(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ext.ma".
+include "basic_2/reduction/lpx_lleq.ma".
+include "basic_2/computation/cpxs_cpys.ma".
include "basic_2/computation/lpxs_ldrop.ma".
include "basic_2/computation/lpxs_cpxs.ma".
(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
-(* Advanced properties ******************************************************)
-
-fact le_repl_sn_aux: ∀x,y,z:nat. x ≤ z → x = y → y ≤ z.
-// qed-.
-
-axiom cpxs_cpys_conf_lpxs: ∀h,g,G,d,e.
- ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡*[h, g] T1 →
- ∀T2. ⦃G, L0⦄ ⊢ T0 ▶*[d, e] T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ▶*[d, e] T & ⦃G, L0⦄ ⊢ T2 ➡*[h, g] T.
-
-axiom cpxs_conf_lpxs_lpys: ∀h,g,G,d,e.
- ∀I,L0,V0,T0,T1. ⦃G, L0.ⓑ{I}V0⦄ ⊢ T0 ➡*[h, g] T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡*[h, g] L1 → ∀V2. ⦃G, L0⦄ ⊢ V0 ▶*[d, e] V2 →
- ∃∃T. ⦃G, L1.ⓑ{I}V0⦄ ⊢ T1 ▶*[⫯d, e] T & ⦃G, L0.ⓑ{I}V2⦄ ⊢ T0 ➡*[h, g] T.
-
-
-include "basic_2/reduction/cpx_cpys.ma".
-
-fact pippo_aux: ∀h,g,G,L1,T,T1,d,e. ⦃G, L1⦄ ⊢ T ▶*[d, e] T1 → e = ∞ →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
- ∃∃T2. ⦃G, L2⦄ ⊢ T ▶*[d, e] T2 & ⦃G, L1⦄ ⊢ T1 ➡*[h, g] T2 &
- L1 ⋕[T, d] L2 ↔ T1 = T2.
-#h #g #G #L1 #T #T1 #d #e #H @(cpys_ind_alt … H) -G -L1 -T -T1 -d -e [ * ]
-[ /5 width=5 by lpxs_fwd_length, lleq_sort, ex3_intro, conj/
-| #i #G #L1 elim (lt_or_ge i (|L1|)) [2: /6 width=6 by lpxs_fwd_length, lleq_free, le_repl_sn_aux, ex3_intro, conj/ ]
- #Hi #d elim (ylt_split i d) [ /5 width=5 by lpxs_fwd_length, lleq_skip, ex3_intro, conj/ ]
- #Hdi #e #He #L2 elim (lleq_dec (#i) L1 L2 d) [ /4 width=5 by lpxs_fwd_length, ex3_intro, conj/ ]
- #HnL12 #HL12 elim (ldrop_O1_lt L1 i) // -Hi #I #K1 #V1 #HLK1
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpxs_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- elim (lift_total V2 0 (i+1)) #W2 #HVW2
- @(ex3_intro … W2) /2 width=7 by cpxs_delta, cpys_subst/ -I -K1 -V1 -Hdi
- @conj #H [ elim HnL12 // | destruct elim (lift_inv_lref2_be … HVW2) // ]
-| /5 width=5 by lpxs_fwd_length, lleq_gref, ex3_intro, conj/
-| #I #G #L1 #K1 #V #V1 #T1 #i #d #e #Hdi #Hide #HLK1 #HV1 #HVT1 #_ #He #L2 #HL12 destruct
- elim (lpxs_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpxs_inv_pair1 … H) -H #K2 #W #HK12 #HVW #H destruct
- elim (cpxs_cpys_conf_lpxs … HVW … HV1 … HK12) -HVW -HV1 -HK12 #W1 #HW1 #VW1
- elim (lift_total W1 0 (i+1)) #U1 #HWU1
- lapply (ldrop_fwd_drop2 … HLK1) -HLK1 #HLK1
- @(ex3_intro … U1) /2 width=10 by cpxs_lift, cpys_subst/
-| #a #I #G #L #V #V1 #T #T1 #d #e #HV1 #_ #IHV1 #IHT1 #He #L2 #HL12
- elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
- elim (IHT1 … (L2.ⓑ{I}V2)) /4 width=3 by lpxs_cpx_trans, lpxs_pair, cpys_cpx/ -IHT1 -He #T2 #HT2 #HT12 * #H1T #H2T
- elim (cpxs_conf_lpxs_lpys … HT12 … HL12 … HV1) -HT12 -HL12 -HV1 #T0 #HT20 #HT10
- @(ex3_intro … (ⓑ{a,I}V2.T0))
- [ @cpys_bind // @(cpys_trans_eq … T2) /3 width=5 by lsuby_cpys_trans, lsuby_succ/
- | /2 width=1 by cpxs_bind/
- | @conj #H destruct
- [ elim (lleq_inv_bind … H) -H #HV #HT >H1V -H1V //
- | @lleq_bind /2 width=1/
-
-
- /3 width=5 by lsuby_cpys_trans, lsuby_succ/
