--- /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/lpx_leq.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_cpxs_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡*[h, g] T2 →
+ ∀L1. L1 ⋕[0, T1] L2 → ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L2 #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+/4 width=6 by lleq_cpx_conf_dx, lleq_cpx_trans, cpxs_strap2/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/ldrop_leq.ma".
+include "basic_2/relocation/lleq_lleq.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_cpx_trans_leq: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L1,d. L1 ⋕[d, T1] L2 → L1 ≃[d, ∞] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
+[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #d #H #HL12 elim (lleq_inv_lref_dx … H … HLK2) -H *
+ [ #K1 #HLK1 #HV1 #Hdi elim (ldrop_leq_conf_be … HL12 … HLK1) -HL12 /2 width=1 by yle_inj/
+ >yminus_Y_inj #J #Y #X #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
+ #H destruct /3 width=7 by cpx_delta/
+ | #J #K1 #V #HLK1 #_ #HV1 #Hid elim (ldrop_leq_conf_lt … HL12 … HLK1) -HL12 /2 width=1 by ylt_inj/
+ #Y #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
+ #H destruct /3 width=7 by cpx_delta/
+ ]
+| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_bind … H) -H
+ /4 width=3 by cpx_bind, leq_succ/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
+ /3 width=3 by cpx_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #H elim (lleq_inv_bind … H) -H
+ /4 width=3 by cpx_zeta, leq_succ/
+| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
+ /3 width=3 by cpx_tau/
+| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #d #H elim (lleq_inv_flat … H) -H
+ /3 width=3 by cpx_ti/
+| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
+ #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_beta, leq_succ/
+| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
+ #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_theta, leq_succ/
+]
+qed-.
+
+(* Note: this can be proved directly *)
+lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L1. L1 ⋕[0, T1] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
+/4 width=6 by lleq_cpx_trans_leq, lleq_fwd_length, leq_O_Y/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/ldrop_leq.ma".
-include "basic_2/relocation/lleq_lleq.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Properties on lazy equivalence for local environments ********************)
-
-lemma lleq_cpx_trans_leq: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1,d. L1 ⋕[d, T1] L2 → L1 ≃[d, ∞] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2 /2 width=2 by cpx_sort/
-[ #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #d #H #HL12 elim (lleq_inv_lref_dx … H … HLK2) -H *
- [ #K1 #HLK1 #HV1 #Hdi elim (ldrop_leq_conf_be … HL12 … HLK1) -HL12 /2 width=1 by yle_inj/
- >yminus_Y_inj #J #Y #X #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
- #H destruct /3 width=7 by cpx_delta/
- | #J #K1 #V #HLK1 #_ #HV1 #Hid elim (ldrop_leq_conf_lt … HL12 … HLK1) -HL12 /2 width=1 by ylt_inj/
- #Y #HK12 #H lapply (ldrop_mono … H … HLK2) -L2
- #H destruct /3 width=7 by cpx_delta/
- ]
-| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_bind … H) -H
- /4 width=3 by cpx_bind, leq_succ/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
- /3 width=3 by cpx_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #d #H elim (lleq_inv_bind … H) -H
- /4 width=3 by cpx_zeta, leq_succ/
-| #G #L2 #W1 #T1 #T2 #_ #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
- /3 width=3 by cpx_tau/
-| #G #L2 #W1 #W2 #T1 #_ #IHW12 #L1 #d #H elim (lleq_inv_flat … H) -H
- /3 width=3 by cpx_ti/
-| #a #G #L1 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
- #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_beta, leq_succ/
-| #a #G #L1 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #d #H elim (lleq_inv_flat … H) -H
- #HV1 #H elim (lleq_inv_bind … H) -H /4 width=3 by cpx_theta, leq_succ/
-]
-qed-.
-
-lemma lleq_cpx_trans: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
- ∀L1. L1 ⋕[0, T1] L2 → ⦃G, L1⦄ ⊢ T1 ➡[h, g] T2.
-/4 width=6 by lleq_cpx_trans_leq, lleq_fwd_length, leq_O_Y/ 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/reduction/cpx_leq.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
+
+(**) (* to be proved later *)
+axiom- lleq_beta: ∀L2s,L2d,V2,W2,T2,d.
+ L2s.ⓛW2 ⋕[d+1, T2] L2d.ⓛW2 →
+ L2s.ⓓⓝW2.V2 ⋕[d+1, T2] L2d.ⓓⓝW2.V2.
