≡[h, o] T2 → ⊥.
/3 width=3 by tdeq_canc_sn/ qed-.
-theorem tndeq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≡[h, o] T → ⊥) → ∀T2. T2 ≡[h, o] T →
+theorem tdneq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≡[h, o] T → ⊥) → ∀T2. T2 ≡[h, o] T →
T1 ≡[h, o] T2 → ⊥.
/3 width=3 by tdeq_trans/ qed-.
(anniversary milestone).
</news>
<news class="alpha" date="2014 January 20.">
- Parametrized slicing of local environments
+ Parametrized slicing on local environments
comprises both versions of this operation
(one from basic_1, the other used in basic_2 till now).
</news>
[ "lpxs ( ⦃?,?⦄ ⊢ ➡*[?,?] ? )" "lpxs_drop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ "cpxs_lreq" + "cpxs_lleq" + "cpxs_aaa" * ]
*)
- [ "lfsx ( ? ⊢ ⬈*[?,?,?] 𝐒⦃?⦄ )" "lfsx_fqup" + "lfsx_lfsx" * ]
+ [ "lfsx ( ? ⊢ ⬈*[?,?,?] 𝐒⦃?⦄ )" "lfsx_fqup" + "lfsx_lfpxs" + "lfsx_lfsx" * ]
[ "csx_vector ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx_vector" + "csx_csx_vector" * ]
[ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_theq" + "csx_drops" + "csx_lsubr" + "csx_gcp" + "csx_gcr" + "csx_lfpx" + "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
- [ "lfpxs ( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_fqup" + "lfpxs_cpxs" * ]
+ [ "lfpxs ( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_fqup" + "lfpxs_lfdeq" + "lfpxs_cpxs" + "lfpxs_lfpxs" * ]
[ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_lsubr" + "cpxs_aaa" + "cpxs_lfpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
}
]
class "water"
[ { "iterated static typing" * } {
[ { "iterated extension on referred entries" * } {
- [ "tc_lfxs ( ? ⦻**[?,?] ? )" * ]
+ [ "tc_lfxs ( ? ⦻**[?,?] ? )" "tc_lfxs_length" + "tc_lfxs_fqup" * ]
}
]
}
projects / helm.git / commitdiff