-| #I #G #L #V #V1 #T #T1 #d #e #HV1 #HT1 #IHV1 #IHT1 #He #L2 #HL12
- elim (IHV1 … HL12) // -IHV1 #V2 #HV2 #HV12 * #H1V #H2V
- elim (IHT1 … HL12) // -IHT1 #T2 #HT2 #HT12 * #H1T #H2T -He -HL12
- @(ex3_intro … (ⓕ{I}V2.T2)) /2 width=1 by cpxs_flat, cpys_flat/
- @conj #H destruct [2: /3 width=1 by lleq_flat/ ]
- elim (lleq_inv_flat … H) -H /3 width=1 by eq_f2/
-]
-
-
-
- [
- | @cpxs_bind //
- @(lpx_cpxs_trans … HT12)
-|
-]
+(* Properties on lazy equivalence for local environments ********************)
-axiom lleq_lpxs_trans: ∀h,g,G,L1,L2,T,d. L1 ⋕[T, d] L2 → ∀K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+lemma lleq_lpxs_trans: ∀h,g,G,L2,K2. ⦃G, L2⦄ ⊢ ➡*[h, g] K2 →
+ ∀L1,T,d. L1 ⋕[T, d] L2 →
∃∃K1. ⦃G, L1⦄ ⊢ ➡*[h, g] K1 & K1 ⋕[T, d] K2.
-(*
-#h #g #G #L1 #L2 #T #d #H @(lleq_ind_alt … H) -L1 -L2 -T -d
-[
-|
-|
-|
-|
-| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #K2 #HLK2
- elim (IHV … HLK2) -IHV #KV #HLKV #HV
- elim (IHT (K2.ⓑ{I}V)) -IHT /2 width=1 by lpxs_pair_refl/ -HLK2 #Y #H #HT
- elim (lpxs_inv_pair1 … H) -H #KT #VT #HLKT #_ #H destruct
-
-#h #g #G #L1 #L2 #T #d * #HL12 #IH #K2 #HLK2
-*)
-
-(* Properties on lazy equivalence for local environments ********************)
+#h #g #G #L2 #K2 #H @(lpxs_ind … H) -K2 /2 width=3 by ex2_intro/
+#K #K2 #_ #HK2 #IH #L1 #T #d #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_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 → K1 ⋕[T1, 0] L1 →
| * #HG #HL #HT destruct /2 width=4 by ex3_intro/
]
qed-.
+
+fact lsuby_lpxs_trans_lleq_aux: ∀h,g,G,L1,L0,d,e. L1 ⊑×[d, e] L0 → e = ∞ →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ⊑×[d, e] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. |L1| = |L0| → |L| = |L2| → L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
+#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
+[ #L1 #d #e #_ #L2 #H >(lpxs_inv_atom1 … H) -H
+ /3 width=5 by ex3_intro, conj/
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
+| #I1 #I0 #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+ elim (lpxs_inv_pair1 … H) -H #L2 #V2 #HL02 #HV02 #H destruct
+ lapply (ysucc_inv_Y_dx … He) -He #He
+ elim (IHL10 … HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
+ @(ex3_intro … (L.ⓑ{I1}V2)) /3 width=3 by lpxs_pair, lsuby_cpxs_trans, lsuby_pair/
+ #T #H1 #H2 lapply (injective_plus_l … H1) lapply (injective_plus_l … H2) -H1 -H2
+ #H1 #H2 elim (IH T) // #HL0dx #HL0sn
+ @conj #H @(lleq_lsuby_repl … H) -H normalize
+ /3 width=1 by lsuby_sym, lsuby_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #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, lsuby_succ/
+ #T #H1 #H2 lapply (injective_plus_l … H1) lapply (injective_plus_l … H2) -H1 -H2
+ #H1 #H2 elim (IH T) // #HL0dx #HL0sn
+ @conj #H @(lleq_lsuby_repl … H) -H
+ /3 width=1 by lsuby_sym, lsuby_succ/ normalize //
+]
+qed-.
+
+lemma lsuby_lpxs_trans_lleq: ∀h,g,G,L1,L0,d. L1 ⊑×[d, ∞] L0 →
+ ∀L2. ⦃G, L0⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃L. L ⊑×[d, ∞] L2 & ⦃G, L1⦄ ⊢ ➡*[h, g] L &
+ (∀T. |L1| = |L0| → |L| = |L2| → L0 ⋕[T, d] L2 ↔ L1 ⋕[T, d] L).
+/2 width=1 by lsuby_lpxs_trans_lleq_aux/ qed-.
projects / helm.git / blobdiff