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties using equivalences for local environments *********************)
+
+(* Note: lemma 1000 *)
+lemma lleq_cpx_conf_leq_dx: ∀h,g,G,L1s,L1d,T1,d. L1s ⋕[d, T1] L1d → L1s ≃[d, ∞] L1d →
+ ∀T2. ⦃G, L1d⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L2s. ⦃G, L1s⦄ ⊢ ➡[h, g] L2s → L1s ≃[0, d] L2s →
+ ∀L2d. ⦃G, L1d⦄ ⊢ ➡[h, g] L2d → L1d ≃[0, d] L2d →
+ L2s ≃[d, ∞] L2d → L2s ⋕[d, T2] L2d.
+#h #g #G #L1s #L1d #T1 #d #H elim H -L1s -L1d -T1 -d
+[ #L1s #L1d #d #k #_ #_ #X #H2 #L2s #_ #_ #L2d #_ #_ #H3
+ lapply (leq_fwd_length … H3) -H3 #HL2sd
+ elim (cpx_inv_sort1 … H2) -H2 [| * #l #_ ]
+ #H destruct /2 width=1 by lleq_sort/
+| #Is #Id #L1s #L1d #K1s #K1d #V1s #V1d #d #i #Hid #HLK1s #HLK1d #_ #_ #_ #IHV1d #H1 #X #H2 #L2s #H1s #H2s #L2d #H1d #H2d #H3
+ elim (ldrop_leq_conf_lt … H1 … HLK1s) -H1 /2 width=1 by ylt_inj/ #Y #H1 #HY
+ lapply (ldrop_mono … HY … HLK1d) -HY #H destruct
+ elim (lpx_ldrop_conf … HLK1s … H1s) -H1s #Y #H #HLK2s
+ elim (lpx_inv_pair1 … H) -H #K2s #V2s #H1s #HV12s #H destruct
+ elim (lpx_ldrop_conf … HLK1d … H1d) -H1d #Y #H #HLK2d
+ elim (lpx_inv_pair1 … H) -H #K2d #V2d #H1d #HV12d #H destruct
+ elim (ldrop_leq_conf_be … H2s … HLK1s) -H2s /2 width=1 by ylt_inj/ #Z #Y #X #HK12s #H
+ lapply (ldrop_mono … H … HLK2s) -H #H destruct
+ elim (ldrop_leq_conf_be … H2d … HLK1d) -H2d /2 width=1 by ylt_inj/ #Z #Y #X #HK12d #H
+ lapply (ldrop_mono … H … HLK2d) -H #H destruct
+ elim (ldrop_leq_conf_lt … H3 … HLK2s) -H3 /2 width=1 by ylt_inj/ #Y #H3 #HY
+ lapply (ldrop_mono … HY … HLK2d) -HY #H destruct
+ elim (cpx_inv_lref1 … H2) -H2 -L1s
+ [ -L1d #H destruct /3 width=15 by lleq_skip/
+ | * #Z #Y #X1 #X2 #H #HX12 #HX2 lapply (ldrop_mono … H … HLK1d) -L1d
+ #H destruct >(plus_minus_m_m d (i+1)) //
+ lapply (ldrop_fwd_ldrop2 … HLK2s) -HLK2s
+ lapply (ldrop_fwd_ldrop2 … HLK2d) -HLK2d
+ /3 width=9 by lleq_lift_ge/
+ ]
+| #I #L1s #L1d #K1s #K1d #V1 #d #i #Hdi #HLK1s #HLK1d #_ #IHV1 #H1 #X #H2 #L2s #H1s #H2s #L2d #H1d #H2d #H3
+ elim (ldrop_leq_conf_be … H1 … HLK1s) -H1 /2 width=1 by ylt_Y, yle_inj/ #Z #Y #X #H1 #HY
+ lapply (ldrop_mono … HY … HLK1d) -HY #H destruct
+ elim (lpx_ldrop_conf … HLK1s … H1s) -H1s #Y #H #HLK2s
+ elim (lpx_inv_pair1 … H) -H #K2s #V2s #H1s #HV12s #H destruct
+ elim (lpx_ldrop_conf … HLK1d … H1d) -H1d #Y #H #HLK2d
+ elim (lpx_inv_pair1 … H) -H #K2d #V2d #H1d #HV12d #H destruct
+ lapply (ldrop_leq_conf_ge … H2s … HLK1s ?) /2 width=1 by yle_inj/ #H
+ lapply (ldrop_mono … H … HLK2s) -H #H destruct
+ lapply (ldrop_leq_conf_ge … H2d … HLK1d ?) /2 width=1 by yle_inj/ #H
+ lapply (ldrop_mono … H … HLK2d) -H #H destruct
+ elim (ldrop_leq_conf_be … H3 … HLK2s) -H3 /2 width=1 by ylt_Y, yle_inj/
+ >yminus_Y_inj #Z #Y #X #H3 #HY
+ lapply (ldrop_mono … HY … HLK2d) -HY #H destruct
+ elim (cpx_inv_lref1 … H2) -H2 -L1s
+ [ -L1d #H destruct /3 width=12 by lleq_lref/
+ | * #Z #Y #X1 #X2 #H #HX12 #HX2 lapply (ldrop_mono … H … HLK1d) -L1d
+ #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2s) -HLK2s #HLK2s
+ lapply (ldrop_fwd_ldrop2 … HLK2d) -HLK2d #HLK2d
+ @(lleq_ge … 0) /3 width=10 by lleq_lift_le/ (**) (* full auto too slow *)
+ ]
+| #L1s #L1d #d #i #HL1s #HL1d #_ #_ #X #H2 #L2s #_ #_ #L2s #_ #H2d #H3
+ lapply (leq_fwd_length … H2d) -H2d
+ lapply (leq_fwd_length … H3) -H3
+ elim (cpx_inv_lref1 … H2) -H2
+ [ #H destruct /2 width=1 by lleq_free/
+ | -L1s * #I #K1d #V1 #V2 #HLK1d
+ lapply (ldrop_fwd_length_lt2 … HLK1d) -HLK1d #H
+ elim (lt_refl_false … i) /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+ ]
+| #L1s #L1d #d #k #_ #_ #X #H2 #L2s #_ #_ #L2d #_ #_ #H3
+ lapply (leq_fwd_length … H3) -H3 #HL2sd
+ lapply (cpx_inv_gref1 … H2) -H2
+ #H destruct /2 width=1 by lleq_gref/
+| #a #I #L1s #L1d #V1 #T1 #d #HV1 #_ #IHV1 #IHT1 #H1 #X #H2 #L2s #H1s #H2s #L2d #H1d #H2d #H3
+ elim (cpx_inv_bind1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=5 by lpx_pair, lleq_cpx_trans_leq, lleq_bind, leq_pair, leq_succ/
+ | #T2 #HT12 #HT2X #H1 #H2 destruct >(minus_plus_m_m d 1)
+ /4 width=9 by lpx_pair, lleq_inv_lift_ge, ldrop_ldrop, leq_pair, leq_succ/
+ ]
+| #I #L1s #L1d #V1 #T1 #d #HV1 #_ #IHV1 #IHT1 #H1 #X #H2 #L2s #H1s #H2s #L2d #H1d #H2d #H3
+ elim (cpx_inv_flat1 … H2) -H2 *
+ [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=1 by lleq_flat/
+ | #HT1X #H destruct /2 width=1 by/
+ | #HV1X #H destruct /2 width=1 by/
+ | #a #V2 #W1 #W2 #T0 #T2 #HV12 #HW12 #HT02 #H1 #H2 #H3 destruct
+ lapply (IHT1 … (ⓛ{a}W2.T2) … L2s … L2d ? ? ?) -IHT1 /2 width=1 by cpx_bind/ #H
+ elim (lleq_inv_bind … H) -H -HW12 -HT02 #HW2 #HT2
+ /4 width=1 by lleq_beta, lleq_flat, lleq_bind/
+ | #a #V0 #V2 #W1 #W2 #T0 #T2 #HV10 #HV02 #HW12 #HT02 #H1 #H2 #H3 destruct
+ lapply (IHT1 … (ⓓ{a}W2.T2) … L2s … L2d ? ? ?) -IHT1 /2 width=1 by cpx_bind/ #H
+ elim (lleq_inv_bind … H) -H -HW12 -HT02
+ /5 width=9 by lleq_lift_ge, lleq_flat, lleq_bind, ldrop_ldrop/
+ ]
+]
+qed-.
+
+lemma lleq_cpx_conf_dx: ∀h,g,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ➡[h, g] T2 →
+ ∀L1. L1 ⋕[0, T1] L2 → L1 ⋕[0, T2] L2.
+#h #g #G #L2 #T1 #T2 #HT12 #L1 #HT1 lapply (lleq_fwd_length … HT1)
+/3 width=13 by lleq_cpx_conf_leq_dx, leq_O_Y/
+qed-.
}
]
[ { "context-sensitive extended reduction" * } {
- [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
- [ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpx_lift" + "cpx_lleq" + "cpx_cix" * ]
+ [ "lpx ( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lpx_leq" + "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
+ [ "cpx ( ⦃?,?⦄ ⊢ ? ➡[?,?] ? )" "cpx_leq" + "cpx_lift" + "cpx_cix" * ]
}
]
[ { "context-sensitive extended irreducible forms" * } {
projects / helm.git / commitdiff